How many conjugacy classes of subgroups does GL(2,p) have?
For instance the dihedral group of order 2*n has τ(n) cyclic normal subgroups and σ(n) "dihedral" subgroups (as in, containing a reflection), but they fall into gcd(2,n)*τ(n)+τ(n/gcd(2,n)) conjugacy classes. Here τ(n) is the number of divisors of n, and σ(n) is their sum.
The formula is relatively compact and can be explicitly evaluated for n in the millions without much work. The description is nice because it even indicates the structure of the subgroups.
The subgroups of GL(2,p) whose order is divisible by p either have a normal Sylow p-subgroup or contain SL(2,p). The former types have conjugacy classes indexed by the subgroups of (p-1) × (p-1), and the latter by subgroups of (p-1). The number of the first type has some reasonable formulas at OEIS: A060724 and the latter is just τ(p-1) again.
Again the description is compact and can be explicitly evaluated for numbers into the millions without any real effort: GL(2,1000003) has 1000008 conjugacy classes of subgroups of order divisible by 1000003 and GL(2,10000019) has 10000024 conjugacy classes of subgroups of order divisible by 10000019, each number computed in under 1ms. Again the description is especially nice because it even indicates the structure of the subgroups.
What about the conjugacy classes of subgroups of GL(2,p) whose order is coprime to p?
Is there a similarly compact and easily evaluated description of their number, and even more nicely, does it also indicate the structure of the subgroups?