Finite Group DatabasebetaFinite Group DatabasebetaOrder 23268.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C7×C277 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | #19154113 | — | 4 | 1 | no | maximal, Hall |
| 3 | C2×C7×C277 | — | 3 | 1 | no | — |
| 4 | C22⋊(C277⋊C3) | — | 7 | 1 | no | maximal, Hall |
| 5 | C7×C277 | A4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C22×C277 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Hall |
| 7 | C277⋊C3 | — | 28 | 1 | no | Hall |
| 8 | C2×C277 | — | 3 | 1 | no | — |
| 9 | C277 | C22⋊ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C22⋊ΓL(1,𝔽8) | — | 277 | 1 | no | maximal, Hall |
| 11 | C22×C7 | C277⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 12 | ΓL(1,𝔽8) | — | 1108 | 1 | no | Hall |
| 13 | C2×C7 | — | 3 | 1 | no | — |
| 14 | A4 | — | 1939 | 1 | no | Hall |
| 15 | C7 | C22⋊(C277⋊C3) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | C22 | #19154113 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C3 | — | 7756 | 1 | no | Sylow |
| 18 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C7×C277 | C3 |
| LCS | G ▷ C22×C7×C277 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×C277 | C3 | semidirect |
| 5. C7×C277 | A4 | semidirect |
| 6. C22×C277 | ΓL(1,𝔽8) | semidirect |
| 9. C277 | C22⋊ΓL(1,𝔽8) | semidirect |
| 11. C22×C7 | C277⋊C3 | semidirect |
| 15. C7 | C22⋊(C277⋊C3) | semidirect |
| 16. C22 | #19154113 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×C277 | 3 |
| #19154113 | 4 |
| C22⋊(C277⋊C3) | 7 |
| C22⋊ΓL(1,𝔽8) | 277 |