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

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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:

## Family contexts

Family name | Parameter values | General discussion of subgroup structure of family |
---|---|---|

projective special linear group of degree two over a finite field of size | , i.e., field:F11, so the group is | 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 -Hall subgroups (of order 12) and -Hall subgroups (of order 60), and -Hall subgroups (of order 55) Note that there are two distinct isomorphism classes of -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) |