For the property $P$ discussed, the usual argument that $P(G_1) \wedge P(G_2) \Rightarrow P(\langle G_1, G_2\rangle)$, where $G_1$ and $G_2$ are groups and $\langle G_1, G_2\rangle$ is the group generated by both, doesn't work. The addition of the parenthetical "in size" indicates that @Klim is aware of this, but is there a different argument that says that there is only one maximal-cardinality group having $P$? The question seems to claim that there is, but I don't see it.

(I'd leave this as a comment if I had the rep, but I don't. Admins, please feel free to convert it into one.)