Abelian groups of order 16

Since \[ 16=2^4, \] the finite abelian groups of order \(16\) correspond to the partitions of \(4\). Thus there are exactly five isomorphism types: \[ C_{16}, \qquad C_8\times C_2, \qquad C_4\times C_4, \qquad C_4\times C_2\times C_2, \qquad C_2^4. \]

These correspond respectively to the partitions \[ 4,\quad 3+1,\quad 2+2,\quad 2+1+1,\quad 1+1+1+1. \] By the classification theorem for finite abelian groups, the list is complete and the five groups are pairwise nonisomorphic.