Subgroup generated by double transposition in symmetric group:S4

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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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.

Cosets

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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, $\{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$.

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
retract	has a normal complement	No
permutably complemented subgroup	has a permutable complement	No
lattice-complemented subgroup	has a lattice complement	Yes

Subgroup properties

Normality-related properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	No
2-subnormal subgroup	normal subgroup of normal subgroup	Yes	Normal in $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$, which is the normal Klein four-subgroup of symmetric group:S4
hypernormalized subgroup	taking normalizers repeatedly reaches the whole group	No	Normalizer is D8 in S4, which is self-normalizing
pronormal subgroup	any conjugate is conjugate to it in their join	No		pronormal and subnormal implies normal
contranormal subgroup	normal closure is whole group	No
self-normalizing subgroup	equals its normalizer in the whole group	No

Resemblance-based properties

Property	Meaning	Satisfied?	Explanation	Comment
order-isomorphic subgroup	isomorphic to any subgroup of the same order	Yes	Any group of order two is isomorphic to cyclic group:Z2
isomorph-automorphic subgroup	automorphic to any subgroup isomorphic to it	No	The subgroup $\{ (), (1,2) \}$ is isomorphic to it but not related to it via an automorphism	See S2 in S4 for this other automorphism class.
automorph-conjugate subgroup	conjugate to any subgroup automorphic to it	Yes