Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
216 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #2206 |
| commutator, Frattini, cosocle | C2×C16 | C23×C53 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ C4×C16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C53×#2263 | C2 | semidirect |
| 2. C53×#2246 | C2 | semidirect |
| 3. C53×#2178 | C2 | semidirect |
| 4. C53×#1422 | C2 | semidirect |
| 5. C53×#1419 | C2 | semidirect |
| 22. Q32×C53 | D4 | semidirect |
| 27. D16×C53 | D4 | semidirect |
| 68. #13692 | C53 | direct (N × H) |
| 77. #2263 | C2×C53 | semidirect |
| 78. #2246 | C2×C53 | semidirect |
| 79. #2178 | C2×C53 | semidirect |
| 80. #1422 | C2×C53 | semidirect |
| 81. #1419 | C2×C53 | semidirect |
| 97. C53 | #13692 | direct (N × H) |
| 103. Q32 | D4×C53 | semidirect |
| 108. D16 | D4×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2263 | 2 |
| C53×#2246 | 2 |
| C53×#2178 | 2 |
| C53×#1422 | 2 |
| C53×#1419 | 2 |
| #13692 | 53 |