Abstract: An algebraic subgroup $H$ of a reductive algebraic group $G$ is spherical if there exists a Borel subgroup $B$ of $G$ such that $BH$ is dense in $G$. Spherical subgroups play an important role in algebraic geometry and representation theory since they are broad enough to include many interesting subgroups (parabolic subgroups, symmetric subgroups, maximal unipotent subgroups) yet restrictive enough to possess a combinatorial structure theory. A spherical subgroup $H$ is strongly solvable if it is contained in a Borel subgroup $G$. Building on work of Avdeev, who gave a new classification of strongly solvable spherical subgroups, and Gandini and Pezzini, who parameterized the $H$-orbits on the flag variety $G/B$, we describe the maximal chains in certain intervals of the weak order for $H$. The primary application is to give the cohomology class of the closure of an $H$-orbit in the Schubert basis of the flag variety.