We investigate the disruption of group and cluster satellite galaxies with total mass (dark matter plus baryons) above 1010 M⊙ in the Hydrangea simulations, a suite of 24 high-resolution cosmological hydrodynamical zoom-in simulations based on the EAGLE model. The simulations predict that ∼50 per cent of satellites survive to redshift z = 0, with higher survival fractions in massive clusters than in groups and only small differences between baryonic and pure N-body simulations. For clusters, up to 90 per cent of galaxy disruption occurs in lower-mass sub-groups (i.e., during pre-processing); 96 per cent of satellites in massive clusters that were accreted at z < 2 and have not been pre-processed survive. Of those satellites that are disrupted, only a few per cent merge with other satellites, even in low-mass groups. The survival fraction changes rapidly from less than 10 per cent of those accreted at high z to more than 90 per cent at low z. This shift, which reflects faster disruption of satellites accreted at higher z, happens at lower z for more massive galaxies and those accreted onto less massive haloes. The disruption of satellite galaxies is found to correlate only weakly with their pre-accretion baryon content, star formation rate, and size, so that surviving galaxies are nearly unbiased in these properties. These results suggest that satellite disruption in massive haloes is uncommon, and that it is predominantly the result of gravitational rather than baryonic processes.