Finite Group DatabasebetaFinite Group DatabasebetaOrder 5888.
1112 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C23 |
| socle | C2×C23 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#3529 | C2 | semidirect |
| 2. C23×#3018 | C2 | semidirect |
| 3. C23×#2203 | C2 | semidirect |
| 4. C23×#2202 | C2 | semidirect |
| 5. C23×#2132 | C2 | semidirect |
| 6. C23×#585 | C22 | semidirect |
| 8. C23×#563 | C22 | semidirect |
| 10. C23×H(3,𝔽4) | C22 | semidirect |
| 15. C23×#456 | C22 | semidirect |
| 20. C23×#413 | C22 | semidirect |
| 21. C23×#356 | C22 | semidirect |
| 22. C23×#355 | C22 | semidirect |
| 114. #33564 | C23 | direct (N × H) |
| 149. #3529 | C2×C23 | semidirect |
| 150. #3018 | C2×C23 | semidirect |
| 151. #2203 | C2×C23 | semidirect |
| 152. #2202 | C2×C23 | semidirect |
| 153. #2132 | C2×C23 | semidirect |
| 169. #585 | C22×C23 | semidirect |
| 171. #563 | C22×C23 | semidirect |
| 173. H(3,𝔽4) | C22×C23 | semidirect |
| 178. #456 | C22×C23 | semidirect |
| 183. #413 | C22×C23 | semidirect |
| 184. #356 | C22×C23 | semidirect |
| 185. #355 | C22×C23 | semidirect |
| 232. C23 | #33564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#3529 | 2 |
| C23×#3018 | 2 |
| C23×#2203 | 2 |
| C23×#2202 | 2 |
| C23×#2132 | 2 |
| #33564 | 23 |