We investigate the influence of supernova (SN) feedback on the satellites of elliptical host galaxies using hydrodynamic simulations. Utilizing a modified version of the GADGET-3 code, we perform cosmological zoom-in simulations of 11 elliptical galaxies with stellar masses in the range 1011M⊙ < M* < 2 × 1011M⊙. We conduct two sets of simulations with identical initial conditions: a fiducial model, which includes a three-phase SN mechanical wind, and a weak SN feedback model, where nearly all SN energy is released as thermal energy with a reduced SN wind velocity. Our comparison shows minimal differences in the elliptical host galaxies, but significant variations in the physical properties of satellite galaxies. The weak SN feedback model produces a larger number of satellite galaxies compared to the fiducial model, and significantly more than observed. For satellite galaxies with stellar masses above 108M⊙, the weak SN feedback model generates approximately 5 times more satellites than observed in the Extending the Satellites Around Galactic Analogs Survey (or xSAGA) survey. Most of these overproduced satellites have small stellar masses, below 1010M⊙. Additionally, satellites in the weak SN feedback model are about 3.5 times more compact than those observed in the SAGA survey and the fiducial model, with metallicities nearly 1 dex higher than observed values. In conclusion, the satellite galaxies in the fiducial model, which includes mechanical SN feedback, exhibit properties that more closely align with observations. This underscores the necessity of incorporating both mechanical active galactic nuclei and SN feedback to reproduce the observed properties of elliptical galaxies and their satellites in simulations.