Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
306 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 94 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×C71 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C71 |
| prime radical | C71×#2246 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C42×C71 ◁ C82×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C71×#2248 | C2 | semidirect |
| 3. C71×#2241 | C2 | semidirect |
| 4. C71×#2240 | C2 | semidirect |
| 5. C71×#1709 | C2 | semidirect |
| 6. C71×#1706 | C2 | semidirect |
| 7. C71×#1312 | C2 | semidirect |
| 107. #12527 | C71 | direct (N × H) |
| 113. #2248 | C2×C71 | semidirect |
| 115. #2241 | C2×C71 | semidirect |
| 116. #2240 | C2×C71 | semidirect |
| 117. #1709 | C2×C71 | semidirect |
| 118. #1706 | C2×C71 | semidirect |
| 119. #1312 | C2×C71 | semidirect |
| 120. C71 | #12527 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C71×#2248 | 2 |
| C71×#2246 | 2 |
| C71×#2241 | 2 |
| C71×#2240 | 2 |
| C71×#1709 | 2 |
| C71×#1706 | 2 |
| C71×#1312 | 2 |
| #12527 | 71 |