We consider factorizations of a finite group G into conjugate subgroups,
G=Ax1⋯Axk for A≤G and x1,…,xk∈G,
where A is nilpotent or solvable. First we exploit the split BN-pair
structure of finite simple groups of Lie type to give a unified self-contained
proof that every such group is a product of four or three unipotent Sylow
subgroups. Then we derive an upper bound on the minimal length of a solvable
conjugate factorization of a general finite group. Finally, using conjugate
factorizations of a general finite solvable group by any of its Carter
subgroups, we obtain an upper bound on the minimal length of a nilpotent
conjugate factorization of a general finite group