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In Fraleigh, there is a question asking if you can say how many subgroups of order 8 an abelian group $G$ of order 72 has, and also if you can say how many subgroups of order 4 that group has. The answer key says there is only one of order 8, but I don't understand why.
I know that $G$ can be decomposed into a direct product of cyclic groups. But both $Z_8 \times Z_9$ and $ Z_2 \times Z_4 \times Z_9$ are of order 72; while the first one has a subgroup $Z_8$, the second one has a subgroup of $Z_2 \times Z_4$, and these two groups are not isomorphic.
What am I missing? Also, why couldn't we say how many groups of order 4 $G$ has?