Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
184 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 184 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 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C23×C53 |
| prime radical | C53×#2221 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ #131×C53 ◁ C53×#504 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C53×SD64 | C2 | semidirect |
| 2. C53×#2258 | C2 | semidirect |
| 3. C53×#2222 | C2 | semidirect |
| 5. C53×#1425 | C2 | semidirect |
| 6. C53×#1420 | C2 | semidirect |
| 7. C53×#1419 | C2 | semidirect |
| 79. #13702 | C53 | direct (N × H) |
| 88. C2×SD64 | C2×C53 | semidirect |
| 89. #2258 | C2×C53 | semidirect |
| 90. #2222 | C2×C53 | semidirect |
| 92. #1425 | C2×C53 | semidirect |
| 93. #1420 | C2×C53 | semidirect |
| 94. #1419 | C2×C53 | semidirect |
| 114. C53 | #13702 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C53×SD64 | 2 |
| C53×#2258 | 2 |
| C53×#2222 | 2 |
| C53×#2221 | 2 |
| C53×#1425 | 2 |
| C53×#1420 | 2 |
| C53×#1419 | 2 |
| #13702 | 53 |