  We study the topological properties of a spin-orbit coupled Hofstadter model
on the Kagome lattice. The model is time-reversal invariant and realizes a
$\mathbb{Z}_2$ topological insulator as a result of artificial gauge fields. We
develop topological arguments to describe this system showing three
inequivalent sites in a unit cell and a flat band in its energy spectrum in
addition to the topological dispersive energy bands. We show the stability of
the topological phase towards spin-flip processes and different types of
on-site potentials. In particular, we also address the situation where on-site
energies may differ inside a unit cell. Moreover, a staggered potential on the
lattice may realize topological phases for the half-filled situation. Another
interesting result is the occurrence of a topological phase for large on-site
energies. To describe topological properties of the system we use a numerical
approach based on the twisted boundary conditions and we develop a mathematical
approach, related to smooth fields.
