Limited compute, memory, and communication capabilities of edge users create a significant bottleneck for federated learning (FL) of large models. Current literature typically tackles the challenge with a heterogeneous client setting or allows training to be offloaded to the server. However, the former requires a fraction of clients to train near-full models, which may not be achievable at the edge; while the latter can compromise privacy with sharing of intermediate representations or labels. In this work, we consider a realistic, but much less explored, cross-device FL setting in which no client has the capacity to train a full large model nor is willing to share any intermediate representations with the server. To this end, we present Principal Sub-Model (PriSM) training methodology, which leverages models low-rank structure and kernel orthogonality to train sub-models in the orthogonal kernel space. More specifically, by applying singular value decomposition to original kernels in the server model, PriSM first obtains a set of principal orthogonal kernels with importance weighed by their singular values. Thereafter, PriSM utilizes a novel sampling strategy that selects different subsets of the principal kernels independently to create sub-models for clients with reduced computation and communication requirements. Importantly, a kernel with a large singular value is assigned with a high sampling probability. Thus, each sub-model is a low-rank approximation of the full large model, and all clients together achieve nearly full coverage of the principal kernels. To further improve memory efficiency, PriSM exploits low-rank structure in intermediate representations and allows each sub-model to learn only a subset of them while still preserving training performance.