  We investigate the problem of intertwined orders in fractional Chern
insulators by considering lattice fractional quantum Hall (FQH) states arising
from pairing of composite fermions in the square-lattice Hofstadter model. At
certain filling fractions, magnetic translation symmetry ensures the composite
fermions form Fermi surfaces with multiple pockets, leading to the formation of
finite-momentum Cooper pairs in the presence of attractive interactions. We
obtain mean-field phase diagrams exhibiting a rich array of striped and
topological phases, establishing paired lattice FQH states as an ideal platform
to investigate the intertwining of topological and conventional broken symmetry
order.
