Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
696 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #392 |
| prime radical | C53×#2397 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#3101 | C2 | semidirect |
| 3. C53×#1995 | C2 | semidirect |
| 4. C53×#1486 | C2 | semidirect |
| 5. C53×#1274 | C2 | semidirect |
| 26. C4⋊Q8×C53 | D4 | semidirect |
| 49. C4×C8×C53 | D4 | semidirect |
| 82. C42×C53 | SD16 | semidirect |
| 115. #10015 | C53 | direct (N × H) |
| 132. #3101 | C2×C53 | semidirect |
| 134. #1995 | C2×C53 | semidirect |
| 135. #1486 | C2×C53 | semidirect |
| 136. #1274 | C2×C53 | semidirect |
| 156. C53 | #10015 | direct (N × H) |
| 162. C4⋊Q8 | D4×C53 | semidirect |
| 185. C4×C8 | D4×C53 | semidirect |
| 218. C42 | SD16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3101 | 2 |
| C53×#2397 | 2 |
| C53×#1995 | 2 |
| C53×#1486 | 2 |
| C53×#1274 | 2 |
| #10015 | 53 |