Products of subgroups, subnormality, and relative orders of elements

Let $G$ be a group. We give an explicit description of the set of elements $x\in G$ such that $x^{\mathrm{\mid }G:H\mathrm{\mid }}\in H$ for every subgroup of finite index $H⩽G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $\mathrm{\mid }HK:H\mathrm{\mid }$ divide $\mathrm{\mid }G:H\mathrm{\mid }$?