Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
180 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 78 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 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C23 |
| prime radical | C2×C23×D16 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ C42×C23 ◁ C4×C8×C23 ◁ C4×C16×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#2253 | C2 | semidirect |
| 3. C23×#1416 | C2 | semidirect |
| 5. C4×C23×C32 | C2 | semidirect |
| 44. #13721 | C23 | direct (N × H) |
| 57. #2253 | C2×C23 | semidirect |
| 59. #1416 | C2×C23 | semidirect |
| 61. C4×C32 | C2×C23 | semidirect |
| 96. C23 | #13721 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#2253 | 2 |
| C23×#2249 | 2 |
| C23×#1416 | 2 |
| C23×#1415 | 2 |
| C4×C23×C32 | 2 |
| #13721 | 23 |