We use cosmological LCDM numerical simulations to model the evolution of the substructure population in sixteen dark matter haloes with resolutions of up to seven million particles within the virial radius. The combined substructure circular velocity distribution function (VDF) for hosts of 10^11 to 10^14 Msun at redshifts from zero to two or higher has a self-similar shape, is independent of host halo mass and redshift, and follows the relation: dn/dv=(1/8)(v_cmax/v_cmax,host)^-4. Halo to halo variance in the VDF is a factor of roughly two to four. At high redshifts, we find preliminary evidence for fewer large substructure haloes (subhaloes). Specific angular momenta are significantly lower for subhaloes nearer the host halo centre where tidal stripping is more effective. The radial distribution of subhaloes is marginally consistent with the mass profile for r >~ 0.3r_vir, where the possibility of artificial numerical disruption of subhaloes can be most reliably excluded by our convergence study, although a subhalo distribution that is shallower than the mass profile is favoured. Subhalo masses but not circular velocities decrease toward the host centre. Subhalo velocity dispersions hint at a positive velocity bias at small radii. There is a weak bias toward more circular orbits at lower redshift, especially at small radii. We additionally model a cluster in several power law cosmologies of P ~ k^n, and demonstrate that a steeper spectral index, n, results in significantly less substructure.
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