For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb{Q}[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arithmetic invariants of the subfields of an algebraic number field with Galois group $G$. On the algorithm side this leads to subfield based algorithms for computing rings of integers, $S$-unit groups and class groups. For the $S$-unit group computation this yields a polynomial-time reduction to the corresponding problem in subfields. We compute class groups of large number fields under GRH, and new unconditional values of class numbers of cyclotomic fields.