Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
1492 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 553 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 | C2×C4×Q8×C181 | C2 |
| hypercenter | C2×C4×Q8 | D181 |
| hypocenter | C181 | #3497 |
| LCS | G ▷ C22×C181 ▷ C181 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19153864 | C2 | semidirect |
| 2. #26041951 | C2 | semidirect |
| 3. Q8×D724 | C2 | semidirect |
| 4. #26041941 | C2 | semidirect |
| 5. #26041860 | C2 | semidirect |
| 6. #26041829 | C2 | semidirect |
| 7. C2×C4×Q8×C181 | C2 | semidirect |
| 39. C2×Q8×C181 | D4 | semidirect |
| 74. C4×C181 | 2−1+4 | semidirect |
| 81. C181 | #3497 | semidirect |
| 89. C2×C4×Q8 | D181 | semidirect |
| 112. C2×Q8 | D724 | semidirect |
| 157. C4 | #19153864 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19153864 | 2 |
| #26041951 | 2 |
| Q8×D724 | 2 |
| #26041941 | 2 |
| #26041860 | 2 |
| #26041829 | 2 |
| C2×C4×Q8×C181 | 2 |
| #3497 | 181 |