Non-normal Klein four-subgroups of symmetric group: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 Young subgroup for the partition $$\{ \{ 1,2 \}, \{ 3,4 \} \}$$. Explicitly, it is the subgroup comprising those permutations that send each of the subsets $$\{ 1,2 \}$$ and $$\{ 3,4 \}$$ to within itself. $$H$$ has a total of three conjugates, listed below:

$$H$$ (and hence each of its conjugate subgroups) is isomorphic to the Klein four-group. However, $$G$$ has another subgroup isomorphic to the Klein four-group that is not one of these conjugate subgroups, and is not automorphic to these either. That is the normal Klein four-subgroup of symmetric group:S4 that comprises the identity element and the three double transpositions. The current article is not about that subgroup.

Cosets
There is a total of 18 cosets, each of which is a left coset for exactly one subgroup and a right coset for exactly one subgroup. Moreover, each coset is parametrized by the way it sends one partition (labeled) to another. Table below is incomplete.

Complements
None of these subgroups has a permutable complement. All the subgroups do have a common lattice complement; in fact, there is a conjugacy class of subgroups each of which is conjugate to each of these subgroups. That conjugacy class is the conjugacy class of A3 in S4, which includes each of these subgroups:

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

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