A connection between $L$-valued ($L$ is a lattice) congruences [4] and $L$-valued normal subgroups of a group is given. It is proved that the corresponding lattices of all such mappings are isomorphic. It is also proved that $L$ is (up to the isomorphism) a sublattice of the lattice of ail $L$-valued normal subgroups of a group, and that the latter is modular (as well as the lattice of all $L$-valued congruences) if $L$ is infinitely distributive.