We use a semi-analytic galaxy catalogue constructed from the Millennium Simulation (MS) to study the satellites of isolated galaxies in the Λ cold dark matter (ΛCDM) cosmogony. The large volume surveyed by the MS (5003 h−3 Mpc3), together with its unprecedented numerical resolution, enable the compilation of a large sample of ∼80 000 bright (Mr < −20.5) primaries, surrounded by ∼178 000 satellites down to the faint magnitude limit (Mr=−17) of our catalogue. This sample allows the characterization, with minimal statistical uncertainty, of the dynamical properties of satellite/primary galaxy systems in a ΛCDM universe. The details of this characterization are sensitive to the details of the modelling, such as its assumptions on galaxy merging and dynamical friction time-scales, but many of its general predictions should be applicable to hierarchical formation models such as ΛCDM. We find that, overall, the satellite population traces the dark matter rather well: its spatial distribution and kinematics may be approximated by a Navarro, Frenk & White 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. Our analysis also highlights a number of difficulties afflicting studies that rely on satellite velocities to constrain the primary halo mass. These arise from variations in the star formation efficiency and assembly history of isolated galaxies, which result in a scatter of up to approximately two decades in halo mass at a fixed primary luminosity. Our isolation criterion (primaries may only have companions at least 2 mag fainter within 1 h−1 Mpc) contributes somewhat to the scatter, since it 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. Much tighter halo mass–luminosity relations are found when splitting the sample by colour: red primaries inhabit haloes more than twice as massive as those surrounding blue primaries, a difference that vanishes, however, when considering stellar mass instead of luminosity. The large scatter in the halo mass–luminosity relation hinders the interpretation of the velocity dispersion of satellites stacked according to the luminosity of the primary. We find L∝σ3 (the natural scaling expected for ΛCDM) for truly isolated primaries, that is, systems where the central galaxy contributes more than 85 per cent of the total luminosity within its virial radius. Less-strict primary selection, however, leads to substantial modification of the scaling relation: blindly stacking satellites of all primaries results in a much shallower L–σ relation that is only poorly approximated by a power law.