We obtain a series of estimates on the number of small integers and small order Farey fractions which belong to a given coset of a subgroup of order t of the group of units of the residue ring modulo a prime p, in the case when t is small compared to p. We give two applications of these results: to the simultaneous distribution of two high degree monomials x k1 and x k2 modulo p and to a question of J. Holden and P. Moree on fixed points of the discrete logarithm.