Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
396 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 78 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 |
| prime radical | C53×#2184 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4⋊C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3250 | C2 | semidirect |
| 2. C53×#2187 | C2 | semidirect |
| 4. C53×#2054 | C2 | semidirect |
| 5. C53×#1348 | C2 | semidirect |
| 6. C53×#1347 | C2 | semidirect |
| 7. C53×#1322 | C2 | semidirect |
| 149. #12160 | C53 | direct (N × H) |
| 174. #3250 | C2×C53 | semidirect |
| 175. #2187 | C2×C53 | semidirect |
| 177. #2054 | C2×C53 | semidirect |
| 178. #1348 | C2×C53 | semidirect |
| 179. #1347 | C2×C53 | semidirect |
| 180. #1322 | C2×C53 | semidirect |
| 207. C53 | #12160 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3250 | 2 |
| C53×#2187 | 2 |
| C53×#2184 | 2 |
| C53×#2054 | 2 |
| C53×#1348 | 2 |
| C53×#1347 | 2 |
| C53×#1322 | 2 |
| #12160 | 53 |