Finite Group DatabasebetaFinite Group Databasebeta← ΓL(1,𝔽8)⋊AGL(1,𝔽7) #11725090
Order 882.
56 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | ΓL(1,𝔽8)2 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | #64425 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #64424 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | ΓL(1,𝔽8)⋊D7 | C3 | 1 | 2 | yes | maximal, complemented |
| 5 | #3862 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | #3861 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C7×ΓL(1,𝔽8) | C2×C3 | 1 | 2 | yes | complemented |
| 8 | C3×AGL(1,𝔽7) | — | 7 | 2 | no | maximal |
| 9 | C7⋊D7 | C32 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | C3×ΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 11 | C72 | C2×C32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C3×D7 | — | 7 | 2 | no | — |
| 13 | AGL(1,𝔽7) | — | 7 | 2 | no | — |
| 14 | AGL(1,𝔽7) | — | 7 | 2 | no | — |
| 15 | AGL(1,𝔽7) | — | 7 | 2 | no | — |
| 16 | AGL(1,𝔽7) | — | 21 | 2 | no | — |
| 17 | C3×C7 | — | 7 | 2 | no | — |
| 18 | ΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 19 | ΓL(1,𝔽8) | — | 7 | 2 | no | — |
| 20 | ΓL(1,𝔽8) | — | 21 | 2 | no | — |
| 21 | ΓL(1,𝔽8) | AGL(1,𝔽7) | 1 | 2 | yes | complemented |
| 22 | C2×C32 | — | 49 | 1 | no | Hall |
| 23 | D7 | — | 7 | 2 | no | — |
| 24 | D7 | — | 21 | 2 | no | — |
| 25 | C32 | — | 49 | 1 | no | Sylow |
| 26 | C7 | — | 3 | 2 | no | — |
| 27 | C7 | C3×AGL(1,𝔽7) | 1 | 2 | yes | complemented |
| 28 | C2×C3 | — | 49 | 2 | no | — |
| 29 | C2×C3 | — | 49 | 1 | no | — |
| 30 | C2×C3 | — | 49 | 1 | no | — |
| 31 | C3 | — | 7 | 2 | no | — |
| 32 | C3 | — | 49 | 1 | no | — |
| 33 | C3 | — | 49 | 1 | no | — |
| 34 | C2 | — | 49 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72 | C2×C32 |
| LCS | G ▷ C72 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)2 | C2 | semidirect |
| 2. #64425 | C3 | semidirect |
| 3. #64424 | C3 | semidirect |
| 4. ΓL(1,𝔽8)⋊D7 | C3 | semidirect |
| 5. #3862 | C2×C3 | semidirect |
| 6. #3861 | C2×C3 | semidirect |
| 7. C7×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 9. C7⋊D7 | C32 | semidirect |
| 11. C72 | C2×C32 | semidirect |
| 21. ΓL(1,𝔽8) | AGL(1,𝔽7) | semidirect |
| 27. C7 | C3×AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)2 | 2 |
| #64425 | 3 |
| #64424 | 3 |
| ΓL(1,𝔽8)⋊D7 | 3 |
| C3×AGL(1,𝔽7) | 7 |