Every smooth manifold is assumed to be Hausdorff and second-countable.

Suppose $G$ is a Lie group, $H$ is a Lie subgroup of $G$, $N$ is a closed Lie subgroup of $G$ such that $N$ is normal, $N\cap H=\{e\}$, and $NH=G$, where $NH=\{ab:a\in N, b\in H\}$.

Is $H$ closed in $G$?