Finite Group DatabasebetaFinite Group DatabasebetaOrder 32718.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C41×#87705 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D5453 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C7×C19×C41 | C2×C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C41⋊(D19⋊C3) | — | 7 | 1 | no | maximal, Hall |
| 5 | C41×C19⋊C3 | — | 7 | 1 | no | Hall |
| 6 | ΓL(1,𝔽8)⋊D41 | — | 19 | 1 | no | maximal, Hall |
| 7 | D779 | — | 7 | 1 | no | Hall |
| 8 | ΓL(1,𝔽8)×C41 | — | 19 | 1 | no | Hall |
| 9 | #11716328 | — | 41 | 1 | no | maximal, Hall |
| 10 | C19×C41 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented, Hall |
| 11 | D287 | — | 19 | 1 | no | Hall |
| 12 | #87705 | D41 | 1 | 1 | yes | characteristic, complemented, Hall |
| 13 | C7×C41 | D19⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 14 | D133 | — | 41 | 1 | no | Hall |
| 15 | C3×D41 | — | 133 | 1 | no | Hall |
| 16 | C7×C19 | C3×D41 | 1 | 1 | yes | characteristic, complemented, Hall |
| 17 | C3×C41 | — | 133 | 1 | no | Hall |
| 18 | D19⋊C3 | — | 287 | 1 | no | Hall |
| 19 | D41 | — | 133 | 1 | no | Hall |
| 20 | C19⋊C3 | — | 7 | 1 | no | Hall |
| 21 | AGL(1,𝔽7) | — | 779 | 1 | no | Hall |
| 22 | C41 | #11716328 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 23 | D19 | — | 287 | 1 | no | Hall |
| 24 | ΓL(1,𝔽8) | — | 19 | 1 | no | Hall |
| 25 | C19 | ΓL(1,𝔽8)⋊D41 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 26 | D7 | — | 779 | 1 | no | Hall |
| 27 | C7 | C41⋊(D19⋊C3) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 28 | C2×C3 | — | 5453 | 1 | no | Hall |
| 29 | C3 | — | 133 | 1 | no | Sylow |
| 30 | C2 | — | 5453 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C19×C41 | C2×C3 |
| LCS | G ▷ C7×C19×C41 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C41×#87705 | C2 | semidirect |
| 2. D5453 | C3 | semidirect |
| 3. C7×C19×C41 | C2×C3 | semidirect |
| 10. C19×C41 | AGL(1,𝔽7) | semidirect |
| 12. #87705 | D41 | semidirect |
| 13. C7×C41 | D19⋊C3 | semidirect |
| 16. C7×C19 | C3×D41 | semidirect |
| 22. C41 | #11716328 | semidirect |
| 25. C19 | ΓL(1,𝔽8)⋊D41 | semidirect |
| 27. C7 | C41⋊(D19⋊C3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C41×#87705 | 2 |
| D5453 | 3 |
| C41⋊(D19⋊C3) | 7 |
| ΓL(1,𝔽8)⋊D41 | 19 |
| #11716328 | 41 |