  Compact groups (CGs) of galaxies appear to be the densest galaxy systems
containing a few luminous galaxies in close proximity to each other, which have
a typical size of a few tens kilopacsec in observation. On the other hand, in
the modern hierarchical structure formation paradigm, galaxies are assembled
and grouped in dark matter haloes, which have a typical size of a few hundreds
of kiloparsec. Few studies have explored the physical connection between the
observation based CGs and halo model based galaxy groups to date. In this
study, by matching the largest local CG catalog of Zheng & Shen (2020) to the
halo based group catalog of Yang et al. (2007), we find that the CGs are
physically heterogenous systems and can be mainly separated into two
categories, the isolated systems and those embedded in rich groups or clusters.
By examining the dynamical features of CGs, we find that the isolated CGs have
systematically lower dynamical masses than that of non-compact ones at the same
group luminosity, indicating a more evolved stage of isolated CGs. On the other
hand, the embedded CGs are mixtures of chance alignments in poor clusters and
recent infalling groups (sub-structures) of rich clusters.
