  By adapting the technique of David, Koukoulopoulos and Smith for computing
sums of Euler products, and using their interpretation of results of Schoof \`a
la Gekeler, we determine the average number of subgroups (or cyclic subgroups)
of an elliptic curve over a fixed finite field of prime size. This is in line
with previous works computing the average number of (cyclic) subgroups of
finite abelian groups of rank at most 2. A required input is a good estimate
for the divisor function in both short interval and arithmetic progressions,
that we obtain by combining ideas of Ivi\'c--Zhai and Blomer.
