stabilizer
If the orbit of a point asks “where can this point go?”, the stabilizer asks the opposite: “which moves leave this point exactly where it is?” Among all the symmetries in the group, the stabilizer collects precisely those that pin a chosen point in place.
Formally, for an action of G on X and a point x, the stabilizer is Stab(x) = { g in G : g·x = x }. It is always a subgroup of G — the identity fixes x, and if g and h fix x then so do gh and g^{-1}. The stabilizer measures how much symmetry the point itself carries; the more group elements fix it, the larger its stabilizer and, by the orbit-stabilizer theorem, the smaller its orbit.
Stabilizers of different points in the same orbit are conjugate: if y = g·x then Stab(y) = g Stab(x) g^{-1}. So an orbit determines a single conjugacy class of subgroups, not a single subgroup, and transitive actions of G correspond (up to equivalence) to subgroups H of G via the coset action on G/H, where the stabilizer of the base coset is exactly H. This dictionary between transitive actions and subgroups is one of the most useful organizing principles in group theory.
Let S_4 act on {1, 2, 3, 4}. The stabilizer of the point 4 consists of all permutations fixing 4, namely the permutations of {1, 2, 3}; this is a copy of S_3 inside S_4, of order 6. The orbit of 4 has size 4, and indeed 24 = |S_4| = 4 · 6.
A point stabilizer inside S_4 is a copy of S_3.