It is well known that the maximal order of Lie-Backlund symmetries for any nth-order ordinary differential equation is equal to n-1, and that the whole set of such symmetries forms an infinite-dimensional Lie algebra. Symmetries of the order pless than or equal ton - 2 span a linear subspace (but not a subalgebra) in this algebra. We call them symmetries of submaximal order. The purpose of the article is to prove that for n less than or equal to 4 this subspace is finite-dimensional and it's dimension cannot be greater than 35 for n=4, 10 for n=3 and 3 for n=2. In the case n=3 this statement follows immediately from Lie's result on contact symmetries of third-order ordinary differential equations. The maximal values of dimensions are reached, e.g., on the simplest equations y((n))=0.