  In this paper we consider the models of the accelerated expanding large scale
universe~(according to general relativity) containing a generalized holographic
dark energy with a Nojiri - Odintsov cut - off. The second component of the
darkness is assumed to be the pressureless cold dark matter according to
observed symmetries of the large scale universe. Moreover, we assume specific
forms of the interaction between these two components and besides the
cosmographic analysis, we discuss appropriate results from $Om$ and $Om3$
analysis and organize a closer look to the models via the statefinder hierarchy
analysis, too. In this way we study mainly impact of the interaction on the
dynamics of the background of our universe~(within specific forms of
interaction). To complete the cosmographic analysis, the present day values of
the statefinder parameters $(r,s)$ and $(\omega^{\prime}_{de}, \omega_{de})$
has been estimated for all cases and the validity of the generalized second law
of thermodynamics is demonstrated. Our study showed that theoretical results
from considered phenomenological models are consistent with the available
observational data and symmetries.
