Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
424 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 90 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| prime radical | D4×D8×C53 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C8×D4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2240 | C2 | semidirect |
| 3. C53×#2224 | C2 | semidirect |
| 4. C53×#1673 | C2 | semidirect |
| 5. C53×#1361 | C2 | semidirect |
| 6. C53×#1331 | C2 | semidirect |
| 7. C53×D4⋊C16 | C2 | semidirect |
| 10. C53×#500 | C22 | semidirect |
| 19. C53×C4⋊C16 | C22 | semidirect |
| 22. C4×C16×C53 | C22 | semidirect |
| 23. C53×#335 | C22 | semidirect |
| 75. D8×C53 | SD16 | semidirect |
| 126. #12368 | C53 | direct (N × H) |
| 147. #2240 | C2×C53 | semidirect |
| 148. #2224 | C2×C53 | semidirect |
| 149. #1673 | C2×C53 | semidirect |
| 150. #1361 | C2×C53 | semidirect |
| 151. #1331 | C2×C53 | semidirect |
| 152. D4⋊C16 | C2×C53 | semidirect |
| 161. #500 | C22×C53 | semidirect |
| 170. C4⋊C16 | C22×C53 | semidirect |
| 173. C4×C16 | C22×C53 | semidirect |
| 174. #335 | C22×C53 | semidirect |
| 176. C53 | #12368 | direct (N × H) |
| 227. D8 | SD16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×D8×C53 | 2 |
| C53×#2240 | 2 |
| C53×#2224 | 2 |
| C53×#1673 | 2 |
| C53×#1361 | 2 |
| C53×#1331 | 2 |
| C53×D4⋊C16 | 2 |
| #12368 | 53 |