Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
570 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 570 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×C22⋊C4 | C23×C23 |
| prime radical | C23×#2021 | C2 |
| 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×#2840 | C2 | semidirect |
| 3. C23×#1601 | C2 | semidirect |
| 4. C23×#1351 | C2 | semidirect |
| 5. C23×#1316 | C2 | semidirect |
| 21. C22×Q8×C23 | D4 | semidirect |
| 45. C23×#95 | D4 | semidirect |
| 52. C24×C23 | SD16 | semidirect |
| 87. #12123 | C23 | direct (N × H) |
| 135. #2840 | C2×C23 | semidirect |
| 137. #1601 | C2×C23 | semidirect |
| 138. #1351 | C2×C23 | semidirect |
| 139. #1316 | C2×C23 | semidirect |
| 191. C22×Q8 | D4×C23 | semidirect |
| 215. #95 | D4×C23 | semidirect |
| 216. C23 | #12123 | direct (N × H) |
| 223. C24 | SD16×C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#2840 | 2 |
| C23×#2021 | 2 |
| C23×#1601 | 2 |
| C23×#1351 | 2 |
| C23×#1316 | 2 |
| #12123 | 23 |