In Isaac's Finite Group Theory Page 50, it states：
A $Sylow$ $2$-subgroup $P$ of $G=PSL(2,7)$ of order 168 is contained in two maximal subgroup of $G$, each of order $24$, and $Z(P)$, which has order $2$, is subnormal in each of these maximal subgroups.
I'm wondering is there a way to check that $P$ is contained in two different subgroup of $G$, in which $Z(P)$ is subnormal, without writing out each element $PSL(2,7)$ as matrices. It took me several hours to do such matrix computations to check the two properties.
In fact, as projective linear groups might be used to construct lots of counterexamples in group theory, I expect for some more efficient ways of checking properties of projective linear groups.
Hope for an answer!