Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
726 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23×C23 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C4⋊C4×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#2679 | C2 | semidirect |
| 2. C23×#2006 | C2 | semidirect |
| 3. C23×#1598 | C2 | semidirect |
| 4. C23×#1345 | C2 | semidirect |
| 5. C23×#1318 | C2 | semidirect |
| 7. C23×#537 | C22 | semidirect |
| 16. C23×#343 | C22 | semidirect |
| 18. C23×#324 | C22 | semidirect |
| 75. C2×D4×C23 | D8 | semidirect |
| 99. #12096 | C23 | direct (N × H) |
| 149. #2679 | C2×C23 | semidirect |
| 150. #2006 | C2×C23 | semidirect |
| 151. #1598 | C2×C23 | semidirect |
| 152. #1345 | C2×C23 | semidirect |
| 153. #1318 | C2×C23 | semidirect |
| 182. #537 | C22×C23 | semidirect |
| 191. #343 | C22×C23 | semidirect |
| 193. #324 | C22×C23 | semidirect |
| 232. C23 | #12096 | direct (N × H) |
| 259. C2×D4 | D8×C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#2679 | 2 |
| C23×#2006 | 2 |
| C23×#1598 | 2 |
| C23×#1345 | 2 |
| C23×#1318 | 2 |
| #12096 | 23 |