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 Sersic 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 Sersic 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.
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