  The set $\mathcal S(G)$ of compact subgroups of a Hausdorff topological group
$G$ can be equipped with the Vietoris topology. A compact subgroup
$K\in\mathcal S(G)$ is isolated up to conjugacy if there is a neighborhood
$\mathcal U\subseteq\mathcal S(G)$ of $K$ such that every $L\in\mathcal U$ is
conjugate to $K$. In this paper, we characterize compact subgroups of a Lie
group that are isolated up to conjugacy. Our characterization depends only on
the intrinsic structure of $K$, the ambient Lie group $G$ and the embedding of
$K$ into $G$ are irrelevant.
  In addition, we prove that any continuous homomorphism from a compact group
$G$ onto a compact Lie group $H$ induces a continuous open map from $\mathcal
S(G)$ onto $\mathcal S(H)$.
