# Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one.

Are there simple formulas if one restricts to low rank for the subgroups? For example, are there formulas for enumerating cyclic subgroups, or subgroups whose minimal number of generators is $2$?

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