Finding subgroups of a particular order

Question

Is it true that a group of order pqr, distinct primes, has a subgroup of order pq, pr, and qr? It seems to me that this should be an "easy" corollary to either the actual Sylow Theorems or the discussions surrounding the proof of them which use centers and normalizers etc.

Answer 1

It is proved in an answer to Groups of order $pqr$ and their normal subgroups that, if $p>q>r$, then the group $G$ has normal subgroups $N$ of order $p$ and $M$ of order $pq$. From Sylow's Theorem in $G/N$, you can then show that $G$ has a subgroup of order $pr$. You can find a subgroup of order $qr$ by applying the Frattini argument to a Sylow $q$-subgroup of $M$. More details on request!