If the number of elements in a group $G$ of order $289$ is $n\geq 273$ then $G$ is not cyclic.

Question

Let $G$ a finite group and $n$ the number of elements in $G$ of order $289$. Show that if $n\geq 273$ then $G$ is not cyclic.

Answer 1

If $G$ is cyclic and has one element of order $289$ then it contains exactly one subgroup of order $289=17^2$, it follows that in $G$ there are exactly $\phi(289)=17\times 16=272$ elements of order $289 $. This contradicts the asumption.