Subgroup generated by double transposition in 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 $$\{ 1,2,3 \}$$.

$$H$$ is the subgroup of $$G$$ generated by the double transposition $$(1,2)(3,4)$$. This is the permutation that interchanges $$1$$ with $$2$$ and $$3$$ with $$4$$. Since the element has order two, $$H$$ is a two-element subgroup, isomorphic to cyclic group:Z2, and its two elements are the identity and $$(1,2)(3,4)$$.

There are two other conjugate subgroups to $$H$$ in $$G$$ (so the total conjugacy class size of subgroups is 3). The three subgroups are given below:

$$H = \{, (1,2)(3,4) \}, \qquad H_1 = \{ , (1,4)(2,3) \}, \qquad H_2 = \{ , (1,3)(2,4) \}$$

See also subgroup structure of symmetric group:S4.

Complements
$$H$$ (and hence also each of its conjugate subgroups) has no permutable complements. However, it does have a lattice complement. Specifically, any S3 in S4 is a lattice complement to $$H$$ and also to each of its conjugates. For instance, $$\{, (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$$.