  Strong local interaction in systems with non-trivial topological bands can
stabilize quantum states such as magnetic topological insulators. We
investigate the influence of the lattice symmetry on the possible emergence of
antiferromagnetic quantum Hall states. We consider the spinful
Harper-Hofstadter model extended by a next-nearest-neighbor (NNN) hopping which
opens a gap at half-filling and allows for the realization of a quantum Hall
insulator. The quantum Hall insulator has the Chern number $\mathcal{C}=2$ as
both spin components are in the same quantum Hall state. We add to the system a
staggered potential $\Delta$ along the $\hat{x}$-direction favoring a normal
insulator and the Hubbard interaction $U$ favoring a Mott insulator. The Mott
insulator is a N\'eel antiferromagnet for small and a stripe antiferromagnet
for large NNN hopping. We investigate the $U$-$\Delta$ phase diagram of the
model for both small and large NNN hoppings. We show that while for large NNN
hopping there exists a $\mathcal{C}=1$ stripe antiferromagnetic quantum Hall
insulator in the phase diagram, there is no equivalent $\mathcal{C}=1$ N\'eel
antiferromagnetic quantum Hall insulator at the small NNN hopping. We discuss
that a $\mathcal{C}=1$ antiferromagnetic quantum Hall insulator can emerge only
if the effect of the spin-flip transformation cannot be compensated by a space
group operation. Our findings can be used as a guideline in future
investigations searching for antiferromagnetic quantum Hall states.
