Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
814 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 814 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 |
| 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×#3207 | C2 | semidirect |
| 2. C23×#3205 | C2 | semidirect |
| 3. C23×#1721 | C2 | semidirect |
| 4. C23×#1717 | C2 | semidirect |
| 5. C23×#1712 | C2 | semidirect |
| 6. C23×#1711 | C2 | semidirect |
| 7. C23×#1454 | C2 | semidirect |
| 12. C23×#493 | C22 | semidirect |
| 13. C23×#492 | C22 | semidirect |
| 14. C23×#487 | C22 | semidirect |
| 17. C23×#486 | C22 | semidirect |
| 23. C23×#322 | C22 | semidirect |
| 24. C82×C23 | C22 | semidirect |
| 75. #19743 | C23 | direct (N × H) |
| 96. #3207 | C2×C23 | semidirect |
| 97. #3205 | C2×C23 | semidirect |
| 98. #1721 | C2×C23 | semidirect |
| 99. #1717 | C2×C23 | semidirect |
| 100. #1712 | C2×C23 | semidirect |
| 101. #1711 | C2×C23 | semidirect |
| 102. #1454 | C2×C23 | semidirect |
| 118. #493 | C22×C23 | semidirect |
| 119. #492 | C22×C23 | semidirect |
| 120. #487 | C22×C23 | semidirect |
| 123. #486 | C22×C23 | semidirect |
| 129. #322 | C22×C23 | semidirect |
| 130. C82 | C22×C23 | semidirect |
| 161. C23 | #19743 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#3207 | 2 |
| C23×#3205 | 2 |
| C23×#1721 | 2 |
| C23×#1717 | 2 |
| C23×#1712 | 2 |
| C23×#1711 | 2 |
| C23×#1454 | 2 |
| #19743 | 23 |