By applying the Cayley--Dickson process to the division algebra of real octonions, one obtains a 16-dimensional real algebra known as (real) sedenions. We denote this algebra by $\bA_4$. It is a flexible quadratic algebra (with unit element 1) but not a division algebra. We classify the subalgebras of $\bA_4$ up to conjugacy (\emph{i.e.,} up to the action of the automorphism group $G$ of $\bA_4$) with one exception: we leave aside the more complicated case of classifying the quaternion subalgebras. Any nonzero subalgebra contains 1 and we show that there are no proper subalgebras of dimension 5, 7 or $>8$. The proper non-division subalgebras have dimensions 3, 6 and 8. We show that in each of these dimensions there is exactly one conjugacy class of such subalgebras. There are infinitely many conjugacy classes of subalgebras in dimensions 2 and 4, but only 4 conjugacy classes in dimension 8.