Bounds on the number of autotopisms and subsquares of a Latin square

A subsquare of a Latin square L is a submatrix that is also a Latin square. An autotopism of L is a triplet of permutations (α,β,γ) such that L is unchanged after the rows are permuted by α, the columns are permuted by β and the symbols are permuted by γ. Let n!(n-1)!R_n be the number of n × n Latin squares. We show that an n × n Latin square has at most n^{O(log k)} subsquares of order k and admits at most n^{O(log n)} autotopisms. This enables us to show that p^{n/(p-1)-O(log² n)} divides R_n for all primes p. We also extend a theorem by McKay and Wanless that gave a factorial divisor of R_n, and give a new proof that R_p = 1 mod p for prime p.