The subgroups He of the R. Thompson group F that are stabilizers of finite sets U of numbers in the interval (0,1) are studied. <...> The algebraic structure of Hi: is described and it is proved that the stabilizer Ни is finitely generated if and only if U consists of rational numbers. <...> It is also shown that such subgroups are isomorphic surprisingly often. <...> In particular, if finite sets U С [0,1] and V С [0,1] consist of rational numbers that are not finite binary fractions, and \U\ = |V'|, then the stabilizers of U and V are isomorphic. <...> In fact these subgroups are conjugate inside a subgroup F < Homeo([0,1]) that is the completion of F with respect to what is called the Hamming metric on F. Moreover the conjugator can be found in a certain subgroup T < P which consists of possibly infinite tree-diagrams with finitely many infinite branches. <...> It is also shown that the group T is non-amenable! <...>