Let $G$ be a non-abelian $p$-group and $p^k\ge p^3$ be a proper divisor of $|G|$.
It is well known that the number of subgroups of order $p^k$ in $G$ is $1\pmod{p}$.
Divide the subgroups of order $p^k$ into two parts: abelian and non-abelian; which of them are $1\pmod{p}$ in number? Or is it possible that none of them are $1\pmod{p}$ in number? (Just stating the result is fine; I will try to prove it.)