← C22×C3×#1136 #13061123
Subgroups — C22×C3×#1136 #13061123
Order 1344.
596 conjugacy classes of subgroups.
The subgroup lattice has 124 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C23×C3×C4×C7 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×#1136 | C2 | 1 | 12 | yes | maximal, direct product factor |
| 3 | C22×C3×C7⋊C8 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C22×#1136 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | C24×C3×C7 | C4 | 1 | 1 | yes | characteristic |
| 6 | C22×C3×C4×C7 | C4 | 1 | 6 | yes | — |
| 7 | C22×C3×C4×C7 | C4 | 1 | 1 | yes | characteristic |
| 8 | C22×C3×C4×C7 | C22 | 1 | 6 | yes | — |
| 9 | C22×C3×C4×C7 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#1136 | C22 | 1 | 16 | yes | direct product factor |
| 11 | C2×C3×C7⋊C8 | C22 | 1 | 12 | yes | complemented |
| 12 | C23×C4×C7 | C2×C3 | 1 | 1 | yes | characteristic |
| 13 | C2×#1136 | C2×C3 | 1 | 12 | yes | direct product factor |
| 14 | C22×C7⋊C8 | C2×C3 | 1 | 2 | yes | complemented |
| 15 | C22×C3×M16 | — | 7 | 1 | no | maximal, Hall |
| 16 | C23×C3×C7 | — | 2 | 4 | no | — |
| 17 | C23×C3×C7 | C2×C4 | 1 | 6 | yes | — |
| 18 | C23×C3×C7 | C2×C4 | 1 | 1 | yes | characteristic |
| 19 | C2×C3×C4×C7 | C2×C4 | 1 | 12 | yes | — |
| 20 | C2×C3×C4×C7 | C2×C4 | 1 | 6 | yes | — |
| 21 | C2×C3×C4×C7 | C2×C4 | 1 | 3 | yes | — |
| 22 | C2×C3×C4×C7 | C23 | 1 | 3 | yes | — |
| 23 | C2×C3×C4×C7 | C23 | 1 | 4 | yes | — |
| 24 | C3×C7⋊C8 | C23 | 1 | 8 | yes | complemented |
| 25 | C24×C7 | C3×C4 | 1 | 1 | yes | characteristic |
| 26 | C22×C4×C7 | C3×C4 | 1 | 6 | yes | — |
| 27 | C22×C4×C7 | C3×C4 | 1 | 1 | yes | characteristic |
| 28 | C22×C4×C7 | C22×C3 | 1 | 1 | yes | characteristic |
| 29 | C22×C4×C7 | C22×C3 | 1 | 6 | yes | — |
| 30 | #1136 | C22×C3 | 1 | 16 | yes | direct product factor |
| 31 | C2×C7⋊C8 | C22×C3 | 1 | 12 | yes | complemented |
| 32 | C23×C3×C4 | D7 | 1 | 1 | yes | characteristic |
| 33 | C2×C3×M16 | — | 7 | 12 | no | — |
| 34 | C22×C3×C8 | — | 7 | 2 | no | — |
| 35 | C22×C3×C7 | — | 2 | 12 | no | — |
| 36 | C22×C3×C7 | M16 | 1 | 4 | yes | complemented |
| 37 | C22×C3×C7 | C22×C4 | 1 | 4 | yes | — |
| 38 | C22×C3×C7 | C22×C4 | 1 | 3 | yes | — |
| 39 | C3×C4×C7 | C22×C4 | 1 | 3 | yes | — |
| 40 | C3×C4×C7 | C22×C4 | 1 | 4 | yes | — |
| 41 | C3×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 42 | C22×M16 | — | 7 | 1 | no | Sylow |
| 43 | C23×C7 | — | 2 | 4 | no | — |
| 44 | C23×C7 | C2×C3×C4 | 1 | 6 | yes | — |
| 45 | C23×C7 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 46 | C2×C4×C7 | C2×C3×C4 | 1 | 12 | yes | — |
| 47 | C2×C4×C7 | C2×C3×C4 | 1 | 3 | yes | — |
| 48 | C2×C4×C7 | C2×C3×C4 | 1 | 6 | yes | — |
| 49 | C2×C4×C7 | C23×C3 | 1 | 3 | yes | — |
| 50 | C2×C4×C7 | C23×C3 | 1 | 4 | yes | — |
| 51 | C7⋊C8 | C23×C3 | 1 | 8 | yes | complemented |
| 52 | C24×C3 | Dic7 | 1 | 1 | yes | characteristic |
| 53 | C22×C3×C4 | Dic7 | 1 | 1 | yes | characteristic |
| 54 | C22×C3×C4 | Dic7 | 1 | 6 | yes | — |
| 55 | C22×C3×C4 | C2×D7 | 1 | 6 | yes | — |
| 56 | C22×C3×C4 | C2×D7 | 1 | 1 | yes | characteristic |
| 57 | C3×M16 | — | 7 | 16 | no | — |
| 58 | C2×C3×C8 | — | 7 | 12 | no | — |
| 59 | C2×C3×C7 | — | 2 | 4 | no | — |
| 60 | C2×C3×C7 | C2×M16 | 1 | 6 | yes | complemented |
| 61 | C2×C3×C7 | C23×C4 | 1 | 1 | yes | characteristic |
| 62 | C23×C4 | C3×D7 | 1 | 1 | yes | characteristic |
| 63 | C2×M16 | — | 7 | 12 | no | — |
| 64 | C22×C8 | — | 7 | 2 | no | — |
| 65 | C22×C7 | — | 2 | 12 | no | — |
| 66 | C22×C7 | C3×M16 | 1 | 4 | yes | complemented |
| 67 | C22×C7 | C22×C3×C4 | 1 | 4 | yes | — |
| 68 | C22×C7 | C22×C3×C4 | 1 | 3 | yes | — |
| 69 | C4×C7 | C22×C3×C4 | 1 | 3 | yes | — |
| 70 | C4×C7 | C22×C3×C4 | 1 | 4 | yes | — |
| 71 | C4×C7 | C24×C3 | 1 | 1 | yes | characteristic |
| 72 | C23×C3 | — | 2 | 4 | no | — |
| 73 | C23×C3 | C2×Dic7 | 1 | 6 | yes | — |
| 74 | C23×C3 | C2×Dic7 | 1 | 1 | yes | characteristic |
| 75 | C2×C3×C4 | C2×Dic7 | 1 | 12 | yes | — |
| 76 | C2×C3×C4 | C2×Dic7 | 1 | 6 | yes | — |
| 77 | C2×C3×C4 | C2×Dic7 | 1 | 3 | yes | — |
| 78 | C2×C3×C4 | C22×D7 | 1 | 3 | yes | — |
| 79 | C2×C3×C4 | C22×D7 | 1 | 4 | yes | — |
| 80 | C3×C8 | — | 7 | 8 | no | — |
| 81 | C3×C7 | C22×M16 | 1 | 1 | yes | characteristic, complemented, Hall |
| 82 | C24 | C3×Dic7 | 1 | 1 | yes | characteristic |
| 83 | C22×C4 | C3×Dic7 | 1 | 6 | yes | — |
| 84 | C22×C4 | C3×Dic7 | 1 | 1 | yes | characteristic |
| 85 | C22×C4 | C2×C3×D7 | 1 | 1 | yes | characteristic |
| 86 | C22×C4 | C2×C3×D7 | 1 | 6 | yes | — |
| 87 | M16 | — | 7 | 16 | no | — |
| 88 | C2×C8 | — | 7 | 12 | no | — |
| 89 | C2×C7 | — | 2 | 4 | no | — |
| 90 | C2×C7 | C2×C3×M16 | 1 | 6 | yes | complemented |
| 91 | C2×C7 | C23×C3×C4 | 1 | 1 | yes | characteristic |
| 92 | C22×C3 | — | 2 | 12 | no | — |
| 93 | C22×C3 | #1136 | 1 | 4 | yes | direct product factor |
| 94 | C22×C3 | C22×Dic7 | 1 | 4 | yes | — |
| 95 | C22×C3 | C22×Dic7 | 1 | 3 | yes | — |
| 96 | C3×C4 | C22×Dic7 | 1 | 4 | yes | — |
| 97 | C3×C4 | C22×Dic7 | 1 | 3 | yes | — |
| 98 | C3×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 99 | C23 | — | 2 | 4 | no | — |
| 100 | C23 | C2×C3×Dic7 | 1 | 6 | yes | — |
| 101 | C23 | C2×C3×Dic7 | 1 | 1 | yes | characteristic |
| 102 | C2×C4 | C2×C3×Dic7 | 1 | 12 | yes | — |
| 103 | C2×C4 | C2×C3×Dic7 | 1 | 3 | yes | — |
| 104 | C2×C4 | C2×C3×Dic7 | 1 | 6 | yes | — |
| 105 | C2×C4 | C22×C3×D7 | 1 | 3 | yes | — |
| 106 | C2×C4 | C22×C3×D7 | 1 | 4 | yes | — |
| 107 | C8 | — | 7 | 8 | no | — |
| 108 | C7 | C22×C3×M16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 109 | C2×C3 | — | 2 | 4 | no | — |
| 110 | C2×C3 | C2×#1136 | 1 | 6 | yes | direct product factor |
| 111 | C2×C3 | C23×Dic7 | 1 | 1 | yes | characteristic |
| 112 | C22 | — | 2 | 12 | no | — |
| 113 | C22 | C3×#1136 | 1 | 4 | yes | direct product factor |
| 114 | C22 | C22×C3×Dic7 | 1 | 4 | yes | — |
| 115 | C22 | C22×C3×Dic7 | 1 | 3 | yes | — |
| 116 | C4 | C22×C3×Dic7 | 1 | 3 | yes | — |
| 117 | C4 | C22×C3×Dic7 | 1 | 4 | yes | — |
| 118 | C4 | C23×C3×D7 | 1 | 1 | yes | characteristic |
| 119 | C3 | C22×#1136 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 120 | C2 | — | 2 | 4 | no | — |
| 121 | C2 | C2×C3×#1136 | 1 | 6 | yes | direct product factor |
| 122 | C2 | C23×C3×Dic7 | 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