G is a group. For a subgroup H of G, note $[H]$ the class of subgroups which are conjugate to H.
Define the binary relation: $[H] \leq [K]$ iff $H_0 \subset K_0$ for some $H_0 \in [H]$ and $K_0 \in [K]$
It is easy to see that this relation is reflexive and transitive. But how to show that it is anti-symmetric?
P.S. In a book, the author claims that this relation defines a partial order on the classes of conjugate subgroups in a context where G is a compact Lie group. But I don't think the compact Lie group condition be essential, right?