We analyze landslides on Ceres using several quantitative approaches to constrain the composition and structure of the top few kilometers of Ceres' crust. We focus on a subset of archetypal landslides classified morphologically as thick, steep-snouted "type 1" (T1) flows and thin spatulate "type 2" (T2) flows (Schmidt et al., 2017, https://doi.org/10.1038/ngeo2936) to explore the landslides' mechanical properties. Our results confirm earlier observations showing that T1 landslides are typically found poleward of 70° latitude and T2 mostly equatorward of 70° latitude. Measurements of landslide drop height and runout length imply effective friction coefficients lower than common friction coefficients in any of Ceres' identified or suggested non-ice surface materials, including saturated clays. Our measurements of the volume and area of landslide scars suggest that T1 landslides can fail to greater depths than T2 for a given scar area, consistent with depth-limited failure in T2 landslides. These results are consistent with a layer of lower shear strength material overlying a stronger layer in Ceres' outer shell at low to middle latitudes and a single layer without an overlying weak layer at polar latitudes. Combining these observations with known constraints on Ceres' near-surface composition, we propose that Ceres' crust at low to middle latitudes consists of a topmost layer with an ice content in excess of the spectral and elemental detection depths, thins out at high latitudes, and overlies a stronger and more ice-rich layer.