Similar to other unfitted finite element methods, the finite cell method is motivated by the desire to circumvent a costly step in classical finite element methods, namely the generation of a boundary-conforming mesh through the embedding of the physical domain in a regular background mesh. The overlap between the physical domain and the background computational domain, in turn, leads to the numerical ill-conditioning of the resultant system of equations, hampering the usability of iterative solvers out of the box. Therefore, the development of specialized solvers is a crucial aspect of obtaining a scalable solution strategy for the finite cell method. In this work, we present adaptive geometric multigrid (GMG) solvers with tailored smoothers as an efficient and scalable choice for the finite cell method. In addition, an end-to-end efficient parallelization is only possible given that the necessary infrastructure such as an appropriate distributed discretization is available to support the simulation pipeline. We discuss the computational complexity and parallel scalability of the solvers as well as other essential algorithms for the solution of large-scale finite cell problems on distributed-memory clusters. Furthermore, several cache policies for the solvers are presented that offer a balance between on-the-fly and cached computation, and the memory footprint and computational efficiency of each one is discussed. Results indicate that the presented solvers are effective for the solution of finite cell systems and can achieve excellent strong and weak scalability in parallel for problems with more than a billion degrees of freedom.