Finite Group DatabasebetaFinite Group DatabasebetaOrder 12112.
22 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×D757 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20258493 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C8×C757 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×D757 | C4 | 1 | 1 | yes | characteristic |
| 5 | C4×C757 | C22 | 1 | 1 | yes | characteristic |
| 6 | Dic757 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×C757 | C2×C4 | 1 | 1 | yes | characteristic |
| 8 | D757 | C8 | 1 | 2 | yes | direct product factor |
| 9 | C757 | C2×C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C2×C8 | — | 757 | 1 | no | maximal, Sylow |
| 11 | C2×C4 | — | 757 | 1 | no | — |
| 12 | C8 | — | 757 | 1 | no | — |
| 13 | C8 | D757 | 1 | 1 | yes | characteristic, direct product factor |
| 14 | C22 | — | 757 | 1 | no | — |
| 15 | C4 | — | 757 | 1 | no | — |
| 16 | C4 | C2×D757 | 1 | 1 | yes | characteristic |
| 17 | C2 | — | 757 | 2 | no | — |
| 18 | C2 | C4×D757 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D757 |
| commutator, hypocenter | C757 | C2×C8 |
| Frattini | C4 | C2×D757 |
| Fitting | C8×C757 | C2 |
| socle | C2×C757 | C2×C4 |
| prime radical | C2×D757 | C4 |
| cosocle | C4×C757 | C22 |
| LCS | G ▷ C757 | |
| UCS | C1 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20258493 | C2 | semidirect |
| 3. C8×C757 | C2 | semidirect |
| 8. D757 | C8 | direct (N × H) |
| 9. C757 | C2×C8 | semidirect |
| 13. C8 | D757 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D757 | 2 |
| #20258493 | 2 |
| C8×C757 | 2 |
| C2×C8 | 757 |