What are the orders of maximal abelian subgroups of the simple groups $F\_4(q)$ and $C\_4(q)$, where $F\_4(q)$ is an exceptional group and $C\_4(q)$ is a symplectic group?



orders of maximal abelian subgroups
For a finite group $G$, denote the maximum of the orders of its Abelian subgroups by $a(G)$. Then we have, for $G=F\_4(q)$ and $q$ even,
$$
q^{11}\le a(G)\le q^{17},
$$
and for $G=F\_4(q)$, $q$ odd,
$$
q^{9}\le a(G)\le q^{14}.
$$
For $G=C\_n(q)$ we have, for $n\neq 2$, 
$$
a(G)=q^{\frac{n(n+1)}{2}}.
$$
For a reference, see the paper *Maximal Orders of Abelian Subgroups
in Finite Chevalley Groups* by E. P. Vdovin, published in $2001$; in particular table $2$.


