Clusters of galaxies evolve and accrete mass, mostly from small galaxy systems. Our aim is to study the velocity field of the galaxy cluster Abell 780, which is known for the powerful radio source Hydra A at its center and where a spectacular X-ray tail associated with the galaxy LEDA 87445 has been discovered. Our analysis is based on the new spectroscopic data for hundreds of galaxies obtained with the Italian Telescopio Nazionale Galileo and the Very Large Telescope. We have constructed a redshift catalog of 623 galaxies and selected a sample of 126 cluster members. We analyze the internal structure of the cluster using a number of techniques. We estimate the mean redshift z=0.0545, the line-of-sight velocity dispersion {sigma}_v_~800km/s, and the dynamical mass M200~5.4x10^14^M_{sun}_. The global properties of Abell 780 are typical of relaxed clusters. On a smaller scale, we can detect the presence of a galaxy group associated with LEDA 87445 in projected phase space. The mean velocity and position of the center of the group agree well with the velocity and position of LEDA 87445. We estimate the following parameters of the collision. The group is characterized by a higher velocity relative to the main system. It is infalling at a rest frame velocity of V_rf_~+870km/s and lies at a projected distance D~1.1Mpc to the south, slightly southeast of the cluster center. The mass ratio between the group and the cluster is ~1:5. We also find evidence of an asymmetry in the velocity distribution of galaxies in the inner cluster region, which might be related to a small low-velocity group detected as a substructure at V_rf_~-750km/s. We conclude that A780, although dynamically relaxed at first sight, contains small substructures that may have some impact on the energetics of the core region.