Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
492 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 74 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23×C53 |
| prime radical | C53×#1595 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2712 | C2 | semidirect |
| 2. C53×#2044 | C2 | semidirect |
| 3. C53×#1607 | C2 | semidirect |
| 5. C53×#1344 | C2 | semidirect |
| 6. C53×#1343 | C2 | semidirect |
| 7. C53×#1326 | C2 | semidirect |
| 117. C22⋊C4×C53 | D8 | semidirect |
| 178. #12151 | C53 | direct (N × H) |
| 212. #2712 | C2×C53 | semidirect |
| 213. #2044 | C2×C53 | semidirect |
| 214. #1607 | C2×C53 | semidirect |
| 216. #1344 | C2×C53 | semidirect |
| 217. #1343 | C2×C53 | semidirect |
| 218. #1326 | C2×C53 | semidirect |
| 247. C53 | #12151 | direct (N × H) |
| 338. C22⋊C4 | D8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2712 | 2 |
| C53×#2044 | 2 |
| C53×#1607 | 2 |
| C53×#1595 | 2 |
| C53×#1344 | 2 |
| C53×#1343 | 2 |
| C53×#1326 | 2 |
| #12151 | 53 |