Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
382 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 382 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 |
| prime radical | C23×#537 | C22 |
| 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×#2612 | C2 | semidirect |
| 5. C23×#1354 | C2 | semidirect |
| 6. C23×#1353 | C2 | semidirect |
| 7. C23×#1324 | C2 | semidirect |
| 104. C42×C23 | SD16 | semidirect |
| 105. #12182 | C23 | direct (N × H) |
| 150. #2612 | C2×C23 | semidirect |
| 154. #1354 | C2×C23 | semidirect |
| 155. #1353 | C2×C23 | semidirect |
| 156. #1324 | C2×C23 | semidirect |
| 237. C23 | #12182 | direct (N × H) |
| 279. C42 | SD16×C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#2612 | 2 |
| C23×#2040 | 2 |
| C23×#1616 | 2 |
| C23×#1604 | 2 |
| C23×#1354 | 2 |
| C23×#1353 | 2 |
| C23×#1324 | 2 |
| #12182 | 23 |