Transverse intersection of two fixed point submanifolds?

Question

Let a finite group $G$ act smoothly on a smooth manifold $M$ and $H_1, H_2$ be subgroups of $G$ such that the set $H_1\cup H_2$ generates $G$. Moreover, assume that $$ \dim(M^{H_1})+\dim(M^{H_2})-\dim(M^G)=\dim(M) $$ (notice that $M^G=M^{H_1}\cap M^{H_2}$ since $H_1\cup H_2$ generates $G$). Is it true that $M^{H_1}$ and $M^{H_2}$ are transverse?