Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
8094 conjugacy classes of subgroups; only the 1243 normal subgroups are shown.
This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 1243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D53 |
| commutator, cosocle | C23×C53 | C25 |
| Frattini | C23 | C24×D53 |
| Fitting | C22×#127×C53 | C2 |
| socle | C24×C53 | C24 |
| hypercenter | C22×#127 | D53 |
| hypocenter | C53 | C2×#2867 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C22×#127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#13133214 | C2 | semidirect |
| 2. C22×#13133213 | C2 | semidirect |
| 3. #20283109 | C2 | direct (N × H) |
| 4. C2×C42×Dic53 | C2 | semidirect |
| 5. C22×#127×C53 | C2 | semidirect |
| 9. C2×#13133214 | C22 | semidirect |
| 11. C2×#13133213 | C22 | semidirect |
| 13. C42×Dic53 | C22 | semidirect |
| 16. C2×#127×C53 | C22 | semidirect |
| 27. C2×C4×Dic53 | D4 | semidirect |
| 35. C2×C42×C53 | D4 | semidirect |
| 57. C42×C53 | C2×D4 | semidirect |
| 58. C4×Dic53 | C2×D4 | semidirect |
| 71. C2×C4×C53 | #127 | semidirect |
| 75. C2×Dic53 | #127 | semidirect |
| 89. C4×C53 | C2×#127 | semidirect |
| 90. Dic53 | C2×#127 | semidirect |
| 93. C22×#127 | D53 | semidirect |
| 97. C2×C53 | #2867 | semidirect |
| 109. C2×#127 | C2×D53 | semidirect |
| 113. C53 | C2×#2867 | semidirect |
| 131. C2×C42 | C22⋊D53 | semidirect |
| 154. C42 | C2×C22⋊D53 | semidirect |
| 171. C2×C4 | #13133213 | semidirect |
| 185. C4 | C2×#13133213 | semidirect |
| 188. C2 | #20283109 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13133214 | 2 |
| C22×#13133213 | 2 |
| #20283109 | 2 |
| C2×C42×Dic53 | 2 |
| C22×#127×C53 | 2 |
| C2×#2867 | 53 |