  We use a semianalytic galaxy catalogue constructed from the Millennium
Simulation to study the satellites of isolated galaxies in the LCDM cosmogony.
This sample (~80,000$ bright primaries, surrounded by ~178,000 satellites)
allows the characterization, with minimal statistical uncertainty, of the
dynamical properties of satellite/primary galaxy systems in a LCDM universe. We
find that, overall, the satellite population traces the dark matter rather
well: its spatial distribution and kinematics may be approximated by an NFW
profile with a mildly anisotropic velocity distribution. Their spatial
distribution is also mildly anisotropic, with a well-defined ``anti-Holmberg''
effect that reflects the misalignment between the major axis and angular
momentum of the host halo. The isolation criteria for our primaries picks not
only galaxies in sparse environments, but also a number of primaries at the
centre of ''fossil'' groups. We find that the abundance and luminosity function
of these unusual systems are in reasonable agreement with the few available
observational constraints. We recover the expected L_{host} \sigma_{sat}^3
relation for LCDM models for truly-isolated primaries. Less strict primary
selection, however, leads to substantial modification of the scaling relation.
Our analysis also highlights a number of difficulties afflicting studies that
rely on blind stacking of satellite systems to constrain the mean halo mass of
the primary galaxies.
