We set an upper bound on the gravitational cutoff in theories with exact quantum numbers of large N periodicity, such as Z(N) discrete symmetries. The bound stems from black hole physics. It is similar to the bound appearing in theories with N particle species, though a priori, a large discrete symmetry does not imply a large number of species. Thus, there emerges a potentially wide class of new theories that address the hierarchy problem by lowering the gravitational cutoff due to the existence of large Z(1032)-type symmetries.