Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
4086 conjugacy classes of subgroups; only the 811 normal subgroups are shown.
This group has 4086 conjugacy classes of subgroups — too many to draw. Only its 811 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×D53 |
| commutator, socle, cosocle | C23×C53 | C25 |
| Fitting | C53×#2867 | C2 |
| hypercenter | #2867 | D53 |
| hypocenter | C53 | #36644 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #2867 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16811171 | C2 | semidirect |
| 2. #20283113 | C2 | semidirect |
| 3. #20283071 | C2 | semidirect |
| 4. #20283057 | C2 | semidirect |
| 5. #127×Dic53 | C2 | semidirect |
| 6. C43×D53 | C2 | semidirect |
| 7. C53×#2867 | C2 | semidirect |
| 39. C42×D53 | D4 | semidirect |
| 64. C42×C53 | #41 | semidirect |
| 87. C4×D53 | #127 | semidirect |
| 97. C4×C53 | #532 | semidirect |
| 103. #2867 | D53 | semidirect |
| 127. C53 | #36644 | semidirect |
| 174. C42 | #11719947 | semidirect |
| 204. C4 | #16811171 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16811171 | 2 |
| #20283113 | 2 |
| #20283071 | 2 |
| #20283057 | 2 |
| #127×Dic53 | 2 |
| C43×D53 | 2 |
| C53×#2867 | 2 |
| #36644 | 53 |