  (ABRIDGED) We present a semi-analytic model to explore merger histories,
destruction rates, and survival probabilities of substructure in dark matter
halos and use it to study the substructure populations of galaxy-sized halos as
a function of the power spectrum. We successfully reproduce the subhalo
velocity function and radial distribution seen in N-body simulations for
standard LCDM. We explore the implications of spectra with normalizations and
tilts spanning sigma_8 = 0.65-1 and n = 0.8-1. We also study a running index
(RI) model with dn/dlnk=-0.03, as discussed in the first year WMAP report, and
several WDM models with masses m_W = 0.75, 1.5, 3.0 keV. The substructure mass
fraction is relatively insensitive to the tilt and overall normalization of the
power spectrum. All CDM-type models yield projected substructure mass fractions
that are consistent with, but on the low side of, estimates from strong lens
systems: f = 0.4-1.5% (64 percentile) in systems M_sub < 10^9 Msun. Truncated
models produce significantly smaller fractions and are disfavored by lensing
results. We compare our predicted subhalo velocity functions to the dwarf
satellite population of the Milky Way. Assuming isotropic velocity dispersions,
we find the standard n=1 model overpredicts the number of MW satellites as
expected. Models with less small-scale power are more successful because there
are fewer subhalos of a given circular velocity and the mapping between
observed velocity dispersion and halo circular velocity is markedly altered.
The RI model, or a fixed tilt with sigma_8=0.75, can account for the MW dwarfs
without the need for differential feedback; however, these comparisons depend
sensitively on the assumption of isotropic velocities in satellite galaxies.
