Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
8094 conjugacy classes of subgroups; only the 1123 normal subgroups are shown.
This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 1123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D59 |
| commutator, socle, cosocle | C23×C59 | C25 |
| Fitting | C59×#2867 | C2 |
| hypercenter | #2867 | D59 |
| hypocenter | C59 | C2×#2867 |
| LCS | G ▷ C23×C59 ▷ C59 | |
| UCS | C1 ◁ C23 ◁ #2867 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#127×D59 | C2 | semidirect |
| 2. #21432491 | C2 | semidirect |
| 3. #21432490 | C2 | semidirect |
| 4. C43×D59 | C2 | semidirect |
| 5. C59×#2867 | C2 | semidirect |
| 7. C2×#14273724 | C22 | semidirect |
| 8. C2×#14273723 | C22 | semidirect |
| 11. C42×Dic59 | C22 | semidirect |
| 12. C2×#127×C59 | C22 | semidirect |
| 13. C43×C59 | C22 | semidirect |
| 22. C42×D59 | D4 | semidirect |
| 37. C42×C59 | C2×D4 | semidirect |
| 38. C4×Dic59 | C2×D4 | semidirect |
| 49. C4×D59 | #127 | semidirect |
| 56. C4×C59 | C2×#127 | semidirect |
| 57. Dic59 | C2×#127 | semidirect |
| 61. #2867 | D59 | direct (N × H) |
| 66. D59 | #2867 | direct (N × H) |
| 71. C2×#127 | C2×D59 | semidirect |
| 74. C43 | C2×D59 | semidirect |
| 75. C59 | C2×#2867 | semidirect |
| 100. C42 | D4×D59 | semidirect |
| 118. C4 | #127×D59 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#127×D59 | 2 |
| #21432491 | 2 |
| #21432490 | 2 |
| C43×D59 | 2 |
| C59×#2867 | 2 |
| C2×#2867 | 59 |