Let G be a finite group and \(\textrm{Irr}(G)\) be the set of irreducible characters of G. The codegree of an irreducible character \(\chi \) of the group G is defined as \(\textrm{cod}(\chi )=|G:\textrm{ker}(\chi )|/\chi (1)\) . Let \(b^c(G)\) be the largest codegree of G. In this paper, we study how the structure of a group G is bounded by its largest codegree \(b^c(G)\) . Firstly, we give a criterion for solvability: If \(b^c(G)<20\) , then G is solvable. Secondly, we consider the nonsolvable groups G with a small \(b^c(G)\) and prove that, if \(b^c(G)<56\) , then G is isomorphic to \(A_5\) , \(S_5\) or \(A_5\times A\) where A is an elementary abelian 2-group.
