The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound

We prove a Lindel"of-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^{\ast }\mathrm{\mid }d$, where $q^{\ast }$ is the least positive integer such that $q^2\mathrm{\mid }(q^{\ast }{)}^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet $L$-functions with no restrictions on the conductor.