We introduce Copernicus Complexio (COCO), a high-resolution cosmological N-body simulation of structure formation in the $Lambda{rm CDM}{}$ model. COCO follows an approximately spherical region of radius $sim 17.4h^{-1},{rm Mpc}$ embedded in a much larger periodic cube that is followed at lower resolution. The high resolution volume has a particle mass of $1.135times10^5h^{-1}{rm M}_{odot}$ (60 times higher than the Millennium-II simulation). COCO gives the dark matter halo mass function over eight orders of magnitude in halo mass; it forms $sim 60$ haloes of galactic size, each resolved with about 10 million particles. We confirm the power-law character of the subhalo mass function, $bar{N}(>mu)proptomu^{-s}$, down to a reduced subhalo mass $M_{sub}/M_{200}equivmu=10^{-6}$, with a best-fit power-law index, $s=0.94$, for hosts of mass $langle M_{200}rangle=10^{12}h^{-1}{rm M}_{odot}$. The concentration-mass relation of COCO haloes deviates from a single power law for masses $M_{200}<textrm{a few}times 10^{8}h^{-1}{rm M}_{odot}$, where it flattens, in agreement with results by Sanchez-Conde et al. The host mass invariance of the reduced maximum circular velocity function of subhaloes, $ uequiv V_{max}/V_{200}$, hinted at in previous simulations, is clearly demonstrated over five orders of magnitude in host mass. Similarly, we find that the average, normalised radial distribution of subhaloes is approximately universal (i.e. independent of subhalo mass), as previously suggested by the Aquarius simulations of individual haloes. Finally, we find that at fixed physical subhalo size, subhaloes in lower mass hosts typically have lower central densities than those in higher mass hosts.
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