Subgroup structure of direct product of Z4 and Z2

The direct product of Z4 and Z2 is an abelian group of order eight obtained as the external direct product of cyclic group:Z4 and cyclic group:Z2.

For simplicity, we denote the elements of this group as ordered pairs where the first entry is an integer taken modulo 4 and the second entry is an integer taken modulo 2, with coordinate-wise addition. The group thus has the following eight elements:

$$\! \{ (0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (3,1) \}$$

Subgroup-defining functions yielding this subgroup

 * Frattini subgroup: The Frattini subgroup of a group of prime power order is the smallest normal subgroup for which the quotient is elementary abelian. For an abelian group of prime power order, the Frattini subgroup is the group comprising the $$p^{th}$$ powers.
 * first agemo subgroup: This is the group generated by the $$p^{th}$$ powers, in this case the squares. The first agemo subgroup equals the Frattini subgroup for all abelian $$p$$-groups.

Subgroup properties satisfied by this subgroup

 * Verbal subgroup: This is on account of its being the first agemo subgroup.
 * Fully invariant subgroup:
 * Image-closed fully invariant subgroup: Under any surjective homomorphism from the whole group, the image of this subgroup is fully invariant in the image.
 * Characteristic subgroup
 * Image-closed characteristic subgroup

Subgroup properties not satisfied by this subgroup

 * Isomorph-free subgroup: There are other isomorphic subgroups, namely the subgroups of type (3).
 * Homomorph-containing subgroup
 * Direct factor
 * Intermediately fully invariant subgroup
 * Intermediately characteristic subgroup

Subgroup-defining functions yielding this subgroup

 * first omega subgroup

Subgroup properties satisfied by this subgroup

 * Homomorph-containing subgroup
 * Isomorph-free subgroup
 * Fully invariant subgroup
 * Intermediately fully invariant subgroup
 * Characteristic subgroup

Subgroup properties not satisfied by this subgroup

 * Verbal subgroup
 * Image-closed fully invariant subgroup
 * Image-closed characteristic subgroup
 * Direct factor

Subgroup properties satisfied by these subgroups

 * Direct factor
 * HEP-subgroup:
 * EEP-subgroup
 * AEP-subgroup

Subgroup properties not satisfied by these subgroups

 * Characteristic subgroup
 * Isomorph-automorphic subgroup

Subgroup properties satisfied by these subgroups

 * Direct factor
 * HEP-subgroup:
 * EEP-subgroup
 * AEP-subgroup
 * Isomorph-automorphic subgroup
 * Cyclic normal subgroup

Subgroup properties not satisfied by these subgroups

 * Characteristic subgroup