  The most usual option to stabilize Dark Matter (DM) is a $Z_2$ symmetry. In
general, though, DM may be stabilized by any $Z_N$ with $N \ge 2$. We consider
the way $Z_N$ is a subgroup of the internal-symmetry group $G$ of a model; we
entertain the possibility that $Z_N$ is the center of $G$, yet $G$ is not of
the form $Z_N \times G^\prime$, where $G^\prime$ is a group smaller (i.e. of
lower order) than $G$. We examine all the discrete groups of order smaller than
2001 and we find that many of them cannot be written as the direct product of a
cyclic group and some other group, yet they have a non-trivial center that
might be used in Model Building to stabilize DM.
