  The classification of dark matter halos as isolated hosts or subhalos is
critical for our understanding of structure formation and the galaxy-halo
connection. Most commonly, subhalos are defined to reside inside a spherical
overdensity boundary such as the virial radius. The resulting host-subhalo
relations depend sensitively on the somewhat arbitrary overdensity threshold,
but the impact of this dependence is rarely quantified. The recently proposed
splashback radius tends to be larger and to include more subhalos than even the
largest spherical overdensity boundaries. We systematically investigate the
dependence of the subhalo fraction on the radius definition and show that it
can vary by factors of unity between different spherical overdensity
definitions. Using splashback radii can yet double the abundance of subhalos
compared to the virial definition. We also quantify the abundance of flyby (or
backsplash) halos, hosts that used to be subhalos in the past. We show that the
majority of these objects are mislabeled satellites that are naturally
classified as subhalos when we use the splashback radius. We show that the
subhalo fraction can be understood as a universal function of only peak height
and the slope of the linear power spectrum. We provide a simple fitting
function that captures our simulation results to 20% accuracy across a wide
range of halo masses, redshifts, and cosmologies. Finally, we demonstrate that
splashback radii significantly change our understanding of satellite and flyby
galaxies in the Local Group.
