  Two-dimensional systems with $C_{2}\mathcal{T}$ ($P\mathcal{T}$) symmetry
exhibit the Euler class topology $E\in\mathbb{Z}$ in each two-band subspace
realizing a fragile topology beyond the symmetry indicators. By systematically
studying the energy levels of Euler insulating phases in the presence of an
external magnetic field, we reveal the robust gaplessness of the Hofstadter
butterfly spectrum in the flat-band limit, while for the dispersive bands the
gapping of the Landau levels is controlled by a hidden symmetry. We also find
that the Euler class $E$ of a two-band subspace gives a lower bound for the
Chern numbers of the magnetic subgaps. Our study provides new fundamental
insights into the fragile topology of flat-band systems going beyond the
special case of $E=1$ as e.g.~in twisted bilayer graphene, thus opening the way
to a very rich, still mainly unexplored, topological landscape with higher
Euler classes.
