Finite Group DatabasebetaFinite Group DatabasebetaOrder 10584.
68 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×SL(2,𝔽8) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | ΣL(2,𝔽8) | — | 7 | 1 | no | maximal |
| 3 | #11924306 | — | 9 | 1 | no | maximal |
| 4 | SL(2,𝔽8) | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 5 | C7×AGL(1,𝔽8) | — | 9 | 1 | no | Hall |
| 6 | ΓL(1,𝔽64) | — | 28 | 1 | no | maximal |
| 7 | #64423 | — | 36 | 1 | no | maximal |
| 8 | #4428 | — | 28 | 1 | no | — |
| 9 | C2×C22⋊ΓL(1,𝔽8) | — | 63 | 1 | no | — |
| 10 | AΓL(1,𝔽8) | — | 63 | 6 | no | — |
| 11 | AΓL(1,𝔽8) | — | 63 | 1 | no | — |
| 12 | #3861 | — | 36 | 1 | no | — |
| 13 | D3×ΓL(1,𝔽8) | — | 84 | 1 | no | — |
| 14 | C7×D9 | — | 28 | 1 | no | — |
| 15 | C7×D7 | — | 36 | 1 | no | — |
| 16 | C22⋊ΓL(1,𝔽8) | — | 63 | 1 | no | — |
| 17 | C3×ΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 18 | C7×C9 | — | 28 | 1 | no | — |
| 19 | C7⋊C9 | — | 56 | 1 | no | — |
| 20 | C23×C7 | — | 9 | 1 | no | — |
| 21 | AGL(1,𝔽8) | — | 9 | 1 | no | — |
| 22 | AGL(1,𝔽8) | — | 9 | 6 | no | — |
| 23 | AGL(1,ℤ/9) | — | 196 | 1 | no | — |
| 24 | C72 | — | 36 | 1 | no | Sylow |
| 25 | D3×C7 | — | 84 | 1 | no | — |
| 26 | C2×ΓL(1,𝔽8) | — | 252 | 1 | no | — |
| 27 | AGL(1,𝔽7) | — | 252 | 1 | no | — |
| 28 | C22×C7 | — | 63 | 1 | no | — |
| 29 | 3−1+2 | — | 196 | 1 | no | Sylow |
| 30 | C2×A4 | — | 441 | 1 | no | — |
| 31 | C3×C7 | — | 28 | 1 | no | — |
| 32 | ΓL(1,𝔽8) | — | 84 | 1 | no | — |
| 33 | ΓL(1,𝔽8) | — | 504 | 3 | no | — |
| 34 | ΓL(1,𝔽8) | — | 252 | 1 | no | — |
| 35 | C3×D3 | — | 588 | 1 | no | — |
| 36 | D9 | — | 28 | 1 | no | — |
| 37 | C2×C7 | — | 63 | 1 | no | — |
| 38 | D7 | — | 36 | 1 | no | — |
| 39 | A4 | — | 441 | 1 | no | — |
| 40 | C32 | — | 196 | 1 | no | — |
| 41 | C9 | — | 28 | 1 | no | — |
| 42 | C9 | — | 392 | 1 | no | — |
| 43 | C23 | — | 9 | 1 | no | Sylow |
| 44 | C7 | — | 36 | 1 | no | — |
| 45 | C7 | — | 72 | 3 | no | — |
| 46 | C7 | ΣL(2,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 47 | C2×C3 | — | 1764 | 1 | no | — |
| 48 | D3 | — | 84 | 1 | no | — |
| 49 | C22 | — | 63 | 1 | no | — |
| 50 | C3 | — | 28 | 1 | no | — |
| 51 | C3 | — | 588 | 1 | no | — |
| 52 | C2 | — | 63 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C7×SL(2,𝔽8) | C3 |
| Fitting | C7 | ΣL(2,𝔽8) |
| LCS | G ▷ C7×SL(2,𝔽8) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×SL(2,𝔽8) | C3 | semidirect |
| 4. SL(2,𝔽8) | ΓL(1,𝔽8) | semidirect |
| 46. C7 | ΣL(2,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×SL(2,𝔽8) | 3 |
| ΣL(2,𝔽8) | 7 |
| #11924306 | 9 |
| ΓL(1,𝔽64) | 28 |
| #64423 | 36 |