Finite Group DatabasebetaFinite Group DatabasebetaOrder 22862.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C23×C71 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D1633 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C23×C71 | C2×C7 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C7×D71 | — | 23 | 1 | no | maximal, Hall |
| 5 | C7×C71 | D23 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C7×D23 | — | 71 | 1 | no | maximal, Hall |
| 7 | C7×C23 | D71 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | D71 | — | 23 | 1 | no | Hall |
| 9 | C71 | C7×D23 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | D23 | — | 71 | 1 | no | Hall |
| 11 | C23 | C7×D71 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C7 | — | 1633 | 1 | no | Hall |
| 13 | C7 | D1633 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C2 | — | 1633 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D1633 |
| commutator, cosocle, hypocenter | C23×C71 | C2×C7 |
| Frattini | C1 | G |
| Fitting, socle | C7×C23×C71 | C2 |
| LCS | G ▷ C23×C71 | |
| UCS | C1 ◁ C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C23×C71 | C2 | semidirect |
| 2. D1633 | C7 | direct (N × H) |
| 3. C23×C71 | C2×C7 | semidirect |
| 5. C7×C71 | D23 | semidirect |
| 7. C7×C23 | D71 | semidirect |
| 9. C71 | C7×D23 | semidirect |
| 11. C23 | C7×D71 | semidirect |
| 13. C7 | D1633 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C23×C71 | 2 |
| D1633 | 7 |
| C7×D71 | 23 |
| C7×D23 | 71 |