Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
596 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 596 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 | C22×C53 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23×C53 |
| prime radical | C53×#2668 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C53×#2029 | C2 | semidirect |
| 3. C53×#1614 | C2 | semidirect |
| 4. C53×#1596 | C2 | semidirect |
| 5. C53×#1352 | C2 | semidirect |
| 6. C53×#1343 | C2 | semidirect |
| 7. C53×#1316 | C2 | semidirect |
| 187. #12262 | C53 | direct (N × H) |
| 228. #2029 | C2×C53 | semidirect |
| 229. #1614 | C2×C53 | semidirect |
| 230. #1596 | C2×C53 | semidirect |
| 231. #1352 | C2×C53 | semidirect |
| 232. #1343 | C2×C53 | semidirect |
| 233. #1316 | C2×C53 | semidirect |
| 262. C53 | #12262 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2668 | 2 |
| C53×#2029 | 2 |
| C53×#1614 | 2 |
| C53×#1596 | 2 |
| C53×#1352 | 2 |
| C53×#1343 | 2 |
| C53×#1316 | 2 |
| #12262 | 53 |