Finite Group DatabasebetaFinite Group DatabasebetaOrder 51496.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic6437 | C2 | 1 | 2 | yes | maximal |
| 2 | C4×C41×C157 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×C41×C157 | C22 | 1 | 1 | yes | characteristic |
| 4 | C41×C157 | Q8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | Dic314 | — | 41 | 1 | no | maximal, Hall |
| 6 | C4×C157 | D41 | 1 | 1 | yes | characteristic |
| 7 | Dic157 | — | 41 | 2 | no | — |
| 8 | Dic82 | — | 157 | 1 | no | maximal, Hall |
| 9 | C2×C157 | C2×D41 | 1 | 1 | yes | characteristic |
| 10 | C4×C41 | D157 | 1 | 1 | yes | characteristic |
| 11 | Dic41 | — | 157 | 2 | no | — |
| 12 | C157 | Dic82 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C2×C41 | C2×D157 | 1 | 1 | yes | characteristic |
| 14 | C41 | Dic314 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | Q8 | — | 6437 | 1 | no | Sylow |
| 16 | C4 | — | 6437 | 2 | no | — |
| 17 | C4 | D6437 | 1 | 1 | yes | characteristic |
| 18 | C2 | C2×D6437 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D6437 |
| commutator, socle, prime radical, cosocle | C2×C41×C157 | C22 |
| Fitting | C4×C41×C157 | C2 |
| hypercenter | C4 | D6437 |
| hypocenter | C41×C157 | Q8 |
| LCS | G ▷ C2×C41×C157 ▷ C41×C157 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C41×C157 | Q8 | semidirect |
| 12. C157 | Dic82 | semidirect |
| 14. C41 | Dic314 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Dic6437 | 2 |
| C4×C41×C157 | 2 |
| Dic314 | 41 |
| Dic82 | 157 |