Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
406 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C23×C53 |
| 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×#3292 | C2 | semidirect |
| 2. C53×#2213 | C2 | semidirect |
| 3. C53×#1712 | C2 | semidirect |
| 4. C53×#1357 | C2 | semidirect |
| 5. C53×#1335 | C2 | semidirect |
| 10. C53×#493 | C22 | semidirect |
| 14. C53×#364 | C22 | semidirect |
| 16. C82×C53 | C22 | semidirect |
| 45. Q16×C53 | D8 | semidirect |
| 54. D8×C53 | D8 | semidirect |
| 81. #12310 | C53 | direct (N × H) |
| 92. #3292 | C2×C53 | semidirect |
| 93. #2213 | C2×C53 | semidirect |
| 94. #1712 | C2×C53 | semidirect |
| 95. #1357 | C2×C53 | semidirect |
| 96. #1335 | C2×C53 | semidirect |
| 106. #493 | C22×C53 | semidirect |
| 110. #364 | C22×C53 | semidirect |
| 112. C82 | C22×C53 | semidirect |
| 113. C53 | #12310 | direct (N × H) |
| 142. Q16 | D8×C53 | semidirect |
| 151. D8 | D8×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3292 | 2 |
| C53×#2213 | 2 |
| C53×#1712 | 2 |
| C53×#1357 | 2 |
| C53×#1335 | 2 |
| #12310 | 53 |