Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
684 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 102 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 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#2657 | C2 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 1. C53×#3215 | C2 | semidirect |
| 2. C53×#3138 | C2 | semidirect |
| 4. C53×#1555 | C2 | semidirect |
| 5. C53×#1527 | C2 | semidirect |
| 6. C53×#1506 | C2 | semidirect |
| 7. C53×#1300 | C2 | semidirect |
| 40. C4⋊Q8×C53 | D4 | semidirect |
| 41. C4⋊Q8×C53 | D4 | semidirect |
| 64. C4⋊C8×C53 | D4 | semidirect |
| 73. #97×C53 | D4 | semidirect |
| 171. #10096 | C53 | direct (N × H) |
| 195. #3215 | C2×C53 | semidirect |
| 196. #3138 | C2×C53 | semidirect |
| 198. #1555 | C2×C53 | semidirect |
| 199. #1527 | C2×C53 | semidirect |
| 200. #1506 | C2×C53 | semidirect |
| 201. #1300 | C2×C53 | semidirect |
| 231. C53 | #10096 | direct (N × H) |
| 241. C4⋊Q8 | D4×C53 | semidirect |
| 242. C4⋊Q8 | D4×C53 | semidirect |
| 265. C4⋊C8 | D4×C53 | semidirect |
| 274. #97 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3215 | 2 |
| C53×#3138 | 2 |
| C53×#2657 | 2 |
| C53×#1555 | 2 |
| C53×#1527 | 2 |
| C53×#1506 | 2 |
| C53×#1300 | 2 |
| #10096 | 53 |