Finite Group DatabasebetaFinite Group DatabasebetaOrder 27006.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | ΓL(1,𝔽8)×C643 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D4501 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C7×C643 | C2×C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×D643 | — | 7 | 1 | no | maximal, Hall |
| 5 | C3×C643 | — | 7 | 1 | no | Hall |
| 6 | D643 | — | 7 | 1 | no | Hall |
| 7 | C643 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | AGL(1,𝔽7) | — | 643 | 1 | no | maximal, Hall |
| 9 | ΓL(1,𝔽8) | D643 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | D7 | — | 643 | 1 | no | Hall |
| 11 | C7 | C3×D643 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C3 | — | 4501 | 1 | no | Hall |
| 13 | C3 | — | 7 | 1 | no | Sylow |
| 14 | C2 | — | 4501 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C643 | C2×C3 |
| LCS | G ▷ C7×C643 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×C643 | C2 | semidirect |
| 2. D4501 | C3 | semidirect |
| 3. C7×C643 | C2×C3 | semidirect |
| 7. C643 | AGL(1,𝔽7) | semidirect |
| 9. ΓL(1,𝔽8) | D643 | semidirect |
| 11. C7 | C3×D643 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×C643 | 2 |
| D4501 | 3 |
| C3×D643 | 7 |
| AGL(1,𝔽7) | 643 |