  We construct generalized Hofstadter models that possess "color-entangled"
flat bands and study interacting many-body states in such bands. For a system
with periodic boundary conditions and appropriate interactions, there exist
gapped states at certain filling factors for which the ground-state degeneracy
depends on the number of unit cells along one particular direction. This
puzzling observation can be understood intuitively by mapping our model to a
single-layer or a multilayer system for a given lattice configuration. We
discuss the relation between these results and the previously proposed
"topological nematic states," in which lattice dislocations have non-Abelian
braiding statistics. Our study also provides a systematic way of stabilizing
various fractional topological states in $C>1$ flat bands and provides some
hints on how to realize such states in experiments.
