Let $G=PQ$ where $P$ and $Q$ are $p$- and $q$-Sylow subgroups of $G$ respectively. In addition, suppose that $P\unlhd G$, $Q\ntrianglelefteq G$, $C_G(P)=Z(G)$ and $C_G(Q)\neq Z(G)$, where $Z(G)$ is the center of $G$.
I want to prove there exist two elements $x,y\in G-Z(G)$ such that $\left|C_G(x)\right| \nmid \left|C_G(y)\right|$ and $\left|C_G(y)\right| \nmid \left|C_G(x)\right|$.
By $C_G(Q)\neq Z(G)$, we know that there exists an element in $G$ which centralizes $Q$. By $C_G(P)=Z(G)$ we obtain that there is no non-central element that centralizes $P$. But if we can find an element that centralizes a big $p$-subgroup and small $q$-subgroup, we're done.
By GAP I have checked all groups of order less than $383$ with this hypothesis and couldn't find any counterexamples.
But I can't prove it!