Order-conjugate not implies order-dominated

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., order-conjugate subgroup) need not satisfy the second subgroup property (i.e., order-dominated subgroup)
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Statement

It is possible to have a finite subgroup $H$ of a group $G$ such that $H$ is conjugate to any subgroup of $G$ of the same order as $H$, but such that there exists a subgroup $K$ of $G$ whose order is a multiple of the order of $H$, but such that $H$ is not conjugate to any subgroup of $K$.

Related facts

• Order-conjugate not implies order-dominating
• Order-conjugate and Hall not implies order-dominating

Proof

Example of the alternating group of degree five

Further information: alternating group:A5, subgroup structure of alternating group:A5

Suppose $G$ is the alternating group on the set $\{1,2,3,4,5\}$. Suppose $H$ is the subgroup of $G$ given by:

$H:=\{(),(1,2,3),(1,3,2),(1,2)(4,5),(2,3)(4,5),(1,3)(4,5)\}$.

$H$ is conjugate to any other subgroup of $G$ of the same order.

On the other hand, suppose $K$ is the stabilizer of $\{5\}$ in $G$. Then, $K$ is the alternating group on $\{1,2,3,4\}$ and is isomorphic to the alternating group of degree four. The order of $K$ is $12$, which is a multiple of the order of $H$. However, no conjugate of $H$ is contained in $K$, since $H$ does not stabilize any element, but every conjugate of $K$ stabilizes some element.