Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
712 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 230 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×#127 |
| commutator, Frattini, cosocle | C42 | C24×C23 |
| prime radical | C2×C23×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#3231 | C2 | semidirect |
| 3. C23×#1719 | C2 | semidirect |
| 4. C23×#1714 | C2 | semidirect |
| 5. C23×#1711 | C2 | semidirect |
| 6. C23×#1705 | C2 | semidirect |
| 7. C23×#1454 | C2 | semidirect |
| 10. C23×#500 | C22 | semidirect |
| 11. C23×#498 | C22 | semidirect |
| 14. C23×#487 | C22 | semidirect |
| 15. C23×#487 | C22 | semidirect |
| 23. C23×#322 | C22 | semidirect |
| 24. C82×C23 | C22 | semidirect |
| 72. #19733 | C23 | direct (N × H) |
| 89. #3231 | C2×C23 | semidirect |
| 91. #1719 | C2×C23 | semidirect |
| 92. #1714 | C2×C23 | semidirect |
| 93. #1711 | C2×C23 | semidirect |
| 94. #1705 | C2×C23 | semidirect |
| 95. #1454 | C2×C23 | semidirect |
| 107. #500 | C22×C23 | semidirect |
| 108. #498 | C22×C23 | semidirect |
| 111. #487 | C22×C23 | semidirect |
| 112. #487 | C22×C23 | semidirect |
| 120. #322 | C22×C23 | semidirect |
| 121. C82 | C22×C23 | semidirect |
| 151. C23 | #19733 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#3231 | 2 |
| C23×#3208 | 2 |
| C23×#1719 | 2 |
| C23×#1714 | 2 |
| C23×#1711 | 2 |
| C23×#1705 | 2 |
| C23×#1454 | 2 |
| #19733 | 23 |