Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
258 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C53 | D4 |
| commutator | C4 | C22×C16×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C22×C53 | C8×D4 |
| prime radical | C2×D8×C53 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C53 ◁ C4×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×C16×C53 | C2 | semidirect |
| 3. C53×#1370 | C2 | semidirect |
| 4. C53×D4⋊C16 | C2 | semidirect |
| 5. C8×C16×C53 | C2 | semidirect |
| 33. D8×C53 | C16 | direct (N × H) |
| 48. C16×C53 | D8 | direct (N × H) |
| 51. D4×C53 | C2×C16 | semidirect |
| 62. C8×C53 | C2×C16 | semidirect |
| 65. C16×D8 | C53 | direct (N × H) |
| 76. D4×C16 | C2×C53 | semidirect |
| 78. #1370 | C2×C53 | semidirect |
| 79. D4⋊C16 | C2×C53 | semidirect |
| 80. C8×C16 | C2×C53 | semidirect |
| 95. C53 | C16×D8 | direct (N × H) |
| 113. D8 | C16×C53 | direct (N × H) |
| 128. C16 | D8×C53 | direct (N × H) |
| 131. D4 | C2×C16×C53 | semidirect |
| 142. C8 | C2×C16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C16×C53 | 2 |
| C8×D8×C53 | 2 |
| C53×#1370 | 2 |
| C53×D4⋊C16 | 2 |
| C8×C16×C53 | 2 |
| C16×D8 | 53 |