Irreducible subgroups of $\text{GL}(2,p)$ I just wonder any sources that provides the list of all minimal irreducible subgroups of $\text{GL}(2,p)$. Here the term "minimal" means there is no proper subgroup that is irreducible. A list of maximal subgroups of $\text{PSL}(2,p)$ is available here. For some case, it is easy to know the subgroup is reducible, but generally I am not sure how to dig information from here. I think it would be good if there is some reference which provides full list of all irreducible subgroups.
I think first we need to know all the maximal subgroups of $\text{GL}(2,q)$. Since $\text{GL}(2,q)'=\text{SL}(2,q)$, so a maximal subgroup $M$ of $\text{GL}(2,q)$ contains $Z$ or $\text{SL}(2,q)$. If $M>Z$, then we look at the subgroups of $\text{PGL}(2,q)$; if $M>\text{SL}(2,q)$, then we look at the subgroups of $\text{PSL}(2,q)$. Then the next thing is that if you know a group is irreducible, how do you analyse the irreducibility of its subgroups? Since sometimes it seems not clear to me that how the subgroup acts on the vectors space. And I am also not too sure how to find the critical subgroups, that is the minimal one. Do we have any trick for treating this?
 A: The subgroups of ${\rm PSL}(2,q)$ for prime powers $q$ were classified by Dickson in 1901. You can find the lists and proofs in Huppert's German book "Endliche Gruppen I". Of course you need the subgroups of ${\rm PGL}(2,p)$, which is twice as big, and then you need to check their inverse images in ${\rm GL}(2,p)$ for irreducibility. So you have some work to do!
I will have a go at listing the minimal irreducible groups, but it is a little complicated, and I may not get it absolutely right.


*

*For odd primes $r$ dividing $p+1$, the cyclic group of order $r$.

*The cyclic group of order $2^r$, where $2^{r-1}$ is the highest power of 2 dividing $p-1$.

*The dihedral group of order $2r$ for odd primes $r$ dividing $p-1$.
I believe that is all when $p \equiv 3 \pmod 4$. (That case is simpler, because all elements of order 4 act irreducibly.) It is more complicated when
$p \equiv 1 \pmod 4$. In that case, you also get:


*

*The dihedral group $D_8$.

*The quaternion group $Q_8$.

*For odd primes $r$ dividing $p-1$, a group of order $4r$ contained in ${\rm SL}(2,p)$.

*This is the complicated one. After doing some computations with $p=17$, I think, for all powers $2^r$ dividing $p-1$ with $r \ge 3$, there is a nonabelian group of order $2^{r+1}$ intersecting the scalar subgroup with order $2^{r-1}$.
Here is a bit more information on methods used. The maximal subgroups of ${\rm GL}(2,p)$ (with $p$ prime) are the imprimitive groups of order  $2(p-1)^2$, the semilinear groups of order $2(p^2-1)$, groups whose intersection with ${\rm SL}(2,p)$ is a double cover of $A_4$, $S_4$ or $A_5$, and groups containing ${\rm SL}(2,p)$. Since ${\rm SL}(2,p)$ itself is never minimal irreducible, we can forget about that, and the ones involving $A_4$, $S_4$ or $A_5$ all have a normal irredcuible subgroup $Q_8$, which is itself imprimitive and/or semilinear, so we can effectively forget about those.
So we can restrict attention to imprimitive and semilinear subgroups. The maximal semilinears have a cyclic subgroup of order $q^2-1$ which acts irreducibly but not absolutely irreducibly, and it is not hard to see that their minimal irreducible subgroups are those in classes 1 and 2 above.
So it remains to find the minimal irreducible subgroups of the maximal imprimitive subgroups. These are all conjugate, and one of them, $H$ say, consists of the monomial matrices, so we can just look for subgroups of $H$. So the group $D$ of diagonal matrices has index 2 in $H$ and is reducible. So a minimal irreducible $K$ will have a subgroup of $D$ of index 2. If $|K \cap D|$ is odd, then get the groups in class 3 above. As I noted before, if $p \equiv 3 \pmod 4$, then elements of order 4 act irreducible, and any other such subgroup would have an element of order 4, so would not be minimal irreducible. But it is more complicated when $p \equiv 1 \pmod 4$.
A: A belated addition: representatives of conjugacy classes of irreducible subgroups of GL(2, GF(q)) for finite fields of characteristic p at least 5 are described explicitly in the following paper: 
Dane Flannery and E.A. O'Brien: Linear groups of small degree over finite fields. Internat. J. Algebra Comput. 15, 467-502, 2005.
The pdf file is available at www.math.auckland.ac.nz/~obrien
The list is also available in Magma via the command

IrreducibleSubgroups (n, q)

The paper also lists representatives for soluble irreducible subgroups of GL(3, q); those for insoluble subgroups of degree 3 were recently added by Barry Hurley. 
Eamonn O'Brien
