**Motivation:** It is obvious that for a finite index subgroup $H$ of a group $G$, there exists a normal subgroup $K$ of $G$, $K\subset H$, with $|G/K|<\infty$.

**Our question:** Let $G$ be a topological group and $H$ be a closed but not necessarily 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 $G$ such that $G/K$ is a compact topological group?

**Remark:** One can consider a similar problem in the context of rings and algebras: see [this follow-up question](https://mathoverflow.net/questions/355271/when-a-finite-codimensional-subalgebra-contains-a-finite-codimension-ideal).