TITLE: Group of order $pqr$ and cyclic subgroup
QUESTION [5 upvotes]: Let $G$ be group of order $pqr$, when $p,q,r$ are different prime numbers. Does $G$ must have normal cyclic subgroup $H$ such that $G/H$ is cyclic too ?
I know that $G$ has normal sylow subgroup of order $p$ or $q$ or $r$, and that $G$ is solvable, but I can't see why there is a cyclic quotient.
Edit: Attempt for solution - 
If $G$ is abelian, then $G\cong \mathbb{Z}_{p}\oplus\mathbb{Z}_{q}\oplus\mathbb{Z}_{r}\cong \mathbb{Z}_{pqr}$, hence cyclic and it's clear.
If not, then $G'\neq \left\{ e\right\}$. If $\left|G'\right|=p$, then $G/G'\cong \mathbb{Z}_{qr}$ (because $G'/G$ is abelian), both cyclic. and the other cases are symetric.
$G'\neq G$ because $G$ is solvable.

REPLY [2 votes]: The answer is yes. Let's suppose that $G$ has a normal Sylow $p$-subgroup $P$.
If $C_G(P) = P$ then, since $G/C_G(P)$ is isomorphic to a subgroup of ${\rm Aut}(P)$, which is cyclic, $G/P$ is cyclic.
If $P < C_G(P) < G$, then $C_G(P)$ and $G/P$ are both cyclic.
Finally, if $C_G(P) = G$ then $G/P$ has a normal Sylow subgroup $K/P$ say, and then $K$ and $G/K$ are cyclic.