← C22×C3×C16×C121 #72200554
Subgroups — C22×C3×C16×C121 #72200554
Order 23232.
294 conjugacy classes of subgroups.
The subgroup lattice has 114 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×C3×C8×C121 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×C16×C121 | C2 | 1 | 6 | yes | maximal, direct product factor |
| 3 | C22×C16×C121 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C22×C3×C4×C121 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×C3×C8×C121 | C4 | 1 | 3 | yes | — |
| 6 | C2×C3×C8×C121 | C22 | 1 | 3 | yes | — |
| 7 | C3×C16×C121 | C22 | 1 | 4 | yes | direct product factor |
| 8 | C22×C8×C121 | C2×C3 | 1 | 1 | yes | characteristic |
| 9 | C2×C16×C121 | C2×C3 | 1 | 6 | yes | direct product factor |
| 10 | C23×C3×C121 | C8 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C4×C121 | C8 | 1 | 3 | yes | — |
| 12 | C2×C3×C4×C121 | C2×C4 | 1 | 3 | yes | — |
| 13 | C3×C8×C121 | C2×C4 | 1 | 3 | yes | — |
| 14 | C3×C8×C121 | C23 | 1 | 1 | yes | characteristic |
| 15 | C22×C3×C11×C16 | C11 | 1 | 1 | yes | maximal, characteristic |
| 16 | C22×C4×C121 | C3×C4 | 1 | 1 | yes | characteristic |
| 17 | C2×C8×C121 | C3×C4 | 1 | 3 | yes | — |
| 18 | C2×C8×C121 | C22×C3 | 1 | 3 | yes | — |
| 19 | C16×C121 | C22×C3 | 1 | 4 | yes | direct product factor |
| 20 | C22×C3×C121 | C16 | 1 | 4 | yes | direct product factor |
| 21 | C22×C3×C121 | C2×C8 | 1 | 3 | yes | — |
| 22 | C3×C4×C121 | C2×C8 | 1 | 3 | yes | — |
| 23 | C3×C4×C121 | C22×C4 | 1 | 1 | yes | characteristic |
| 24 | C22×C3×C8×C11 | C2×C11 | 1 | 1 | yes | characteristic |
| 25 | C2×C3×C11×C16 | C2×C11 | 1 | 6 | yes | — |
| 26 | C23×C121 | C3×C8 | 1 | 1 | yes | characteristic |
| 27 | C2×C4×C121 | C3×C8 | 1 | 3 | yes | — |
| 28 | C2×C4×C121 | C2×C3×C4 | 1 | 3 | yes | — |
| 29 | C8×C121 | C2×C3×C4 | 1 | 3 | yes | — |
| 30 | C8×C121 | C23×C3 | 1 | 1 | yes | characteristic |
| 31 | C2×C3×C121 | C2×C16 | 1 | 6 | yes | direct product factor |
| 32 | C2×C3×C121 | C22×C8 | 1 | 1 | yes | characteristic |
| 33 | C22×C11×C16 | C3×C11 | 1 | 1 | yes | characteristic |
| 34 | C22×C3×C4×C11 | C4×C11 | 1 | 1 | yes | characteristic |
| 35 | C2×C3×C8×C11 | C4×C11 | 1 | 3 | yes | — |
| 36 | C2×C3×C8×C11 | C22×C11 | 1 | 3 | yes | — |
| 37 | C3×C11×C16 | C22×C11 | 1 | 4 | yes | — |
| 38 | C22×C121 | C3×C16 | 1 | 4 | yes | direct product factor |
| 39 | C22×C121 | C2×C3×C8 | 1 | 3 | yes | — |
| 40 | C4×C121 | C2×C3×C8 | 1 | 3 | yes | — |
| 41 | C4×C121 | C22×C3×C4 | 1 | 1 | yes | characteristic |
| 42 | C3×C121 | C22×C16 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 43 | C22×C8×C11 | C2×C3×C11 | 1 | 1 | yes | characteristic |
| 44 | C2×C11×C16 | C2×C3×C11 | 1 | 6 | yes | — |
| 45 | C23×C3×C11 | C8×C11 | 1 | 1 | yes | characteristic |
| 46 | C2×C3×C4×C11 | C8×C11 | 1 | 3 | yes | — |
| 47 | C2×C3×C4×C11 | C2×C4×C11 | 1 | 3 | yes | — |
| 48 | C3×C8×C11 | C2×C4×C11 | 1 | 3 | yes | — |
| 49 | C3×C8×C11 | C23×C11 | 1 | 1 | yes | characteristic |
| 50 | C2×C121 | C2×C3×C16 | 1 | 6 | yes | direct product factor |
| 51 | C2×C121 | C22×C3×C8 | 1 | 1 | yes | characteristic |
| 52 | C22×C3×C16 | C121 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 53 | C22×C4×C11 | C3×C4×C11 | 1 | 1 | yes | characteristic |
| 54 | C2×C8×C11 | C3×C4×C11 | 1 | 3 | yes | — |
| 55 | C2×C8×C11 | C22×C3×C11 | 1 | 3 | yes | — |
| 56 | C11×C16 | C22×C3×C11 | 1 | 4 | yes | — |
| 57 | C22×C3×C11 | C11×C16 | 1 | 4 | yes | — |
| 58 | C22×C3×C11 | C2×C8×C11 | 1 | 3 | yes | — |
| 59 | C3×C4×C11 | C2×C8×C11 | 1 | 3 | yes | — |
| 60 | C3×C4×C11 | C22×C4×C11 | 1 | 1 | yes | characteristic |
| 61 | C121 | C22×C3×C16 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 62 | C22×C3×C8 | C2×C121 | 1 | 1 | yes | characteristic |
| 63 | C2×C3×C16 | C2×C121 | 1 | 6 | yes | direct product factor |
| 64 | C23×C11 | C3×C8×C11 | 1 | 1 | yes | characteristic |
| 65 | C2×C4×C11 | C3×C8×C11 | 1 | 3 | yes | — |
| 66 | C2×C4×C11 | C2×C3×C4×C11 | 1 | 3 | yes | — |
| 67 | C8×C11 | C2×C3×C4×C11 | 1 | 3 | yes | — |
| 68 | C8×C11 | C23×C3×C11 | 1 | 1 | yes | characteristic |
| 69 | C2×C3×C11 | C2×C11×C16 | 1 | 6 | yes | — |
| 70 | C2×C3×C11 | C22×C8×C11 | 1 | 1 | yes | characteristic |
| 71 | C22×C16 | C3×C121 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 72 | C22×C3×C4 | C4×C121 | 1 | 1 | yes | characteristic |
| 73 | C2×C3×C8 | C4×C121 | 1 | 3 | yes | — |
| 74 | C2×C3×C8 | C22×C121 | 1 | 3 | yes | — |
| 75 | C3×C16 | C22×C121 | 1 | 4 | yes | direct product factor |
| 76 | C22×C11 | C3×C11×C16 | 1 | 4 | yes | — |
| 77 | C22×C11 | C2×C3×C8×C11 | 1 | 3 | yes | — |
| 78 | C4×C11 | C2×C3×C8×C11 | 1 | 3 | yes | — |
| 79 | C4×C11 | C22×C3×C4×C11 | 1 | 1 | yes | characteristic |
| 80 | C3×C11 | C22×C11×C16 | 1 | 1 | yes | characteristic |
| 81 | C22×C8 | C2×C3×C121 | 1 | 1 | yes | characteristic |
| 82 | C2×C16 | C2×C3×C121 | 1 | 6 | yes | direct product factor |
| 83 | C23×C3 | C8×C121 | 1 | 1 | yes | characteristic |
| 84 | C2×C3×C4 | C8×C121 | 1 | 3 | yes | — |
| 85 | C2×C3×C4 | C2×C4×C121 | 1 | 3 | yes | — |
| 86 | C3×C8 | C2×C4×C121 | 1 | 3 | yes | — |
| 87 | C3×C8 | C23×C121 | 1 | 1 | yes | characteristic |
| 88 | C2×C11 | C2×C3×C11×C16 | 1 | 6 | yes | — |
| 89 | C2×C11 | C22×C3×C8×C11 | 1 | 1 | yes | characteristic |
| 90 | C22×C4 | C3×C4×C121 | 1 | 1 | yes | characteristic |
| 91 | C2×C8 | C3×C4×C121 | 1 | 3 | yes | — |
| 92 | C2×C8 | C22×C3×C121 | 1 | 3 | yes | — |
| 93 | C16 | C22×C3×C121 | 1 | 4 | yes | direct product factor |
| 94 | C22×C3 | C16×C121 | 1 | 4 | yes | direct product factor |
| 95 | C22×C3 | C2×C8×C121 | 1 | 3 | yes | — |
| 96 | C3×C4 | C2×C8×C121 | 1 | 3 | yes | — |
| 97 | C3×C4 | C22×C4×C121 | 1 | 1 | yes | characteristic |
| 98 | C11 | C22×C3×C11×C16 | 1 | 1 | yes | characteristic |
| 99 | C23 | C3×C8×C121 | 1 | 1 | yes | characteristic |
| 100 | C2×C4 | C3×C8×C121 | 1 | 3 | yes | — |
| 101 | C2×C4 | C2×C3×C4×C121 | 1 | 3 | yes | — |
| 102 | C8 | C2×C3×C4×C121 | 1 | 3 | yes | — |
| 103 | C8 | C23×C3×C121 | 1 | 1 | yes | characteristic |
| 104 | C2×C3 | C2×C16×C121 | 1 | 6 | yes | direct product factor |
| 105 | C2×C3 | C22×C8×C121 | 1 | 1 | yes | characteristic |
| 106 | C22 | C3×C16×C121 | 1 | 4 | yes | direct product factor |
| 107 | C22 | C2×C3×C8×C121 | 1 | 3 | yes | — |
| 108 | C4 | C2×C3×C8×C121 | 1 | 3 | yes | — |
| 109 | C4 | C22×C3×C4×C121 | 1 | 1 | yes | characteristic |
| 110 | C3 | C22×C16×C121 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 111 | C2 | C2×C3×C16×C121 | 1 | 6 | yes | direct product factor |
| 112 | C2 | C22×C3×C8×C121 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta