Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
708 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C23 |
| prime radical | C22×D8×C23 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C23 ◁ C43×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#2860 | C2 | semidirect |
| 2. C23×#2086 | C2 | semidirect |
| 3. C23×#2067 | C2 | semidirect |
| 5. C23×#1842 | C2 | semidirect |
| 54. C42×C23 | D8 | semidirect |
| 55. #32858 | C23 | direct (N × H) |
| 74. #2860 | C2×C23 | semidirect |
| 75. #2086 | C2×C23 | semidirect |
| 76. #2067 | C2×C23 | semidirect |
| 78. #1842 | C2×C23 | semidirect |
| 123. C23 | #32858 | direct (N × H) |
| 143. C42 | D8×C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#2860 | 2 |
| C23×#2086 | 2 |
| C23×#2067 | 2 |
| C23×#2054 | 2 |
| C23×#1842 | 2 |
| #32858 | 23 |