Finite Group DatabasebetaFinite Group DatabasebetaOrder 29127.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C19×C73 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C19×C73⋊C3 | — | 7 | 1 | no | maximal, Hall |
| 3 | #13095276 | C19 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C19×C73 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C7×C73 | C3×C19 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C19×ΓL(1,𝔽8) | — | 73 | 1 | no | maximal, Hall |
| 7 | C73⋊C3 | — | 7 | 1 | no | Hall |
| 8 | C7×C19 | C73⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 9 | C73 | C19×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C3×C19 | — | 511 | 1 | no | Hall |
| 11 | ΓL(1,𝔽8) | — | 73 | 1 | no | Hall |
| 12 | C19 | #13095276 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 13 | C7 | C19×C73⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C3 | — | 511 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C19 | #13095276 |
| commutator, cosocle, hypocenter | C7×C73 | C3×C19 |
| Frattini | C1 | G |
| Fitting, socle | C7×C19×C73 | C3 |
| LCS | G ▷ C7×C73 | |
| UCS | C1 ◁ C19 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C19×C73 | C3 | semidirect |
| 3. #13095276 | C19 | direct (N × H) |
| 4. C19×C73 | ΓL(1,𝔽8) | semidirect |
| 5. C7×C73 | C3×C19 | semidirect |
| 8. C7×C19 | C73⋊C3 | semidirect |
| 9. C73 | C19×ΓL(1,𝔽8) | semidirect |
| 12. C19 | #13095276 | direct (N × H) |
| 13. C7 | C19×C73⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C19×C73 | 3 |
| C19×C73⋊C3 | 7 |
| #13095276 | 19 |
| C19×ΓL(1,𝔽8) | 73 |