Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
766 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C97 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C97 |
| socle | C2×C97 | C2×#120 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C97 ◁ C22×C4×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #585×C97 | C2 | semidirect |
| 2. #525×C97 | C2 | semidirect |
| 3. #458×C97 | C2 | semidirect |
| 4. B(4,𝔽2)×C97 | C2 | semidirect |
| 5. #410×C97 | C2 | semidirect |
| 11. C23×C4×C97 | C22 | semidirect |
| 12. #123×C97 | C22 | semidirect |
| 15. #120×C97 | C22 | semidirect |
| 17. #117×C97 | C22 | semidirect |
| 18. #99×C97 | C22 | semidirect |
| 26. C2×Q8×C97 | D4 | semidirect |
| 30. C2×D4×C97 | D4 | semidirect |
| 35. C22×C4×C97 | D4 | semidirect |
| 82. #3025 | C97 | direct (N × H) |
| 83. C97 | #3025 | direct (N × H) |
| 84. #585 | C2×C97 | semidirect |
| 85. #525 | C2×C97 | semidirect |
| 86. #458 | C2×C97 | semidirect |
| 87. B(4,𝔽2) | C2×C97 | semidirect |
| 88. #410 | C2×C97 | semidirect |
| 94. C23×C4 | C22×C97 | semidirect |
| 95. #123 | C22×C97 | semidirect |
| 98. #120 | C22×C97 | semidirect |
| 100. #117 | C22×C97 | semidirect |
| 101. #99 | C22×C97 | semidirect |
| 109. C2×Q8 | D4×C97 | semidirect |
| 113. C2×D4 | D4×C97 | semidirect |
| 118. C22×C4 | D4×C97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #585×C97 | 2 |
| #525×C97 | 2 |
| #458×C97 | 2 |
| B(4,𝔽2)×C97 | 2 |
| #410×C97 | 2 |
| #3025 | 97 |