S3 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 $$\{ 4 \}$$. In particular, $$H$$ is the symmetric group on $$\{ 1, 2,3 \}$$, embedded naturally in $$G$$. It is isomorphic to symmetric group:S3. $$H$$ has order $$6$$.

There are three other conjugate subgroups to $$H$$ in $$G$$ (so the total conjugacy class size of subgroups is 4). The other subgroups are the subgroups fixing $$\{ 1 \}$$, $$\{ 2 \}$$, and $$\{ 3 \}$$ respectively.

The four conjugates are:

$$\! H = H_4 = \{, (1,2), (1,3), (2,3), (1,2,3), (1,3,2) \}$$

$$\! H_1 = \{, (2,3), (3,4), (2,4), (2,3,4), (2,4,3) \}$$

$$\! H_2 = \{, (1,3), (3,4), (1,4), (1,3,4), (1,4,3) \}$$

$$\! H_3 = \{, (1,2), (2,4), (1,4), (1,2,4), (1,4,2) \}$$

See also subgroup structure of symmetric group:S4.

Cosets
There are four left cosets and four right cosets of each subgroup. Each left coset of a subgroup is a right coset of one of its conjugate subgroups. This gives a total of 16 subsets.

The cosets are parametrized by ordered pairs $$(i,j) \in \{ 1,2,3,4 \} \times \{ 1,2,3,4 \}$$. The coset parametrized by $$(i,j)$$ is the set of all elements that send $$i$$ to $$j$$. This is a left coset of $$H_i$$ and a right coset of $$H_j$$.

Complements
There is a unique normal complement that is common to all the subgroups. This is the subgroup normal Klein four-subgroup of symmetric group:S4:

$$\! K := \{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$

There is also a conjugacy class of subgroups each of which is a permutable complement to each of the $$H_i$$s. These are cyclic four-subgroups of symmetric group:S4, and these are:

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

Note that the fact that these are permutable complements can be understood as a special case of Cayley's theorem. See also every group of given order is a permutable complement for symmetric groups, which says that any finite group of order $$n$$ is, via the Cayley embedding, a permutable complement to $$S_{n-1}$$ in $$S_n$$.

Apart from these, each of the $$H_i$$s has a number of lattice complements:


 * Any subgroup generated by double transposition in S4 is a lattice complement to each $$H_i$$ in the whole group. Thus, each $$H_i$$ has three such lattice complements.
 * For each $$H_i$$, a subgroup of order three not contained in that $$H_i$$ is a lattice complement to it. Thus, each $$H_i$$ has three such lattice complements, because one of the four subgroups of order three is contained in that $$H_i$$.