Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
662 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #120 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C2×C53×#496 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2379 | C2 | semidirect |
| 3. C53×#1961 | C2 | semidirect |
| 4. C2×C53×#329 | C2 | semidirect |
| 5. C53×#1295 | C2 | semidirect |
| 11. C53×#496 | C4 | semidirect |
| 37. #127×C53 | C2×C4 | semidirect |
| 54. #97×C53 | C2×C4 | semidirect |
| 149. #9001 | C53 | direct (N × H) |
| 173. #2379 | C2×C53 | semidirect |
| 174. #1961 | C2×C53 | semidirect |
| 175. C2×#329 | C2×C53 | semidirect |
| 176. #1295 | C2×C53 | semidirect |
| 189. #496 | C4×C53 | semidirect |
| 200. C53 | #9001 | direct (N × H) |
| 216. #127 | C2×C4×C53 | semidirect |
| 233. #97 | C2×C4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#496 | 2 |
| C53×#2379 | 2 |
| C53×#1961 | 2 |
| C2×C53×#329 | 2 |
| C53×#1295 | 2 |
| #9001 | 53 |