Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
394 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 74 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×C4⋊C4 | C23×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C53×#1617 | C2 | semidirect |
| 4. C53×#1605 | C2 | semidirect |
| 5. C53×#1354 | C2 | semidirect |
| 6. C53×#1353 | C2 | semidirect |
| 70. C2×D4×C53 | Q16 | semidirect |
| 150. #12223 | C53 | direct (N × H) |
| 176. #1617 | C2×C53 | semidirect |
| 177. #1605 | C2×C53 | semidirect |
| 178. #1354 | C2×C53 | semidirect |
| 179. #1353 | C2×C53 | semidirect |
| 206. C53 | #12223 | direct (N × H) |
| 250. C2×D4 | Q16×C53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C53×#2784 | 2 |
| C53×#2027 | 2 |
| C53×#1617 | 2 |
| C53×#1605 | 2 |
| C53×#1354 | 2 |
| C53×#1353 | 2 |
| C53×#1318 | 2 |
| #12223 | 53 |