Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
186 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 186 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 | C2×C53 | #1677 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | #135×C53 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2245 | C2 | semidirect |
| 5. C53×#1369 | C2 | semidirect |
| 6. C53×#1366 | C2 | semidirect |
| 7. C53×#1358 | C2 | semidirect |
| 81. #12466 | C53 | direct (N × H) |
| 90. #2245 | C2×C53 | semidirect |
| 94. #1369 | C2×C53 | semidirect |
| 95. #1366 | C2×C53 | semidirect |
| 96. #1358 | C2×C53 | semidirect |
| 115. C53 | #12466 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2245 | 2 |
| C53×#2231 | 2 |
| C53×#2214 | 2 |
| C53×#2171 | 2 |
| C53×#1369 | 2 |
| C53×#1366 | 2 |
| C53×#1358 | 2 |
| #12466 | 53 |