Structure theorem for finitely generated abelian groups

How can we use fundamental theorem of finitely generated abelian groups to list all abelian groups of order 16 up to isomorphism.

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This is a minor complement to Arturo’s answer.

The number of abelian groups of order $p_1^{n_1}\cdots p_k^{n_k}$ (where the $p_i$ are distinct primes and the $n_i$ are positive integers), considered up to isomorphism, is
$$
P(n_1)\cdots P(n_k),
$$
where $P(n)$ is the number of partitions of $n$.

As explained by Arturo, the Structure Theorem tells us which group corresponds to a given partition (and a given prime).

To wit: If the prime is $p$ and the parts of the partition are $n_1,\dots,n_k$, then the group is the product of the cyclic groups of order $p^{n_i}$.

(I’m using implicitly the fact that a finite abelian group can we written in a unique way as a product of $p$-groups.)

The Theorem tells you that every finitely generated abelian group can be written as a direct sum of cyclic groups which, under certain conditions, is unique. So find all possible direct sums of cyclic groups that satisfy those conditions and give you groups of order $16$ to get the answer.

(There are two versions of the structure theorem, but the conditions coincide for groups of prime power order).