In this paper, we determinate the structure of groups of order 4pq, p2q2 ( p < q are oddprimes )by utilizing the characters of Fitting height of a group and the theory of group extension .

We say that the conjugacy class graphΓ~* has bounded Fitting height if there is a bound on the Fitting heights of the solvable groups whose conjugacy class graph isΓ~*.

By a definition of Lewis in [3], a degree graphΔhas bounded Fitting height if there is a bound on the Fitting height for the solvable group G withΔ(G) =Δ.

Some properties about the class of π-close-Sylow-tower groups are obtained as follows by deducing the characters of π-close-Sylow-tower groups: Class of π-closed-Sylow-tower groups is S-closed, Sn-closed, H-closed, D-closed, R0-closed and N0-closed, so the class of π-closed-Sylow-tower groups is a saturated formation and a Fitting class.

The classing property of three classes of groups is discussed in this paper, moreover we prove that they are all Fitting classes, and we furtherly discuss p-nilpotent class of group N p-one of three classes of groups, as a result, we prove that N p is a saturated formation and N p class of groups is a p-split formation.

On the base of paper[1],this paper decide the generation relation of group,by finding the order of automorphism group of Fitting subgroup of group G.

A sufficientcondition forsupersolvable group depending on the propertyπ-quasinormal of its Fitting subgroup isgiven.

By the characters of Fitting subgroup and extending theorems of subgroup ,the author shows the structure of a kind of nonAbelian groups.