Order-conjugate not implies order-dominating

Definition
We can have a group $$G$$ with an order-conjugate subgroup $$H$$ (i.e., a subgroup $$H$$ that is conjugate to any other subgroup of the same order) that is not an order-dominating subgroup: in other words, there exists a subgroup $$K$$ of $$G$$ whose order divides the order of $$H$$, but such that $$K$$ is not contained in any conjugate of $$H$$.

Related facts

 * Order-conjugate not implies order-dominated
 * Order-conjugate and Hall not implies order-dominating

Example of the alternating group of degree five
Suppose $$G$$ is the alternating group on the set $$S = \{ 1,2,3,4,5 \}$$. Suppose $$H$$ is the subgroup of $$G$$ that is the alternating group on $$\{ 1,2,3,4 \}$$. In other words, $$H$$ is the stabilizer of the point $$\{ 5 \}$$. Then, $$H$$ is order-conjugate in $$G$$: the subgroups of the same order as $$H$$ are precisely the stabilizers of points in $$S$$, and these are conjugate to $$H$$ by suitable $$5$$-cycles.

On the other hand, consider the subgroup $$K$$:

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

$$K$$ is a group of order six, isomorphic to the symmetric group of degree three. However, $$K$$ is not contained in any conjugate of $$H$$, because any conjugate of $$H$$ stabilizes some element, and $$K$$ does not stabilize any element.