  A major question in $\Lambda$CDM is what this theory actually predicts for
the properties of subhalo populations. Subhalos are difficult to simulate and
to find within simulations, and this propagates into uncertainty in theoretical
predictions for satellite galaxies. We present Symfind, a new
particle-tracking-based subhalo finder, and demonstrate that it can track
subhalos to orders-of-magnitude lower masses than commonly used halo-finding
tools, with a focus on Rockstar and consistent-trees. These longer survival
mean that at a fixed peak subhalo mass, we find $\approx 15\%{-}40\%$ more
subhalos within the virial radius, $R_\textrm{vir}$, and $\approx 35\%-120\%$
more subhalos within $R_\textrm{vir}/4$ in the Symphony dark-matter-only
simulation suite. More subhalos are found as resolution is increased. We
perform extensive numerical testing. In agreement with idealized simulations,
we show that the $v_{\rm max}$ of subhalos is only resolved at high resolutions
($n_\textrm{peak}\gtrsim3\times 10^4$), but that mass loss itself can be
resolved at much more modest particle counts ($n_\textrm{peak}\gtrsim4\times
10^3$). We show that Rockstar converges to false solutions for the mass
function, radial distribution, and disruption masses of subhalos. We argue that
our new method can trace resolved subhalos until the point of typical galaxy
disruption without invoking ``orphan'' modeling. We outline a concrete set of
steps for determining whether other subhalo finders meet the same criteria. We
publicly release Symfind catalogs and particle data for the Symphony simulation
suite at \url{http://web.stanford.edu/group/gfc/symphony}.
