Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
1310 conjugacy classes of subgroups; only the 286 normal subgroups are shown.
This group has 1310 conjugacy classes of subgroups — too many to draw. Only its 286 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C43×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C53×#442 | C2 | semidirect |
| 2. C4×#127×C53 | C2 | semidirect |
| 3. C53×#2094 | C2 | semidirect |
| 4. C53×#2011 | C2 | semidirect |
| 5. C53×#1745 | C2 | semidirect |
| 37. #97×C53 | D4 | semidirect |
| 82. #23802 | C53 | direct (N × H) |
| 95. C2×#442 | C2×C53 | semidirect |
| 96. C4×#127 | C2×C53 | semidirect |
| 97. #2094 | C2×C53 | semidirect |
| 98. #2011 | C2×C53 | semidirect |
| 99. #1745 | C2×C53 | semidirect |
| 117. C53 | #23802 | direct (N × H) |
| 136. #97 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×#442 | 2 |
| C4×#127×C53 | 2 |
| C53×#2094 | 2 |
| C53×#2011 | 2 |
| C53×#1745 | 2 |
| #23802 | 53 |