Let $varepsilon_#(X)$ be the subgroups of $varepsilon(X)$ consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of X and $varepsilon_*(X) $ be the subgroup of $varepsilon(X)$ that fix homology groups for all dimension. In this paper, we establish some connections between the homotopy group of X and the subgroup $varepsilon_#(X)capvarepsilon_*(X);of;varepsilon(X)$. We also give some relations between $pi_n(W)$, as well as a generalized Gottlieb group $G_n^f(W,X)$, and a subset $M_{#N}^f(X,W)$ of [X, W]. Finally we establish a connection between the coGottlieb group of X and the subgroup of $varepsilon(X)$ consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups.