Z2 in V4

Definition
We are interested in the Klein four-group:

$$G := \{ e,a,b,c \}$$

where $$e$$ is the identity element and $$a,b,c$$ are all non-identity elements of order 2.

We are interested in the subgroup:

$$H = H_0 = \{ e, a\}$$

as well as the other two subgroups automorphic to $$H$$:

$$H_1 = \{ e,b \}, H_2 = \{ e, c \}$$

Direct product and direct factor interpretation
This article describes the subgroup $$H$$ in the group $$G$$, where $$G$$ is the Klein four-group:

$$G := \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$$

Denoted additively, the four elements of $$G$$ are:

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

where the addition is modulo 2 in each coordinate separately.

and $$H = H_0$$ is the two-element subgroup:

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

$$H$$ is a normal subgroup (since $$G$$ is an abelian group), so has no other conjugate subgroups. There are two other automorphic subgroups to $$H$$ in $$G$$:

$$\! H_1 = \{ (0,0), (0,1) \}, H_2 = \{ (0,0), (1,1) \}$$

The quotient group $$G/H$$ is isomorphic to cyclic group:Z2.

Cosets
Each of $$H$$, $$H_1$$ and $$H_2$$ is a normal subgroup and each has two cosets: the subgroup itself and the rest of the group. There is a total of six cosets. In fact, every subset of size two in the whole group (total of $$\binom{4}{2} = 6$$ such subsets) is a coset of exactly one of these three subgroups.

Complements
Any two of the three subgroups $$H, H_1, H_2$$ are permutable complements of each other. Since they're all normal, this means that $$G$$ is an internal direct product of any two of them.