Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
3866 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 3866 conjugacy classes of subgroups — too many to draw. Only its 366 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×C59 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C59 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C59×#585 | C2 | semidirect |
| 2. C2×C59×B(4,𝔽2) | C2 | semidirect |
| 3. C59×#2404 | C2 | semidirect |
| 4. C59×#1790 | C2 | semidirect |
| 5. C59×#1782 | C2 | semidirect |
| 12. C2×#120×C59 | C22 | semidirect |
| 13. C22×C22⋊C4×C59 | C22 | semidirect |
| 15. C2×#99×C59 | C22 | semidirect |
| 19. C59×#386 | C22 | semidirect |
| 20. C59×#381 | C22 | semidirect |
| 25. H(5,𝔽2)×C59 | D4 | semidirect |
| 35. C22×Q8×C59 | D4 | semidirect |
| 39. C22×D4×C59 | D4 | semidirect |
| 42. C23×C4×C59 | D4 | semidirect |
| 141. #32899 | C59 | direct (N × H) |
| 163. C2×#585 | C2×C59 | semidirect |
| 164. C2×B(4,𝔽2) | C2×C59 | semidirect |
| 165. #2404 | C2×C59 | semidirect |
| 166. #1790 | C2×C59 | semidirect |
| 167. #1782 | C2×C59 | semidirect |
| 180. C2×#120 | C22×C59 | semidirect |
| 181. C22×C22⋊C4 | C22×C59 | semidirect |
| 183. C2×#99 | C22×C59 | semidirect |
| 187. #386 | C22×C59 | semidirect |
| 188. #381 | C22×C59 | semidirect |
| 189. C59 | #32899 | direct (N × H) |
| 194. H(5,𝔽2) | D4×C59 | semidirect |
| 204. C22×Q8 | D4×C59 | semidirect |
| 208. C22×D4 | D4×C59 | semidirect |
| 211. C23×C4 | D4×C59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C59×#585 | 2 |
| C2×C59×B(4,𝔽2) | 2 |
| C59×#2404 | 2 |
| C59×#1790 | 2 |
| C59×#1782 | 2 |
| #32899 | 59 |