properties on groups of order $p^2qr$

Question

I read somewhere that if $|G|=p^2qr$, $H\subseteq G: |H|= p^2q$, $p>q>r$ primes, then if only $H$ is maximal subgroup, then $H$ is Abelian. Is this problem correct? Are there any same properties in groups of order $p^2q^2r$, $p>q>r$?

Answer 1

I'm not certain that I'm interpreting your question correctly, so this may be rubbish. I think you are asking whether $H$ must be abelian if $H$ is the only subgroup of $G$ of order $p^2q$, where $G$ has order $p^2qr$ with $p,q,r$ primes and $p>q>r$.

In this case, the answer is "no". Take the non-abelian group $H$ of order $75 = 5^23$ (so $p=5$ and $q=3$), and put $G = H\times C_2$, where $C_2$ is cyclic of order $2$ (so $r=2$). Then $H$ is the only subgroup of order $75$ in $G$, but of course, $H$ is not abelian by construction.