Let $G$ be a finite group, say $|G|=p^am$, with $(p,m)=1$, Let $n$ be the number of $p$-Sylow subgroups of $G$. Call them $P_1,\dots,P_n$.
Is true that every subgroup of $G$ of order $p^b$ with $b\le a$ must be contained in some $P_i$?
I conjectured this and I think it's true, but I'm not able to prove it.
Any hint will be welcome, thanks!