Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
342 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 342 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 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23×C53 |
| prime radical | C53×#3199 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×M16×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2238 | C2 | semidirect |
| 3. C53×#2098 | C2 | semidirect |
| 4. C53×#1339 | C2 | semidirect |
| 5. C53×#1321 | C2 | semidirect |
| 11. C53×#430 | C22 | semidirect |
| 17. C53×#349 | C22 | semidirect |
| 18. C53×H(3,ℤ/4) | C22 | semidirect |
| 100. #12301 | C53 | direct (N × H) |
| 117. #2238 | C2×C53 | semidirect |
| 118. #2098 | C2×C53 | semidirect |
| 119. #1339 | C2×C53 | semidirect |
| 120. #1321 | C2×C53 | semidirect |
| 131. #430 | C22×C53 | semidirect |
| 137. #349 | C22×C53 | semidirect |
| 138. H(3,ℤ/4) | C22×C53 | semidirect |
| 139. C53 | #12301 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#3199 | 2 |
| C53×#2238 | 2 |
| C53×#2098 | 2 |
| C53×#1339 | 2 |
| C53×#1321 | 2 |
| #12301 | 53 |