A module M over an associative ring with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating concepts related to these modules out of which summability and h-purity are very significant. The concepts of summable and σ -summable modules were defined by Naji but there were some questions which remained unanswered. This motivated us to define weakly n-summable and n- layered QTAG-modules with the help of their nice modules. We also study α -modules and ℵ1 -U -modules with some of their interesting properties.