Quiescent galaxies are predominantly observed in local galaxy clusters. However, the fraction of quiescent galaxies in high-redshift clusters significantly varies among different clusters. In this study, we present the results of an analysis of the star formation (SF) properties of clusters and groups from a cosmological hydrodynamical simulation Horizon Run 5. We investigate the correlation between the quiescent galaxy fraction (QF) of these model clusters/groups and their various internal or external properties. We find that halo mass is one of the most important characteristics as higher mass clusters and groups have higher QFs. We also find that other properties such as stellar-mass ratio and Friends-of-Friends fraction, which measures the proportion of the area around a cluster occupied by dense structures, may mildly affect the QFs of clusters and groups. This may indicate that the evolutionary history as well as the large-scale environment of clusters and groups also play a certain role in determining the SF status of high-redshift galaxy clusters and groups.