Products of subgroups, subnormality, and relative orders of elements

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