I know that each finite abelian group is isomorphic to a direct product of cyclic groups of prime orders $> 1$. This means taking a finite abelian group of order $n$, I can find the prime factorization of $n$, and the group will be isomorphic to this direct product. Are these the isomorphic classes of abelian groups of order $n$?
For example, for $n = 200$, I know the prime factorization is $2\times2\times2\times5\times5$. How would I know how many isomorphic classes of abelian groups of order $200$ there are? Am I looking for cyclic subgroups of the abelian group of order $200$, with order $2$ or order $5$?
EDIT: I think I figured out where my misunderstanding of the fundamental theorem of abelian groups was. Basically, since $200 = 2^3 \cdot 5^2$, I am looking for all abelian groups of order $8$ and all abelian groups of order $25$, and then multiply the two numbers together to get the number of abelian groups isomorphic to order $200$.