In this paper, we present the idea of interval valued fuzzy subgroup defined over a certain t-conorm (Γ-IVFSG) and prove that every IVFSG is Γ-IVFSG. We use this ideology to define the concepts of Γ-IVF cosets, Γ-IVFNSG and formulate their various important algebraic characteristics. We also propose the study of the notion of level subgroups of Γ-IVFSG and investigate the condition under which a Γ-IVFS is Γ-IVFSG. Moreover, we extend the study of this phenomenon to introduce the concept of quotient group of a group Ζ relative to the Γ-IVFNSG and acquire a correspondence between each Γ-IVF(N)SG of a group Ζ and Γ-IVF(N)SG of its quotient group. Furthermore, we define the index of Γ-IVFSG and establish the Γ-interval valued fuzzification of Lagrange’s theorem of any Γ-IVFSG of a finite group Ζ.