S3 in S5

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) symmetric group:S3 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
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Definition

The group $G$ is taken as symmetric group:S5: the symmetric group of degree five. For concreteness, we take $G$ as the symmetric group on the set $\{1,2,3,4,5\}$ .

We take $H=H_{4,5}$ as the subgroup fixing $\{4,5\}$ , so $H$ is symmetric group:S3 acting on the set $\{1,2,3\}$ .

$H$ has nine other conjugate subgroups, each corresponding to a different pair of fixed points:

$H_{1,2}$ is the subgroup fixing $\{1,2\}$ , and is the symmetric group on the set $\{3,4,5\}$ .
$H_{1,3}$ is the subgroup fixing $\{1,3\}$ , and is the symmetric group on the set $\{2,4,5\}$ .
$H_{1,4}$ is the subgroup fixing $\{1,4\}$ , and is the symmetric group on the set $\{2,3,5\}$ .
$H_{1,5}$ is the subgroup fixing $\{1,5\}$ , and is the symmetric group on the set $\{2,3,4\}$ .
$H_{2,3}$ is the subgroup fixing $\{2,3\}$ , and is the symmetric group on the set $\{1,4,5\}$ .
$H_{2,4}$ is the subgroup fixing $\{2,4\}$ , and is the symmetric group on the set $\{1,3,5\}$ .
$H_{2,5}$ is the subgroup fixing $\{2,5\}$ , and is the symmetric group on the set $\{1,3,4\}$ .
$H_{3,4}$ is the subgroup fixing $\{3,4\}$ , and is the symmetric group on the set $\{1,2,5\}$ .
$H_{3,5}$ is the subgroup fixing $\{3,5\}$ , and is the symmetric group on the set $\{1,2,4\}$ .
$H=H_{4,5}$ is the subgroup fixing $\{4,5\}$ , and is the symmetric group on the set $\{1,2,3\}$ .