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With this Paper we complete a comprehensive study of substructure in darkmatter haloes. In Paper I we derived the radial distribution and mass function(MF) of accreted subhaloes (scaled to the radius and mass of the host halo) andshowed they are essentially universal. This is not the case, however, for thoseof stripped subhaloes, which depend on halo mass and assembly history. In PaperII we derived these latter properties in the simplest case of purely accretinghaloes. Here we extend the study to ordinary haloes having suffered majormergers. After showing that all the properties of substructure are encoded inthe mean truncated-to-original subhalo mass ratio profile, we demonstrate thatthe dependence of the subhalo MF on halo mass arises from their mass-dependentconcentration, while the shape of the subhalo radial distribution depends onthe time of the last major merger of the host halo. In this sense, the latterproperty is a better probe of halo formation time than the former.Unfortunately, this is not the case for the radial distribution of satellitesas this profile is essentially disconnected from subhalo stripping and theproperties of accreted subhaloes are independent of the halo formation time.

An Accurate Comprehensive Approach to Substructure: III. Masses and Formation Times of the Host Haloes