Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
944 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 944 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C97 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C97 |
| socle | C23×C97 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#41×C97 | C2 | semidirect |
| 2. C2×C4×D4×C97 | C2 | semidirect |
| 3. #395×C97 | C2 | semidirect |
| 4. #394×C97 | C2 | semidirect |
| 5. #393×C97 | C2 | semidirect |
| 6. #392×C97 | C2 | semidirect |
| 7. C4×C22⋊C4×C97 | C2 | semidirect |
| 33. C22×C4×C97 | D4 | semidirect |
| 51. C23×C97 | #41 | semidirect |
| 66. C2×C4×C97 | #41 | semidirect |
| 88. #2388 | C97 | direct (N × H) |
| 89. C97 | #2388 | direct (N × H) |
| 90. C22×#41 | C2×C97 | semidirect |
| 91. C2×C4×D4 | C2×C97 | semidirect |
| 92. #395 | C2×C97 | semidirect |
| 93. #394 | C2×C97 | semidirect |
| 94. #393 | C2×C97 | semidirect |
| 95. #392 | C2×C97 | semidirect |
| 96. C4×C22⋊C4 | C2×C97 | semidirect |
| 122. C22×C4 | D4×C97 | semidirect |
| 140. C23 | #41×C97 | semidirect |
| 155. C2×C4 | #41×C97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41×C97 | 2 |
| C2×C4×D4×C97 | 2 |
| #395×C97 | 2 |
| #394×C97 | 2 |
| #393×C97 | 2 |
| #392×C97 | 2 |
| C4×C22⋊C4×C97 | 2 |
| #2388 | 97 |