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Both observations and cosmological simulations have recently shown that thereis a large scatter in the number of satellites of Milky Way (MW)-like galaxies.In this study, we investigate the relation between the satellite number andgalaxy group assembly history, using the $r-$band magnitude gap ($\Deltam_{12}$) between the first and the second brightest galaxy as an indicator.From 20 deg$^2$ of Hyper Suprime-Cam Subaru Strategic Program Wide layer, weidentify 17 dwarf satellite candidates around NGC 4437, a spiral galaxy withabout one-fourth of the MW stellar mass. We estimate their distances using thesurface brightness fluctuation (SBF) method. Then we confirm five candidates asmembers of the NGC 4437 group, resulting in a total of seven group members.Combining the NGC 4437 group (with $\Delta m_{12} = 2.5$ mag) with other groupsin the literature, we find a stratification of the satellite number by $\Deltam_{12}$ for a given host stellar mass. The satellite number for given hoststellar mass decreases as $\Delta m_{12}$ increases. The same trend is found insimulated galaxy groups in IllustrisTNG50 simulations. We also find that thehost galaxies in groups with a smaller $\Delta m_{12}$ (like NGC 4437) haveassembled their halo mass more recently than those in larger gap groups, andthat their stellar-to-halo mass ratios (SHMRs) increase as $\Delta m_{12}$increases. These results show that the large scatter in the satellite number isconsistent with a large range of $\Delta m_{12}$, indicating diverse groupassembly histories.

A Rich Satellite Population of the NGC 4437 Group and Implications of a Magnitude Gap for Galaxy Group Assembly History