Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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×C4⋊C4 | C23×C53 |
| prime radical | C53×#1598 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2782 | C2 | semidirect |
| 2. C53×#2040 | C2 | semidirect |
| 3. C53×#1616 | C2 | semidirect |
| 5. C53×#1353 | C2 | semidirect |
| 6. C53×#1345 | C2 | semidirect |
| 7. C53×#1324 | C2 | semidirect |
| 159. #12141 | C53 | direct (N × H) |
| 185. #2782 | C2×C53 | semidirect |
| 186. #2040 | C2×C53 | semidirect |
| 187. #1616 | C2×C53 | semidirect |
| 189. #1353 | C2×C53 | semidirect |
| 190. #1345 | C2×C53 | semidirect |
| 191. #1324 | C2×C53 | semidirect |
| 218. C53 | #12141 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2782 | 2 |
| C53×#2040 | 2 |
| C53×#1616 | 2 |
| C53×#1598 | 2 |
| C53×#1353 | 2 |
| C53×#1345 | 2 |
| C53×#1324 | 2 |
| #12141 | 53 |