# Subgroup structure of projective special linear group:PSL(2,11)

## Contents

This article gives specific information, namely, subgroup structure, about a particular group, namely: projective special linear group:PSL(2,11).
View subgroup structure of particular groups | View other specific information about projective special linear group:PSL(2,11)

This article discusses the subgroup structure of projective special linear group:PSL(2,11), which is the projective special linear group of degree two over field:F11. The group has order 660, with prime factorization: $660 = 2^2 \cdot 3^1 \cdot 5^1 \cdot 11^1 = 4 \cdot 3 \cdot 5 \cdot 11$

## Family contexts

Family name Parameter values General discussion of subgroup structure of family
projective special linear group of degree two $PSL(2,q)$ over a finite field of size $q$ $q = 11$, i.e., field:F11, so the group is $PSL(2,11)$ subgroup structure of projective special linear group of degree two over a finite field

## Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate

### Quick summary

Item Value
Number of subgroups 620
Number of conjugacy classes of subgroups 16
Number of automorphism classes of subgroups 14
Isomorphism classes of Sylow subgroups and the corresponding fusion systems 2-Sylow: Klein four-group, fusion system is simple fusion system for Klein four-group, Sylow number is 55
3-Sylow: cyclic group:Z3, fusion system is the non-inner one, Sylow number is 55
5-Sylow: cyclic group:Z5, Sylow number is 66
11-Sylow: cyclic group:Z11, Sylow number is 12
Hall subgroups Other than the whole group, the trivial subgroup, and the Sylow subgroups, there are $\{ 2,3 \}$-Hall subgroups (of order 12) and $\{ 2,3,5 \}$-Hall subgroups (of order 60), and $\{ 5,11 \}$-Hall subgroups (of order 55)
Note that there are two distinct isomorphism classes of $\{ 2,3 \}$-Hall subgroups: dihedral group:D12 and alternating group:A4. This gives the smallest example illustrating that Hall not implies order-isomorphic
maximal subgroups maximal subgroups have orders 12, 55, 60.
normal subgroups only the whole group and the trivial subgroup, because the group is simple. See projective special linear group is simple (with a couple of small exceptions, but this isn't one of them)
subgroups that are simple non-abelian groups (apart from the whole group itself) alternating group:A5 (order 60)