Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
5294 conjugacy classes of subgroups; only the 1134 normal subgroups are shown.
This group has 5294 conjugacy classes of subgroups — too many to draw. Only its 1134 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 | C23 | C25×C23 |
| prime radical | C22×C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C23 ◁ C25×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C23×#123 | C2 | semidirect |
| 3. C2×C23×#458 | C2 | direct (N × H) |
| 4. C22×C23×#99 | C2 | semidirect |
| 10. C24×C4×C23 | C22 | semidirect |
| 12. C2×C23×#123 | C22 | semidirect |
| 15. C22×C22⋊C4×C23 | C22 | semidirect |
| 16. C23×#458 | C22 | direct (N × H) |
| 17. C2×C23×#99 | C22 | semidirect |
| 39. C23×C4×C23 | C23 | semidirect |
| 41. C23×#123 | C23 | semidirect |
| 45. C2×C22⋊C4×C23 | C23 | semidirect |
| 46. C23×#99 | C23 | semidirect |
| 80. C22×C4×C23 | C24 | semidirect |
| 84. C22⋊C4×C23 | C24 | semidirect |
| 85. C22×#458 | C23 | direct (N × H) |
| 117. C22×#123 | C2×C23 | semidirect |
| 118. C2×#458 | C2×C23 | direct (N × H) |
| 119. C22×#99 | C2×C23 | semidirect |
| 130. C22×C23 | #458 | direct (N × H) |
| 142. C24×C4 | C22×C23 | semidirect |
| 144. C2×#123 | C22×C23 | semidirect |
| 147. C22×C22⋊C4 | C22×C23 | semidirect |
| 148. #458 | C22×C23 | direct (N × H) |
| 149. C2×#99 | C22×C23 | semidirect |
| 153. C2×C23 | C2×#458 | direct (N × H) |
| 176. C23×C4 | C23×C23 | semidirect |
| 178. #123 | C23×C23 | semidirect |
| 182. C2×C22⋊C4 | C23×C23 | semidirect |
| 183. #99 | C23×C23 | semidirect |
| 184. C23 | C22×#458 | direct (N × H) |
| 218. C22×C4 | C24×C23 | semidirect |
| 222. C22⋊C4 | C24×C23 | semidirect |
| 263. C22 | C23×#458 | direct (N × H) |
| 273. C2 | C2×C23×#458 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C23×H(5,𝔽2) | 2 |
| C22×C23×#123 | 2 |
| C2×C23×#458 | 2 |
| C22×C23×#99 | 2 |
| C22×#458 | 23 |