Finite group having the same order statistics as a cyclic group is cyclic

Statement
Suppose $$G$$ and $$H$$ are finite groups that are order statistics-equivalent, i.e., $$G$$ and $$H$$ have the same order statistics. Then, if $$G$$ is a fact about::cyclic group (in particular, a fact about::finite cyclic group), so is $$H$$. In particular, $$G$$ and $$H$$ are isomorphic groups.

Related facts

 * Order statistics of a finite group determine whether it is nilpotent
 * Finite abelian groups with the same order statistics are isomorphic
 * Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring