TITLE: partial order on conjugate classes of subgroups
QUESTION [5 upvotes]: 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?

REPLY [5 votes]: Here is a countable example. Take the Baumslag-Solitar group $\langle a, b \mid bab^{-1}=a^{4}\rangle$. Let $K=\langle a\rangle$, $H=\langle a^2\rangle$. Then $[K]\ne [H]$ (the standard properties of HNN extensions, see the book of Lyndon and Schupp), but $K > H$ while $bKb^{-1} < H$.