Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
204 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #442 |
| commutator | C8 | C22×C8×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C2×C53 | #1580 |
| prime radical | C53×#508 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C8×C53 ◁ C53×#346 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2171 | C2 | semidirect |
| 3. C53×#1369 | C2 | semidirect |
| 4. C53×#1336 | C2 | semidirect |
| 5. C53×#1313 | C2 | semidirect |
| 18. Q32×C53 | C8 | semidirect |
| 19. SD32×C53 | C8 | semidirect |
| 20. D16×C53 | C8 | semidirect |
| 58. #12068 | C53 | direct (N × H) |
| 66. #2171 | C2×C53 | semidirect |
| 67. #1369 | C2×C53 | semidirect |
| 68. #1336 | C2×C53 | semidirect |
| 69. #1313 | C2×C53 | semidirect |
| 83. C53 | #12068 | direct (N × H) |
| 86. Q32 | C8×C53 | semidirect |
| 87. SD32 | C8×C53 | semidirect |
| 88. D16 | C8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2179 | 2 |
| C53×#2171 | 2 |
| C53×#1369 | 2 |
| C53×#1336 | 2 |
| C53×#1313 | 2 |
| #12068 | 53 |