Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
234 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 234 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 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | C53×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C82×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2227 | C2 | semidirect |
| 4. C53×#1591 | C2 | semidirect |
| 5. C53×#1370 | C2 | semidirect |
| 6. C53×#1365 | C2 | semidirect |
| 92. #12486 | C53 | direct (N × H) |
| 105. #2227 | C2×C53 | semidirect |
| 108. #1591 | C2×C53 | semidirect |
| 109. #1370 | C2×C53 | semidirect |
| 110. #1365 | C2×C53 | semidirect |
| 131. C53 | #12486 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2227 | 2 |
| C53×#2207 | 2 |
| C53×#1706 | 2 |
| C53×#1591 | 2 |
| C53×#1370 | 2 |
| C53×#1365 | 2 |
| C53×#1361 | 2 |
| #12486 | 53 |