= Solution
The <Fundamental theorem of finitely generated abelian groups> expresses every <finite abelian group> of this order as a product of <cyclic groups> of prime-power order. Their exponents correspond to the <partitions of an integer> $4$:
$$
\boxed{C_{p^4},\quad C_{p^3}\times C_p,\quad C_{p^2}\times C_{p^2},\quad C_{p^2}\times C_p\times C_p,\quad C_p^4.}
$$
These are pairwise nonisomorphic: the largest <order of a group element> distinguishes all except the two groups with largest element order $p^2$, and those have respectively $p^2$ and $p^3$ elements annihilated by multiplication by $p$. The classification exhausts the possibilities, so \b[there are five <isomorphism> classes].
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