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Is there a theorem that can help us count easily the number of subgroups of any given finite group?

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    @Rasmus: You would then have $a=b^i$ and $a^j=b$, so $a$ and $b$ both have order finite. In an infinite group you either have an element of infinite order, or infinitely many elements of finite order. This means that every infinite group has infinitely many subgroups.2012-10-25

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Here is a partial answer by G.A. Miller: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077761/pdf/pnas01604-0036.pdf

It turns out that there is a way to count the subgroups of any given Abelian group.