  We study the phase diagram and phase transitions of the three dimensional
multicomponent lattice Higgs model with non-compact Abelian discrete groups.
The model with non-compact U(1) gauge group is known to undergo, for a
sufficiently large number of scalar fields $N$, a continuous transition
associated to the charged fixed point of the continuous Abelian Higgs field
theory. We show that in the model with gauge group $\mathbb{Z}_q^{(nc)}\equiv
2\pi\mathbb{Z}/q$ only critical transitions in the orthogonal universality
classes are present for small values of $N$, while a symmetry enlargement to
the continuous Abelian Higgs universality class happens when $q\ge 5$ and $N$
is large enough.
