The radicals of semigroups were described in accordance with the semigroup theory.The concepts of strong(band) C-radicals of semigroups and the semigroups with strong band C-radicals were defined.It was indicated that the semilattice of left(right) bands of groups and left C-semigroups are contained in the class of semigroups.The properties of the strong band C-radicals and the semigroups with strong band C-radicals were studied.The structure theorem of this class of semigroups was proved.The results in the article generalize the results of semilattice of left(right) bands of groups and left C-semigroups.So,the theory of radicals of semigroup is an effective ways to study the structure of semigroups.