Finite Group DatabasebetaFinite Group DatabasebetaOrder 30422.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C41×C53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C41×D53 | — | 7 | 1 | no | maximal, Hall |
| 3 | C41×C53 | D7 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | D371 | C41 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | D7×C41 | — | 53 | 1 | no | maximal, Hall |
| 6 | C7×C53 | C2×C41 | 1 | 1 | yes | characteristic, complemented, Hall |
| 7 | C7×C41 | D53 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | D53 | — | 7 | 1 | no | Hall |
| 9 | C2×C41 | — | 371 | 1 | no | Hall |
| 10 | C53 | D7×C41 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 11 | C41 | D371 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 12 | D7 | — | 53 | 1 | no | Hall |
| 13 | C7 | C41×D53 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 371 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C41 | D371 |
| commutator, cosocle, hypocenter | C7×C53 | C2×C41 |
| Frattini | C1 | G |
| Fitting, socle | C7×C41×C53 | C2 |
| LCS | G ▷ C7×C53 | |
| UCS | C1 ◁ C41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C41×C53 | C2 | semidirect |
| 3. C41×C53 | D7 | semidirect |
| 4. D371 | C41 | direct (N × H) |
| 6. C7×C53 | C2×C41 | semidirect |
| 7. C7×C41 | D53 | semidirect |
| 10. C53 | D7×C41 | semidirect |
| 11. C41 | D371 | direct (N × H) |
| 13. C7 | C41×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C41×C53 | 2 |
| C41×D53 | 7 |
| D371 | 41 |
| D7×C41 | 53 |