Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$.
Is it true that the family of all such subsets $\{E^H\}$ is finite when $H$ runs over all compact subsgroups of $G$?
The quotient $M/G$ carries a stratification by orbit type (see e.g. this MO question for references). More precisely, for any closed subgroup $F\subseteq G$ the stratum $(M/G)_{(F)}$ is the set of all orbits which are isomorphic to $G/F$. The set $E^H$ is the union of all strata such that $H$ is conjugate to a subgroup of $F$. Since $M$ is compact, the stratification is finite. So also only finitely many subsets of the form $E^H$ are possible.
