In this article, properties of the biprimary subgroups of even order are used to characterize the following series of finite simple groups: the projective special linear groups $PSL(2,q)$ , ( $q>3$ ), the projective special unitary groups $PSU(3,2^{2n})$ ( $n>1$ ), the Suzuki groups $\operatorname{Sz}(q)$ , the Janko group $J_1$ of order 175560, and the Ree-type groups.