We present a general framework for studying the multilevel structure of lattice network coding (LNC), which serves as the theoretical fundamental for solving the ring-based LNC problem in practice, with greatly reduced decoding complexity. Building on the framework developed, we propose a novel lattice-based network coding solution, termed layered integer forcing (LIF), which applies to any lattices having multilevel structure. The theoretic foundations of the developed multilevel framework lead to a new general lattice construction approach, the elementary divisor construction (EDC), which shows its strength in improving the overall rate over multiple access channels (MAC) with low computational cost. We prove that the EDC lattices subsume the traditional complex construction approaches. Then a soft detector is developed for lattice network relaying, based on the multilevel structure of EDC. This makes it possible to employ iterative decoding in lattice network coding, and simulation results show the large potential of using iterative multistage decoding to approach the capacity.