  We use the Halo Model to explore the implications of assuming that galaxy
luminosities in groups are randomly drawn from an underlying luminosity
function. We show that even the simplest of such order statistics models -- one
in which this luminosity function $p(L)$ is universal -- naturally produces a
number of features associated with previous analyses based on the `central plus
Poisson satellites' hypothesis. These include the monotonic relation of mean
central luminosity with halo mass, the Lognormal distribution around this mean,
and the tight relation between the central and satellite mass scales. In stark
contrast to observations of galaxy clustering, however, this model predicts
$\textit{no}$ luminosity dependence of large scale clustering. We then show
that an extended version of this model, based on the order statistics of a
$\textit{halo mass dependent}$ luminosity function $p(L|m)$, is in much better
agreement with the clustering data as well as satellite luminosities, but
systematically under-predicts central luminosities. This brings into focus the
idea that central galaxies constitute a distinct population that is affected by
different physical processes than are the satellites. We model this physical
difference as a statistical brightening of the central luminosities, over and
above the order statistics prediction. The magnitude gap between the brightest
and second brightest group galaxy is predicted as a by-product, and is also in
good agreement with observations. We propose that this order statistics
framework provides a useful language in which to compare the Halo Model for
galaxies with more physically motivated galaxy formation models
