There is a well-known asymptotic formula, due to W. M. Schmidt [Duke Math. J., 35 (1968), pp. 327--339], for the number of full-rank integer lattices of index at most $V$ in ${\mathbb{Z}}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group ${\mathbb{Z}}^n/L$. Accordingly, we count the number of full-rank integer lattices $L \subseteq {\mathbb{Z}}^n$ such that ${\mathbb{Z}}^n/L$ is cyclic and of order at most $V$, and deduce that these co-cyclic lattices are dominant among all integer lattices: their natural density is $(\zeta(6) \prod_{k=4}^n \zeta(k))^{-1} \approx 85\%$. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.