Are there any finite non-abelian group with one subgroup of each size ?

Question

Let $G$ be a finite group with at most one subgroup of any size , then is it true that $G$ is cyclic ? I can prove that the answer is "yes" with the additional assumption of abelian ness on $G$ but without it I cannot make any headway , Please help . Thanks in advance

Answer 1

Hint: Use the Sylow Theorems to reduce to the case when $|G| = p^n$, $p$ a prime.

Answer 2

Alternative hint:

Use the formula $\sum_{d \mid n}\varphi(d)=n$ and the fact that cyclic group of order $n$ has $\varphi(d)$ elements of order $d$ for every $d \mid n$ (in particular, if there is an element of order $d$ in $G$, there are exactly $\varphi(d)$ generators of the same cyclic subgroup). Here $\varphi$ denotes the Euler's totient function.