  In his work on representations of Thompson's group $F$, Vaughan Jones defined
and studied the $3$-\emph{colorable subgroup} $\mathcal{F}$ of $F$. Later, Ren
showed that it is isomorphic with the Brown-Thompson group $F_4$. In this paper
we continue with the study of the $3$-colorable subgroup and prove that the
quasi-regular representation of $F$ associated with the $3$-colorable subgroup
is irreducible. We show moreover that the preimage of $\mathcal{F}$ under a
certain injective endomorphism of $F$ is contained in three (explicit) maximal
subgroups of $F$ of infinite index. These subgroups are different from the
previously known infinite index maximal subgroups of $F$, namely the parabolic
subgroups that fix a point in $(0,1)$, (up to isomorphism) the Jones' oriented
subgroup $\vec{F}$, and the explicit examples found by Golan.
