  We study the phase transitions induced by a lateral superlattice potential (a
metallic grid) placed on top of a two-dimensional electron gas (2DEG)formed in
a semiconductor quantum well. In a quantizing magnetic field and at filling
factor $\nu =1,$ the ground state of the 2DEG depends on the strength $V_{g}$
of the superlattice potential as well as on the number of flux quanta piercing
the unit cell of the external potential. It was recently shown[1] that in the
case of a square lateral superlattice, the potential modulates both the
electronic and spin density and in some range of $V_{g}$, the ground state is a
two-sublattice spin meron crystal where adjacent merons have the global phase
of their spin texture shifted by $\pi$, i.e. they are "antiferromagnetically"
ordered. In this work, we evaluate the importance of Landau-level mixing on the
phase diagram obtained previously for the square lattice [1] and derive the
phase diagram of the 2DEG modulated by a triangular superlattice. When Landau
level mixing is considered, we find in this case that, in some range of
$V_{g},$ the ground state is a three-sublattice spin meron crystal where
adjacent merons of the same vorticity have the global phase of their spin
texture rotated by $120^{\circ }$ with respect to one another. This meron
crystal is preceded in the phase diagram by another meron lattice phase with a
very different spin texture that does not appear, at first glance, to resolve
the spin frustration inherent to an antiferromagnetic ordering on a triangular
lattice.
  [1] R. C\^ot\'e and Xavier Bazier-Matte, Phys. Rev. B 94, 205303 (2016).
