Analysing/studying simple groups with Sylow-$2$ subgroups

Question

In the classification of finite simple groups, some classification is done by considering structure of Sylow-$2$ subgroups (for example, here; it is more than 250 page paper!)

Now in the world of finite groups, $2$-groups appear very large in number and their classification is difficult (incomplete!).

Then, looking for classifying simple groups with knowing structure of its Sylow-$2$ subgroup could even be difficult, which I feel after thinking this way. I don't have idea what could be motivation for classification in this way.

Since there is no pattern (in my opinion) in the classification of finite $2$-groups, is there any pattern of Sylow-$2$ subgroups in the family of finite simple groups?

In other words, let's say we have all simple groups in a box; can we partition these groups according to Sylow-$2$ subgroups in nice way?

(Beside above question, I would be happy for posting known results on following: what families of simple groups have been characterized from their Sylow-$2$ subgroups?)

Answer 1

I think classifications of nonabelian simple groups according to the structure of the Sylow $2$-subgroups were a feature of the earlier parts of the full classification, and the generally concerned the classical groups of small Lie rank. So, for example, ${\rm PSL}(2,q)$ with $q$ odd together with $A_7$ are the simple groups with dihedral Sylow $2$-subgroups, and ${\rm PSL}(2,q)$ with $q$ even or $q \equiv \pm3 \bmod 8$, the groups of Ree type $^2G_2(3^{2n+1})$ and with the Janko group $J_1$ are the simple groups with abelian Sylow $2$-subgroups. There are also characterizations of ${\rm PSL}(3,q)$ and ${\rm PSU}(3,q)$, and possibly some other low rank examples, according to the structure of their Sylow $2$-subgroups. (IIRC, the Janko group $J_1$, which was the first sporadic group to be found since the Mathieu groups in the 19th century was found as a result of this classification.)

The point is that these examples came about not because somebody thought let's look at some possible $2$-groups and classify simple groups having them as Sylow $2$-subgroups, but because they observed that the simple groups of low rank had very specific easily described Sylow $2$-subgroups which,. apart from a couple of exceptions, were not shared by other types of simple groups, and so it was natural to ask whether they could be classified in that way.

The later parts of the classification generally used the structure of the centralizer of some involution rather than the Sylow $2$-subgroup as the classification criterion, becuase this was more convenient when dealing with groups of arbitrarily high Lie rank, and it was more natural for inductive proofs, particularly in odd characteristic. (An involution centralizer in ${\rm GL}(n,q)$ for odd $q$ has the structure ${\rm GL}(k,q) \times {\rm GL}(n-k,q)$.)