Are fossil groups a challenge of the Cold Dark Matter paradigm?

We study six groups and clusters of galaxies suggested in the literature to be `fossil' systems (i.e. to have luminous diffuse X-ray emission and a magnitude gap of at least 2 mag-R between the first and the second ranked member within half of the virial radius), each having good quality X-ray data and SDSS spectroscopic or photometric coverage out to the virial radius. The poor cluster AWM4 is clearly established as a fossil system, and we confirm the fossil nature of four other systems (RXJ1331.5+1108, RXJ1340.6+4018, RXJ1256.0+2556 and RXJ1416.4+2315), while the cluster RXJ1552.2+2013 is disqualified as fossil system. For all systems we present the luminosity functions within 0.5 and 1 virial radius that are consistent, within the uncertainties, with the universal luminosity function of clusters. For the five bona fide fossil systems, having a mass range 2x10^13-3x10^14 M_Sun, we compute accurate cumulative substructure distribution functions (CSDFs) and compare them with the CSDFs of observed and simulated groups/clusters available in the literature. We demonstrate that the CSDFs of fossil systems are consistent with those of normal observed clusters and do not lack any substructure with respect to simulated galaxy systems in the cosmological LambdaCDM framework. In particular, this holds for the archetype fossil group RXJ1340.6+4018 as well, contrary to earlier claims.