Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
528 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C22⋊C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3186 | C2 | semidirect |
| 2. C53×#2184 | C2 | semidirect |
| 3. C53×#1999 | C2 | semidirect |
| 4. C53×#1347 | C2 | semidirect |
| 5. C53×#1314 | C2 | semidirect |
| 6. C22×D8×C53 | C22 | semidirect |
| 17. C53×C22⋊C16 | C22 | semidirect |
| 18. C53×#340 | C22 | semidirect |
| 27. C2×D8×C53 | D4 | semidirect |
| 43. #107×C53 | D4 | semidirect |
| 107. C23×C53 | D16 | semidirect |
| 136. #12102 | C53 | direct (N × H) |
| 164. #3186 | C2×C53 | semidirect |
| 165. #2184 | C2×C53 | semidirect |
| 166. #1999 | C2×C53 | semidirect |
| 167. #1347 | C2×C53 | semidirect |
| 168. #1314 | C2×C53 | semidirect |
| 177. C22×D8 | C22×C53 | semidirect |
| 188. C22⋊C16 | C22×C53 | semidirect |
| 189. #340 | C22×C53 | semidirect |
| 190. C53 | #12102 | direct (N × H) |
| 199. C2×D8 | D4×C53 | semidirect |
| 215. #107 | D4×C53 | semidirect |
| 279. C23 | D16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3186 | 2 |
| C53×#2184 | 2 |
| C53×#1999 | 2 |
| C53×#1347 | 2 |
| C53×#1314 | 2 |
| #12102 | 53 |