Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
1330 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1330 conjugacy classes of subgroups — too many to draw. Only its 238 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 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C23×H(3,𝔽4) | C2 | semidirect |
| 2. C2×C23×#496 | C2 | semidirect |
| 3. C23×#1655 | C2 | direct (N × H) |
| 4. C2×C23×#329 | C2 | semidirect |
| 8. C23×H(3,𝔽4) | C22 | semidirect |
| 9. C2×C23×#127 | C22 | semidirect |
| 14. C23×#496 | C22 | semidirect |
| 15. C2×C23×#97 | C22 | semidirect |
| 16. C23×#329 | C22 | semidirect |
| 28. C23×#127 | C23 | semidirect |
| 41. C23×#97 | C23 | semidirect |
| 70. C2×#1655 | C23 | direct (N × H) |
| 93. C2×H(3,𝔽4) | C2×C23 | semidirect |
| 94. C2×#496 | C2×C23 | semidirect |
| 95. #1655 | C2×C23 | direct (N × H) |
| 96. C2×#329 | C2×C23 | semidirect |
| 111. H(3,𝔽4) | C22×C23 | semidirect |
| 112. C2×#127 | C22×C23 | semidirect |
| 117. #496 | C22×C23 | semidirect |
| 118. C2×#97 | C22×C23 | semidirect |
| 119. #329 | C22×C23 | semidirect |
| 122. C2×C23 | #1655 | direct (N × H) |
| 135. #127 | C23×C23 | semidirect |
| 148. #97 | C23×C23 | semidirect |
| 149. C23 | C2×#1655 | direct (N × H) |
| 213. C2 | C23×#1655 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C23×H(3,𝔽4) | 2 |
| C2×C23×#496 | 2 |
| C23×#1655 | 2 |
| C2×C23×#329 | 2 |
| C2×#1655 | 23 |