Finite cyclic and homomorph-containing implies order-unique

Statement
Suppose $$G$$ is a group and $$H$$ is a finite cyclic subgroup of $$G$$ that is also a fact about::homomorph-containing subgroup. Then, $$H$$ is an fact about::order-unique subgroup of $$G$$: there is no other subgroup of $$G$$ isomorphic to $$H$$.

When $$G$$ is a finite group, this is equivalent to saying that $$H$$ is an fact about::index-unique subgroup of $$G$$.