Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
424 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 70 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×C22⋊C4 | C23×C23 |
| prime radical | C23×#534 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#2523 | C2 | semidirect |
| 5. C23×#1351 | C2 | semidirect |
| 6. C23×#1344 | C2 | semidirect |
| 7. C23×#1326 | C2 | semidirect |
| 108. #12134 | C23 | direct (N × H) |
| 161. #2523 | C2×C23 | semidirect |
| 165. #1351 | C2×C23 | semidirect |
| 166. #1344 | C2×C23 | semidirect |
| 167. #1326 | C2×C23 | semidirect |
| 258. C23 | #12134 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#2523 | 2 |
| C23×#2044 | 2 |
| C23×#1607 | 2 |
| C23×#1601 | 2 |
| C23×#1351 | 2 |
| C23×#1344 | 2 |
| C23×#1326 | 2 |
| #12134 | 23 |