Motivated by the recent discovery of several dwarfs near the Large Magellanic Cloud (LMC), we study the accretion of massive satellites onto Milky Way (MW)/M31-like haloes using the ELVIS suite of N-body simulations. We identify 25 surviving LMC-mass subhaloes, and investigate the lower-mass satellites that were associated with these subhaloes before they fell into the MW/M31 haloes. Typically, 7 per cent of the overall z = 0 satellite population of MW/M31 haloes were in a surviving LMC-group before falling into the MW/M31 halo. This fraction can vary between 1 and 25 per cent, being higher for groups with higher mass and/or more recent infall times. Groups of satellites disperse rapidly in phase space after infall, and their distances and velocities relative to the group centre become statistically similar to the overall satellite population after 4-8 Gyr. We quantify the likelihood that satellites were associated with an LMC-mass group as a function of both distance and velocity relative to the LMC at z = 0. The close proximity in distance of the nine Dark Energy Survey candidate dwarf galaxies to the LMC suggest that ~2-4 are likely associated with the LMC. Furthermore, if several of these dwarfs are genuine members, then the LMC-group probably fell into the MW very recently, ≲2 Gyr ago. If the connection with the LMC is established with follow-up velocity measurements, these 'satellites of satellites' represent prime candidates to study the effects of group pre-processing on lower mass dwarfs.