Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
1476 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1476 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D181 |
| commutator, socle, cosocle | C22×C181 | C25 |
| Fitting | C4×#41×C181 | C2 |
| hypercenter | C4×#41 | D181 |
| hypocenter | C181 | #3499 |
| LCS | G ▷ C22×C181 ▷ C181 | |
| UCS | C1 ◁ C22 ◁ C4×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19153863 | C2 | semidirect |
| 2. Q8×D724 | C2 | semidirect |
| 3. #26041933 | C2 | semidirect |
| 4. #26041855 | C2 | semidirect |
| 5. #26041829 | C2 | semidirect |
| 6. C2×#19153816 | C2 | semidirect |
| 7. C4×#41×C181 | C2 | semidirect |
| 42. #41×C181 | D4 | semidirect |
| 78. C4×C181 | 2−1+4 | semidirect |
| 84. C181 | #3499 | semidirect |
| 92. C4×#41 | D181 | semidirect |
| 115. #41 | D724 | semidirect |
| 164. C4 | #19153863 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19153863 | 2 |
| Q8×D724 | 2 |
| #26041933 | 2 |
| #26041855 | 2 |
| #26041829 | 2 |
| C2×#19153816 | 2 |
| C4×#41×C181 | 2 |
| #3499 | 181 |