We use the Hubble Space Telescope to obtain WFC3/F390W imaging of the supergroup SG1120-1202 at z=0.37, mapping the UV emission of 138 spectroscopically confirmed members. We measure total (F390W-F814W) colors and visually classify the UV morphology of individual galaxies as "clumpy" or "smooth." Approximately 30% of the members have pockets of UV emission (clumpy) and we identify for the first time in the group environment galaxies with UV morphologies similar to the "jellyfish" galaxies observed in massive clusters. We stack the clumpy UV members and measure a shallow internal color gradient, which indicates that unobscured star formation is occurring throughout these galaxies. We also stack the four galaxy groups and measure a strong trend of decreasing UV emission with decreasing projected group distance (R_proj_). We find that the strong correlation between decreasing UV emission and increasing stellar mass can fully account for the observed trend in (F390W-F814W)-R_proj_, i.e., mass-quenching is the dominant mechanism for extinguishing UV emission in group galaxies. Our extensive multi-wavelength analysis of SG1120-1202 indicates that stellar mass is the primary predictor of UV emission, but that the increasing fraction of massive (red/smooth) galaxies at R_proj_<~2R_200_ and existence of jellyfish candidates is due to the group environment.