International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.6, pp. 856857
Section 3.6.2.2.5. Symbol of the subgroup of index 2 of^{a}Department of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA 
For magnetic group types , the magnetic group type symbol of the subgroup is given in the third column of the survey of magnetic groups, see e.g. Table 3.6.2.2. If is a group , then the subgroup is defined by the translational group of and the unprimed coset representatives of .
Example
Consider the threedimensional magnetic spacegroup type 16.3.101 . The representative group is defined by the translational subgroup denoted by the letter P generated by the translationsand the standard set of coset representativesThe subgroup of index 2 of the representative group is defined by the translational group denoted by the letter P and the cosets {1∣0} and {2_{001}∣0}, and is a group of type P2.
If is a group , then the subgroup is defined by the nonprimed translational group of and all the cosets of the standard set of coset representatives of the group .
Example
Consider the threedimensional magnetic spacegroup type 16.4.102 P_{2a}222. The representative group P_{2a}222 is defined by the translational group denoted by the symbol P_{2a} generated by the translationsand the standard set of coset representativesThe subgroup of index 2 of the representative group is defined by the translational subgroup denoted by the symbol P_{2a}, i.e. the translations generated byand the standard set of cosets of P_{2a}222. The group is a group of type P222.
While the group type symbol of is given, the coset representatives of the subgroup of derived from the standard set of coset representatives of may not be identical with the standard set of coset representatives of the representative group of type found in the survey of magnetic group types. Consequently, to show the relationship between this subgroup and the listed representative group of groups of type additional information is provided: a new coordinate system is defined in which the coset representatives of this subgroup are identical with the standard set of coset representatives listed for the representative group of groups of type : Let (O; a, b, c) be the coordinate system in which the group is defined. O is the origin of the coordinate system, and a, b and c are the basis vectors of the coordinate system. a, b and c represent a set of basis vectors of a primitive cell for primitive lattices and of a conventional cell for centred lattices. A second coordinate system, defined by (O + p; a′, b′, c′), is given in which the coset representatives of this subgroup are identical with the standard set of coset representatives listed for the representative group of groups of type . O + p is referred to as the location of the subgroup in the coordinate system of the group (Kopský, 2011). The origin is first translated from O to O + p. On translating the origin from O to O + p, a coset representative {R∣τ} becomes {R∣τ + Rp − p} (Litvin, 2005, 2008b; see also Section 1.5.2.3 ). This is followed by changing the basis vectors a, b and c to a′, b′ and c′, respectively. The basis vectors a′, b′, c′ define the conventional unit cell of the nonprimed subgroup of in the coordinate system (O; a, b, c) in which is defined. (O + p; a′, b′, c′) is given immediately following the group type symbol for the subgroup of . [In Litvin (2013), for typographical simplicity, the symbols `O +' are omitted.]
Example
For the threedimensional magnetic spacegroup type 10.4.52, , one finds in Litvin (2013)^{2}

The translational subgroup of the subgroup = P2 of is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 1} and the coset representatives of this group are {1∣0} and {2_{010}∣0}, the unprimed coset representatives on the right. This subgroup is of type P2. In Litvin (2013), listed for the group type 3.1.8, P2, one finds the identical two coset representatives. Consequently, there is no change in the coordinate system, i.e. and a′ = a, b′ = b and c′ = c. In the coordinate system of the magnetic group , the coset representatives of its subgroup are identical with the standard set of coset representatives of the group type P2.
Example
For the threedimensional magnetic spacegroup type 16.7.105, one has

The translational subgroup of the subgroup of is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 2}, and the coset representatives of this group are all those coset representatives on the right. This subgroup is of type P222_{1}. Listed for the group type 17.1.106 P222_{1}, one finds a different set of coset representatives:Consequently, to show the relationship between this subgroup of and the listed representative group of the group type P222_{1} we change the coordinate system in which is defined to (0, 0, 0; a, b, 2c). In this new coordinate system the coset representatives of the subgroup are identical with the coset representatives of the representative group of the group type P222_{1}.
Example
For the threedimensional magnetic spacegroup type 18.4.116, P2_{1}2_{1}′2′, one has

The translational subgroup of is generated by the translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 1} and the coset representatives of this group are {1∣0} and {2_{100}∣½, ½, 0}, the unprimed coset representatives on the right. The group is of type P2_{1}. For the magnetic group type 4.1.15 P2_{1} one finds a different set of coset representatives: {1∣0} and {2_{010}∣0, ½, 0}. Consequently, to show the relationship between the subgroup of and the listed representative group of the group type P2_{1}, we change the coordinate system in which the subgroup is defined to (0, ¼, 0; c, a, b). The origin is first translated from O to O + p, where p = (0, ¼, 0), and then a new set of basis vectors, a′ = c, b′ = a and c′ = b, is defined. In this new coordinate system the coset representatives of the subgroup are identical with the standard set of coset representatives of the representative group of the group type P2_{1}.
References
Kopský, V. (2011). Private communication.Litvin, D. B. (2005). Tables of properties of magnetic subperiodic groups. Acta Cryst. A61, 382–385.
Litvin, D. B. (2008b). Tables of crystallographic properties of magnetic space groups. Acta Cryst. A64, 419–424.
Litvin, D. B. (2013). Magnetic Group Tables, 1, 2 and 3Dimensional Magnetic Subperiodic Groups and Space Groups. Chester: International Union of Crystallography. Freely available from http://www.iucr.org/publ/978–0–9553602–2–0 .