S2 in S4

We consider the subgroup $$H$$ in the group $$G$$ defined as follows.

$$G$$ is the symmetric group of degree four, which, for concreteness, we take as the symmetric group on the set $$\{ 1,2,3,4 \}$$.

$$H$$ is the subgroup of $$G$$ comprising those permutations that fix $$\{ 3,4 \}$$ pointwise. In particular, $$H$$ is the symmetric group on $$\{ 1, 2\}$$, embedded naturally in $$G$$. It is isomorphic to cyclic group:Z2. As a set, $$H$$ contains two elements: $$$$ and $$(1,2)$$.

There are five other conjugate subgroups to $$H$$ in $$G$$ (so the total conjugacy class size of subgroups is 3). Each subgroup fixed pointwise a subset of size two. Equivalently, each subgroup comprises the identity element and a 2-transposition. Specifically, $$H$$ and its two other conjugate subgroups are:

$$\{, (1,2) \}, \qquad \{ , (2,3) \}, \qquad \{ , (3,4) \}, \qquad \{ , (1,3) \}, \qquad \{ , (2,4) \}, \qquad \{ , (1,4) \}$$

See also subgroup structure of symmetric group:S4.

Cosets
Each of these subgroups has 12 left cosets and 12 right cosets. Further, every left coset of one subgroup is a right coset of one of its conjugate subgroups. Overall, there are thus 72 cosets. Each coset is characterized by a fixed behavior on two of the four points.

Complements
There is a unique permutable complement to all these subgroups, which is also a normal subgroup and hence a normal complement. In particular, each of the subgroups is a retract. The unique permutable complement is A4 in S4, i.e., the alternating group of degree four.

Effect of subgroup operators
In the table below, we provide values specific to $$H$$.

Intermediate subgroups
The values given here are specific to $$H$$.

Smaller subgroups
The subgroup has order two, hence is minimal, and has no smaller nontrivial subgroups.