Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.
As mentioned at http://buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be non-Euclidian, e.g. the pendulum lives on an infinite cylinder.