Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
360 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #418 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C2×C53 | #1490 |
| prime radical | C53×#575 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C8×C53 ◁ C4×M16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2165 | C2 | semidirect |
| 3. C53×#1455 | C2 | semidirect |
| 4. C53×#1336 | C2 | semidirect |
| 5. C53×#1335 | C2 | semidirect |
| 16. C53×#364 | C22 | semidirect |
| 20. C53×#322 | C22 | semidirect |
| 21. C82×C53 | C22 | semidirect |
| 59. Q16×C53 | M16 | semidirect |
| 63. D8×C53 | M16 | semidirect |
| 109. #11669 | C53 | direct (N × H) |
| 122. #2165 | C2×C53 | semidirect |
| 123. #1455 | C2×C53 | semidirect |
| 124. #1336 | C2×C53 | semidirect |
| 125. #1335 | C2×C53 | semidirect |
| 140. #364 | C22×C53 | semidirect |
| 144. #322 | C22×C53 | semidirect |
| 145. C82 | C22×C53 | semidirect |
| 146. C53 | #11669 | direct (N × H) |
| 184. Q16 | M16×C53 | semidirect |
| 188. D8 | M16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2956 | 2 |
| C53×#2165 | 2 |
| C53×#1455 | 2 |
| C53×#1336 | 2 |
| C53×#1335 | 2 |
| #11669 | 53 |