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 such subsets $\{E^H\}$ is finite when $H$ runs over all compact subsgroups of $G$?

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 $G$?