Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
194 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 82 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 | #2177 |
| commutator | C16 | C22×C4×C53 |
| Frattini, cosocle | C2×C16 | C23×C53 |
| prime radical | C53×#2263 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×C8×C53 ◁ C53×#346 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2179 | C2 | semidirect |
| 3. C53×#2177 | C2 | semidirect |
| 4. C53×#1426 | C2 | semidirect |
| 5. C53×#1419 | C2 | semidirect |
| 6. C53×#1418 | C2 | semidirect |
| 7. C53×#1398 | C2 | semidirect |
| 12. C53×SD64 | C4 | semidirect |
| 13. C53×D32 | C4 | semidirect |
| 25. Q32×C53 | C2×C4 | semidirect |
| 29. D16×C53 | C2×C4 | semidirect |
| 38. C32×C53 | C2×C4 | semidirect |
| 73. #13658 | C53 | direct (N × H) |
| 85. #2179 | C2×C53 | semidirect |
| 86. #2177 | C2×C53 | semidirect |
| 87. #1426 | C2×C53 | semidirect |
| 88. #1419 | C2×C53 | semidirect |
| 89. #1418 | C2×C53 | semidirect |
| 90. #1398 | C2×C53 | semidirect |
| 99. SD64 | C4×C53 | semidirect |
| 100. D32 | C4×C53 | semidirect |
| 108. C53 | #13658 | direct (N × H) |
| 113. Q32 | C2×C4×C53 | semidirect |
| 117. D16 | C2×C4×C53 | semidirect |
| 126. C32 | C2×C4×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×#2179 | 2 |
| C53×#2177 | 2 |
| C53×#1426 | 2 |
| C53×#1419 | 2 |
| C53×#1418 | 2 |
| C53×#1398 | 2 |
| #13658 | 53 |