We report the discovery of 40 new satellite dwarf galaxy candidates in the sphere of influence of the Sombrero Galaxy (M104), the most luminous galaxy in the Local Volume. Using the Subaru Hyper Suprime-Cam, we surveyed 14.4 deg(2) of its surroundings, extending to the virial radius. Visual inspection of the deep images and GALFIT modelling yielded a galaxy sample highly complete down to M-g similar to -9 (L-g similar to 3x10(5) L-circle dot) and spanning magnitudes -16.4 < M-g < -8 and half-light radii 50 pc < r(e) < 1600 pc assuming the distance of M104. These 40 new candidates, out of which 27 are group members with high confidence, double the number of potential satellites of M104 within the virial radius, placing it among the richest hosts in the Local Volume. Using a principal component analysis, we find that the entire sample of candidates is consistent with an almost circular on-sky distribution, more circular than any comparable environment found in the Illustris TNG100-1 (The Next Generation) simulation. However, the distribution of the high-probability sample is more oblate and consistent with the simulation. The cumulative satellite luminosity function is broadly consistent with analogues from the simulation, albeit it contains no bright satellite with M-g < -16.4 (L-g similar to 3 x 10(8) L-circle dot), a 2.3 sigma occurrence. Follow-up spectroscopy to confirm group membership will begin to demonstrate how these systems can act as probes of the structure and formation history of the halo of M104.