In this paper, we prove that if $G$ is nonabelian and $ G >p^4$, then $G$ has a unique cyclic subgroup of order $p^m$ with $m\geq 3$ if and only if $G$ has a unique abelian subgroup of order $p^3$ if and only if $G$ is a $2$-group of maximal class.
Zhao, libo; li, yangming; and GONG, LU
"On the finite $p$-groups with unique cyclic subgroup of given order,"
Turkish Journal of Mathematics: Vol. 40:
2, Article 2.
Available at: https://journals.tubitak.gov.tr/math/vol40/iss2/2