So if a Lie group $G$ acts transitively on a manifold $X$, then $X$ is diffeomorphic to the quotient $G/H$ where $H$ is the stabilizer subgroup of a point $x$ in $X$.
Now if $N$ is a normal subgroup of $G$, then $N\cdot x\cong N/(N\cap H)$.
Can we imitate the second isomorphism theorem, i.e, is it true that $N/(N\cap H) \cong NH/H$ as manifolds?