Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1072 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 230 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 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3549 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#3152 | C2 | semidirect |
| 3. C53×#3141 | C2 | semidirect |
| 4. C53×#1659 | C2 | semidirect |
| 5. C53×#1525 | C2 | semidirect |
| 6. C53×#1525 | C2 | semidirect |
| 17. C2×C4⋊Q8×C53 | C22 | semidirect |
| 18. C53×#497 | C22 | semidirect |
| 21. C53×#489 | C22 | semidirect |
| 25. C4×M16×C53 | C22 | semidirect |
| 26. C4×M16×C53 | C22 | semidirect |
| 27. C53×#329 | C22 | semidirect |
| 28. C53×#329 | C22 | semidirect |
| 149. #19074 | C53 | direct (N × H) |
| 167. #3152 | C2×C53 | semidirect |
| 168. #3141 | C2×C53 | semidirect |
| 169. #1659 | C2×C53 | semidirect |
| 170. #1525 | C2×C53 | semidirect |
| 171. #1525 | C2×C53 | semidirect |
| 187. C2×C4⋊Q8 | C22×C53 | semidirect |
| 188. #497 | C22×C53 | semidirect |
| 191. #489 | C22×C53 | semidirect |
| 195. C4×M16 | C22×C53 | semidirect |
| 196. C4×M16 | C22×C53 | semidirect |
| 197. #329 | C22×C53 | semidirect |
| 198. #329 | C22×C53 | semidirect |
| 199. C53 | #19074 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3549 | 2 |
| C53×#3152 | 2 |
| C53×#3141 | 2 |
| C53×#1659 | 2 |
| C53×#1525 | 2 |
| #19074 | 53 |