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The abundance, distribution and inner structure of satellites of galaxyclusters can be sensitive probes of the properties of dark matter. We run 30cosmological zoom-in simulations with self-interacting dark matter (SIDM), witha velocity-dependent cross-section, to study the properties of subhalos withincluster-mass hosts. We find that the abundance of subhalos that survive in theSIDM simulations are suppressed relative to their cold dark matter (CDM)counterparts. Once the population of disrupted subhalos -- which may hostorphan galaxies -- are taken into account, satellite galaxy populations in CDMand SIDM models can be reconciled. However, even in this case, the innerstructure of subhalos are significantly different in the two dark mattermodels. We study the feasibility of using the weak lensing signal from thesubhalo density profiles to distinguish between the cold and self-interactingdark matter while accounting for the potential contribution of orphan galaxies.We find that the effects of self-interactions on the density profile ofsubhalos can appear degenerate with subhalo disruption in CDM, when orphans areaccounted for. With current error bars from the Subaru Hyper Suprime-CamStrategic Program, we find that subhalos in the outskirts of clusters (wheredisruption is less prevalent) can be used to constrain dark matter physics. Inthe future, the Vera C. Rubin Observatory Legacy Survey of Space and Time willgive precise measurements of the weak lensing profile and can be used toconstrain $\sigma_T/m$ at the $\sim 1$ cm$^2$ g$^{-1}$ level at $v\sim 2000$ kms$^{-1}$.

The Signatures of Self-Interacting Dark Matter and Subhalo Disruption on Cluster Substructure