Strong pseudo Frobenius numbers

By Sylow's theorem, the number of Sylow p-subgroups of a finite group is congruent to $1$ modulo p, but not every such integer occurs. Frobenius extended the congruence to the number of subgroups of order pᵃ for every a≥0. Kulakoff and Hall showed that this number is congruent to $1$ or $1+p$ modulo p² whenever pᵃ⁺¹ divides the group order. A positive integer is a Frobenius p-number if it is the number of subgroups of order pᵃ of some finite group for some a. There are integers congruent to $1$ modulo p² which are not Frobenius p-numbers, and it remained open whether there are also such integers congruent to $1+p$ modulo p². We show that no finite group has exactly $106$ subgroups of order 5ᵃ for any a≥0.