Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
4382 conjugacy classes of subgroups; only the 590 normal subgroups are shown.
This group has 4382 conjugacy classes of subgroups — too many to draw. Only its 590 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C53 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C53 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#127×C53 | C2 | semidirect |
| 2. C2×C53×#392 | C2 | semidirect |
| 3. C2×C53×#390 | C2 | semidirect |
| 4. C4×#95×C53 | C2 | semidirect |
| 24. C22×D4×C53 | D4 | semidirect |
| 33. C2×C42×C53 | D4 | semidirect |
| 34. #95×C53 | D4 | semidirect |
| 91. C2×C4×C53 | #127 | semidirect |
| 92. #15142 | C53 | direct (N × H) |
| 109. C4×C53 | #392 | semidirect |
| 110. C22×#127 | C2×C53 | semidirect |
| 111. C2×#392 | C2×C53 | semidirect |
| 112. C2×#390 | C2×C53 | semidirect |
| 113. C4×#95 | C2×C53 | semidirect |
| 130. C53 | #15142 | direct (N × H) |
| 140. C22×D4 | D4×C53 | semidirect |
| 149. C2×C42 | D4×C53 | semidirect |
| 150. #95 | D4×C53 | semidirect |
| 207. C2×C4 | #127×C53 | semidirect |
| 224. C4 | C53×#392 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127×C53 | 2 |
| C2×C53×#392 | 2 |
| C2×C53×#390 | 2 |
| C4×#95×C53 | 2 |
| #15142 | 53 |