Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
784 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C181 | C24 |
| commutator | C22 | C23×C4×C181 |
| Frattini, cosocle | C23 | C24×C181 |
| prime radical | C24×C181 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C181 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #391×C181 | C2 | semidirect |
| 6. C22×Q8×C181 | C4 | semidirect |
| 57. C181 | #2310 | direct (N × H) |
| 58. #2310 | C181 | direct (N × H) |
| 61. #391 | C2×C181 | semidirect |
| 64. C22×Q8 | C4×C181 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C181 | 2 |
| C2×#117×C181 | 2 |
| #391×C181 | 2 |
| #381×C181 | 2 |
| #2310 | 181 |