Finite Group DatabasebetaFinite Group DatabasebetaOrder 31996.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D7999 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C19×D421 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | D19×C421 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C19×C421 | C22 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C2×D421 | — | 19 | 1 | no | maximal, Hall |
| 6 | C2×C421 | — | 19 | 1 | no | — |
| 7 | D421 | — | 19 | 1 | no | — |
| 8 | D421 | D19 | 1 | 1 | yes | characteristic, direct product factor |
| 9 | C421 | C2×D19 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C2×D19 | — | 421 | 1 | no | maximal, Hall |
| 11 | C2×C19 | — | 421 | 1 | no | — |
| 12 | D19 | — | 421 | 1 | no | — |
| 13 | D19 | D421 | 1 | 1 | yes | characteristic, direct product factor |
| 14 | C19 | C2×D421 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C22 | — | 7999 | 1 | no | Sylow |
| 16 | C2 | — | 19 | 1 | no | — |
| 17 | C2 | — | 7999 | 1 | no | — |
| 18 | C2 | — | 421 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C19×C421 | C22 |
| LCS | G ▷ C19×C421 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D7999 | C2 | semidirect |
| 2. C19×D421 | C2 | semidirect |
| 3. D19×C421 | C2 | semidirect |
| 4. C19×C421 | C22 | semidirect |
| 8. D421 | D19 | direct (N × H) |
| 9. C421 | C2×D19 | semidirect |
| 13. D19 | D421 | direct (N × H) |
| 14. C19 | C2×D421 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D7999 | 2 |
| C19×D421 | 2 |
| D19×C421 | 2 |
| C2×D421 | 19 |
| C2×D19 | 421 |