Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
690 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 690 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 | C22×C53 | #386 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C53×#2617 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2312 | C2 | semidirect |
| 3. C53×#1519 | C2 | semidirect |
| 4. C2×C53×#323 | C2 | semidirect |
| 5. C53×#1289 | C2 | semidirect |
| 10. C53×#397 | C4 | semidirect |
| 12. C53×#392 | C4 | semidirect |
| 51. #95×C53 | D4 | semidirect |
| 147. #9205 | C53 | direct (N × H) |
| 180. #2312 | C2×C53 | semidirect |
| 181. #1519 | C2×C53 | semidirect |
| 182. C2×#323 | C2×C53 | semidirect |
| 183. #1289 | C2×C53 | semidirect |
| 197. #397 | C4×C53 | semidirect |
| 199. #392 | C4×C53 | semidirect |
| 208. C53 | #9205 | direct (N × H) |
| 239. #95 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2617 | 2 |
| C53×#2312 | 2 |
| C53×#1519 | 2 |
| C2×C53×#323 | 2 |
| C53×#1289 | 2 |
| #9205 | 53 |