  The demographics of dark matter substructure depend sensitively on the nature
of dark matter. Optimally leveraging this probe requires accurate theoretical
predictions regarding the abundance of subhaloes. These predictions are
hampered by artificial disruption in numerical simulations, by large
halo-to-halo variance, and by the fact that the results depend on the baryonic
physics of galaxy formation. In particular, numerical simulations have shown
that the formation of a central disc can drastically reduce the abundance of
substructure compared to a dark matter-only simulation, which has been
attributed to enhanced destruction of substructure due to disc shocking. We
examine the impact of discs on substructure using the semi-analytical subhalo
model SatGen, which accurately models the tidal evolution of substructure free
of the numerical disruption that still hampers $N$-body simulations. Using a
sample of 10,000 merger trees of Milky Way-like haloes, we study the
demographics of subhaloes that are evolved under a range of composite halo-disc
potentials with unprecedented statistical power. We find that the overall
subhalo abundance is relatively insensitive to properties of the disc aside
from its total mass. For a disc that contains $5\%$ of $M_\mathrm{vir}$, the
mean subhalo abundance within $r_\mathrm{vir}$ is suppressed by
${\lesssim}10\%$ relative to the no-disc case, a difference that is dwarfed by
halo-to-halo variance. For the same disc mass, the abundance of subhaloes
within 50 kpc is reduced by ${\sim}30\%$. We argue that the disc mainly drives
excess mass loss for subhaloes with small pericentric radii and that the impact
of disc shocking is negligible.
