  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
$1.135\times10^5h^{-1}{\rm M}_{\odot}$ (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, $\bar{N}(>\mu)\propto\mu^{-s}$, down to a reduced
subhalo mass $M_{sub}/M_{200}\equiv\mu=10^{-6}$, 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, $\nu\equiv V_{max}/V_{200}$, 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.
