Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
594 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 122 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 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×#97×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C53×#509 | C2 | semidirect |
| 2. C53×#3150 | C2 | semidirect |
| 3. C2×C53×#360 | C2 | semidirect |
| 4. C53×#1972 | C2 | semidirect |
| 5. C53×#1377 | C2 | semidirect |
| 39. C2×D8×C53 | D4 | semidirect |
| 53. M32×C53 | D4 | semidirect |
| 137. #13341 | C53 | direct (N × H) |
| 156. C2×#509 | C2×C53 | semidirect |
| 157. #3150 | C2×C53 | semidirect |
| 158. C2×#360 | C2×C53 | semidirect |
| 159. #1972 | C2×C53 | semidirect |
| 160. #1377 | C2×C53 | semidirect |
| 183. C53 | #13341 | direct (N × H) |
| 201. C2×D8 | D4×C53 | semidirect |
| 215. M32 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#509 | 2 |
| C53×#3150 | 2 |
| C2×C53×#360 | 2 |
| C53×#1972 | 2 |
| C53×#1377 | 2 |
| #13341 | 53 |