With the advent of the Vera C. Rubin Observatory, the discovery rate of supernovae (SNe) will surpass the rate of SNe with real time spectroscopic follow-up by 3 orders of magnitude. Accurate photometric classifiers are essential to both select interesting events for follow-up in real time and for archival population-level studies. In this work, we investigate the impact of observable host-galaxy information on the classification of SNe, both with and without additional light-curve and redshift information. We find that host-galaxy information alone can successfully isolate relatively pure (>90%) samples of Type Ia SNe with or without redshift information. With redshift information, we can additionally produce somewhat pure (>70%) samples of Type II SNe and superluminous SNe. Additionally with redshift information, host- galaxy properties do not significantly improve the accuracy of SN classification when paired with complete light curves. In the absence of redshift information, however, galaxy properties significantly increase the accuracy of photometric classification. As a part of this analysis, we present the first formal application of a new objective function, the weighted hierarchical cross entropy, to the problem of SN classification. This objective function more naturally accounts for the hierarchical nature of SN classes and, more broadly, transients. Finally, we present a new set of SN classifications for the Pan-STARRS Medium Deep Survey of SNe that lack spectroscopic redshift, increasing the full photometric sample to >4400 events.