Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
692 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 692 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 |
| 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×#3213 | C2 | semidirect |
| 2. C53×#3109 | C2 | semidirect |
| 3. C53×#2540 | C2 | semidirect |
| 4. C53×#1980 | C2 | semidirect |
| 5. C53×#1527 | C2 | semidirect |
| 6. C53×#1490 | C2 | semidirect |
| 7. C53×#1285 | C2 | semidirect |
| 43. C4⋊Q8×C53 | D4 | semidirect |
| 44. #127×C53 | D4 | semidirect |
| 72. C4⋊C8×C53 | D4 | semidirect |
| 82. #97×C53 | D4 | semidirect |
| 200. #10004 | C53 | direct (N × H) |
| 231. #3213 | C2×C53 | semidirect |
| 232. #3109 | C2×C53 | semidirect |
| 233. #2540 | C2×C53 | semidirect |
| 234. #1980 | C2×C53 | semidirect |
| 235. #1527 | C2×C53 | semidirect |
| 236. #1490 | C2×C53 | semidirect |
| 237. #1285 | C2×C53 | semidirect |
| 271. C53 | #10004 | direct (N × H) |
| 282. C4⋊Q8 | D4×C53 | semidirect |
| 283. #127 | D4×C53 | semidirect |
| 311. C4⋊C8 | D4×C53 | semidirect |
| 321. #97 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3213 | 2 |
| C53×#3109 | 2 |
| C53×#2540 | 2 |
| C53×#1980 | 2 |
| C53×#1527 | 2 |
| C53×#1490 | 2 |
| C53×#1285 | 2 |
| #10004 | 53 |