Motivation: It is obvious that for a finite index subgroup $H$ of a group $G$, there exist a normal subgroup $K$ of $G$ of finite index with $K\subset G$.

Our question: Let $G$ be a topological group and $H$ be a closed but not necessarilly normal subgroup of $G$ such that the quotient topological space $G/H$ is a compact space. Does there exist a closed subgroup $K\subset H$ which is normal in $G$ and $G/K$ is a compact topological group.