  We study the interplay between quantum Hall ordering and spontaneous
sublattice symmetry breaking in multiple Chern number bands at fractional
fillings. Primarily we study fermions with repulsive interactions near half
filling in a family of square lattice models with flat C=2 bands and a wide
band gap. By perturbing about the particularly transparent limit of two
decoupled C=1 bands and by exact diagonalization studies of small systems in
the more general case, we show that the system generically breaks sublattice
symmetry with a transition temperature $T_c>0$ and additionally exhibits a
quantized Hall conductance of $e^2/h$ as $T \rightarrow 0$. We note the close
analogy to quantum Hall ferromagnetism in the multi-component problem and the
connection to topological Mott insulators. We also discuss generalizations to
other fillings and higher Chern numbers.
