Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
696 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D181 |
| commutator, socle, cosocle | C23×C181 | C24 |
| Fitting | #382×C181 | C2 |
| prime radical | C2×#19153791 | C2 |
| hypercenter | #382 | D181 |
| hypocenter | C181 | #2850 |
| LCS | G ▷ C23×C181 ▷ C181 | |
| UCS | C1 ◁ C23 ◁ #382 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19153804 | C2 | semidirect |
| 2. C2×#19153796 | C2 | semidirect |
| 4. C2×C42×D181 | C2 | semidirect |
| 5. #26041184 | C2 | semidirect |
| 6. #26041180 | C2 | semidirect |
| 7. #26041171 | C2 | semidirect |
| 8. #26041157 | C2 | semidirect |
| 9. #382×C181 | C2 | semidirect |
| 63. C42×C181 | D4 | semidirect |
| 113. C2×D181 | #124 | semidirect |
| 128. C181 | #2850 | semidirect |
| 138. #382 | D181 | semidirect |
| 192. C42 | D724 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19153804 | 2 |
| C2×#19153796 | 2 |
| C2×#19153791 | 2 |
| C2×C42×D181 | 2 |
| #26041184 | 2 |
| #26041180 | 2 |
| #26041171 | 2 |
| #26041157 | 2 |
| #382×C181 | 2 |
| #2850 | 181 |