  Let $F$ be a non-archimedean local field of residual characteristic $p>3$ and
residue degree $f>1$. We study a certain type of diagram, called \emph{cyclic
diagrams}, and use them to show that the universal supersingular modules of
$\mathrm{GL}_{2}(F)$ admit infinitely many non-isomorphic irreducible
admissible quotients.
