  The total luminosity of satellite galaxies around a central galaxy,
L$_{sat}$, is a powerful metric for probing dark matter halos. In this paper we
use data from the Sloan Digital Sky Survey and DESI Legacy Imaging Surveys to
explore the relationship between L$_{sat}$ and various observable galaxy
properties for a sample of 117,966 central galaxies out to $z = 0.15$. At fixed
stellar mass, every galaxy property we explore shows a correlation with
L$_{sat}$. This implies that dark matter halos play a possibly significant role
in determining these secondary galaxy properties. We quantify these
correlations by computing the mutual information between L$_{sat}$ and
secondary properties and explore how this mutual information varies as a
function of stellar mass and when separating the sample into star-forming and
quiescent central galaxies. We find that absolute r-band magnitude correlates
more strongly with L$_{sat}$ than stellar mass across all galaxy populations;
and that effective radius, velocity dispersion, and S\'ersic index do so as
well for star-forming and quiescent galaxies. L$_{sat}$ is sensitive to both
the mass of the host halo as well as the halo formation history, with younger
halos having higher L$_{sat}$. L$_{sat}$ by itself cannot distinguish between
these two effects, but measurements of galaxy large-scale environment can break
this degeneracy. For star-forming central galaxies, we find that r$_{\rm eff}$,
$\sigma_v$, and S\'ersic index all correlate with large-scale density, implying
that these halo age plays a role in determining these properties. For quiescent
galaxies, we find that all secondary properties are independent of environment,
implying that correlations with L$_{sat}$ are driven only by halo mass. These
results are a significant step forward in quantifying the full extent of the
galaxy-halo connection, and present a new test of galaxy formation models.
