  At and near charge neutrality, monolayer graphene in a perpendicular magnetic
field is a quantum Hall ferromagnet. In addition to the highly symmetric
Coulomb interaction, residual lattice-scale interactions, Zeeman, and
sublattice couplings determine the fate of the ground state. Going beyond the
simplest model with ultra-short-range residual couplings to more generic
couplings, one finds integer phases that show the coexistence of magnetic and
lattice order parameters. Here we show that fractional quantum Hall states in
the vicinity of charge neutrality have even richer phase diagrams, with a
plethora of phases with simultaneous magnetic and lattice symmetry breaking.
