The multigroup neutron transport equations have been widely used to study the motion of neutrons and their interactions with the background materials. Numerical simulation of the multigroup neutron transport equations is computationally challenging because the equations is defined on a high dimensional phase space (1D in energy, 2D in angle, and 3D in spatial space), and furthermore, for realistic applications, the computational spatial domain is complex and the materials are heterogeneous. The multilevel domain decomposition methods is one of the most popular algorithms for solving the multigroup neutron transport equations, but the construction of coarse spaces is expensive and often not strongly scalable when the number of processor cores is large. In this paper, we study a highly parallel multilevel Newton-Krylov-Schwarz method equipped with several novel components, such as subspace-based coarsening, partition-based balancing and hierarchical mesh partitioning, that enable the overall simulation strongly scalable in terms of the compute time. Compared with the traditional coarsening method, the subspace-based coarsening algorithm significantly reduces the cost of the preconditioner setup that is often unscalable. In addition, the partition-based balancing strategy enhances the parallel efficiency of the overall solver by assigning a nearly-equal amount of work to each processor core. The hierarchical mesh partitioning is able to generate a large number of subdomains and meanwhile minimizes the off-node communication. We numerically show that the proposed algorithm is scalable with more than 10,000 processor cores for a realistic application with a few billions unknowns on 3D unstructured meshes.