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A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}_2$ with this property?

Finite groups of order $n$ having exactly $n$ subgroups

Is it known a characterization of finite groups of order $n$ having exactly $n$ subgroups?

A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}_2$ with this property?