Let $G$ be a real Lie group. What conditions must $G$ satisfy so that the following is true:

For any finite group $\Gamma$ there exist finitely many conjugacy classes of subgroups of $G$ that are isomorphic to $\Gamma$.

I believe that for $G=GL(n,\mathbb{R})$ this is true because: subgroups of $GL(n,\mathbb{R})$ are conjugate if and only if the restriction of the standard representation of $GL(n,\mathbb{R})$ to the subgroups are isomorphic representations. Finite groups have finitely many irreducible representations, and that proves the claim.

I also believe that for $G=SO(n)$ this is true because of this answer: https://mathoverflow.net/a/17074/164084.

It would be enough for my application to know that every real, compact, connected Lie group that has a faithful representation has the property stated above ("For any finite group $\Gamma$...").



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