  Denote by E(Y) the group of homotopy classes of self-homotopy equivalences of
a finite-dimensional complex Y. We give a selection of results about certain
subgroups of E(Y). We establish a connection between the Gottlieb groups of Y
and the subgroup of E(Y) consisting of homotopy classes of self-homotopy
equivalences that fix homotopy groups through the dimension of Y, denoted by
E_#(Y). We give an upper bound for the solvability class of E_#(Y) in terms of
a cone decomposition of Y. We dualize the latter result to obtain an upper
bound for the solvability class of the subgroup of E(Y) consisting of homotopy
classes of self-homotopy equivalences that fix cohomology groups with various
coefficients. We also show that with integer coefficients, the latter group is
nilpotent.
