← C67×#20338 #30699589
Subgroups — C67×#20338 #30699589
Order 17152.
2036 conjugacy classes of subgroups.
The subgroup lattice has 188 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#568×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C67×#3428 | C2 | 1 | 2 | yes | maximal |
| 3 | C2×H(3,ℤ/4)×C67 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | C22×#41×C67 | C22 | 1 | 1 | yes | characteristic |
| 5 | #568×C67 | — | 2 | 8 | no | — |
| 6 | C2×#131×C67 | C4 | 1 | 8 | yes | complemented |
| 7 | C22×M16×C67 | C22 | 1 | 6 | yes | — |
| 8 | #520×C67 | — | 2 | 8 | no | — |
| 9 | #519×C67 | — | 2 | 2 | no | — |
| 10 | #518×C67 | — | 2 | 6 | no | — |
| 11 | C4×#41×C67 | C4 | 1 | 16 | yes | complemented |
| 12 | C2×#117×C67 | C22 | 1 | 12 | yes | — |
| 13 | H(3,ℤ/4)×C67 | C22 | 1 | 16 | yes | — |
| 14 | C2×#41×C67 | C2×C4 | 1 | 8 | yes | — |
| 15 | C2×#41×C67 | C2×C4 | 1 | 4 | yes | — |
| 16 | C2×#41×C67 | D4 | 1 | 8 | yes | — |
| 17 | C2×#41×C67 | Q8 | 1 | 4 | yes | — |
| 18 | C22×Q8×C67 | D4 | 1 | 1 | yes | characteristic |
| 19 | C22×D4×C67 | D4 | 1 | 3 | yes | — |
| 20 | C23×C4×C67 | C23 | 1 | 3 | yes | — |
| 21 | #131×C67 | — | 2 | 32 | no | — |
| 22 | C2×M16×C67 | — | 2 | 24 | no | — |
| 23 | C2×M16×C67 | C2×C4 | 1 | 12 | yes | complemented |
| 24 | C2×M16×C67 | C23 | 1 | 4 | yes | — |
| 25 | C22×C8×C67 | C2×C4 | 1 | 12 | yes | complemented |
| 26 | C4×Q8×C67 | — | 2 | 16 | no | — |
| 27 | C4×D4×C67 | — | 2 | 48 | no | — |
| 28 | #117×C67 | — | 2 | 12 | no | — |
| 29 | #117×C67 | — | 2 | 12 | no | — |
| 30 | #117×C67 | C2×C4 | 1 | 24 | yes | complemented |
| 31 | #117×C67 | C23 | 1 | 8 | yes | — |
| 32 | C2×C4⋊C4×C67 | — | 2 | 12 | no | — |
| 33 | C2×C22⋊C4×C67 | — | 2 | 12 | no | — |
| 34 | C2×C42×C67 | C2×C4 | 1 | 24 | yes | complemented |
| 35 | C24×C67 | C2×D4 | 1 | 3 | yes | — |
| 36 | #41×C67 | C42 | 1 | 16 | yes | — |
| 37 | #41×C67 | C22⋊C4 | 1 | 16 | yes | — |
| 38 | #41×C67 | C4⋊C4 | 1 | 32 | yes | — |
| 39 | C2×Q8×C67 | C22⋊C4 | 1 | 8 | yes | — |
| 40 | C2×Q8×C67 | C4⋊C4 | 1 | 4 | yes | — |
| 41 | C2×D4×C67 | C22⋊C4 | 1 | 24 | yes | — |
| 42 | C2×D4×C67 | C4⋊C4 | 1 | 12 | yes | — |
| 43 | C22×C4×C67 | — | 2 | 24 | no | — |
| 44 | C22×C4×C67 | C22×C4 | 1 | 6 | yes | — |
| 45 | C22×C4×C67 | C22×C4 | 1 | 12 | yes | — |
| 46 | C22×C4×C67 | C2×D4 | 1 | 3 | yes | — |
| 47 | C22×C4×C67 | C2×D4 | 1 | 12 | yes | — |
| 48 | C22×C4×C67 | C2×Q8 | 1 | 6 | yes | — |
| 49 | C22×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 50 | M16×C67 | — | 2 | 24 | no | — |
| 51 | M16×C67 | — | 2 | 8 | no | — |
| 52 | C2×C8×C67 | — | 2 | 24 | no | — |
| 53 | C2×C8×C67 | C22×C4 | 1 | 8 | yes | — |
| 54 | C4⋊C4×C67 | — | 2 | 24 | no | — |
| 55 | C4⋊C4×C67 | — | 2 | 8 | no | — |
| 56 | C22⋊C4×C67 | — | 2 | 24 | no | — |
| 57 | C22⋊C4×C67 | — | 2 | 8 | no | — |
| 58 | C42×C67 | — | 2 | 24 | no | — |
| 59 | C42×C67 | C22×C4 | 1 | 16 | yes | — |
| 60 | C23×C67 | — | 8 | 3 | no | — |
| 61 | C23×C67 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 62 | C23×C67 | C2×C4⋊C4 | 1 | 6 | yes | — |
| 63 | C23×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 64 | Q8×C67 | #95 | 1 | 16 | yes | — |
| 65 | D4×C67 | #95 | 1 | 48 | yes | — |
| 66 | C2×C4×C67 | — | 2 | 16 | no | — |
| 67 | C2×C4×C67 | — | 8 | 24 | no | — |
| 68 | C2×C4×C67 | C2×C42 | 1 | 12 | yes | — |
| 69 | C2×C4×C67 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 70 | C2×C4×C67 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 71 | C2×C4×C67 | C2×C4⋊C4 | 1 | 24 | yes | — |
| 72 | C2×C4×C67 | C2×C4⋊C4 | 1 | 6 | yes | — |
| 73 | C2×C4×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 74 | C2×C4×C67 | C23×C4 | 1 | 2 | yes | — |
| 75 | C2×C4×C67 | C22×D4 | 1 | 2 | yes | — |
| 76 | C2×C4×C67 | C22×Q8 | 1 | 1 | yes | characteristic |
| 77 | C8×C67 | — | 2 | 8 | no | — |
| 78 | C22×C67 | — | 4 | 12 | no | — |
| 79 | C22×C67 | — | 4 | 1 | no | — |
| 80 | C22×C67 | — | 4 | 6 | no | — |
| 81 | C22×C67 | C2×#95 | 1 | 12 | yes | — |
| 82 | C22×C67 | C22×C22⋊C4 | 1 | 2 | yes | — |
| 83 | C22×C67 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C4×C67 | — | 4 | 16 | no | — |
| 85 | C4×C67 | C2×#95 | 1 | 12 | yes | — |
| 86 | C4×C67 | C22×C42 | 1 | 1 | yes | characteristic |
| 87 | C4×C67 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 88 | C4×C67 | C22×C4⋊C4 | 1 | 2 | yes | — |
| 89 | #20338 | C67 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 90 | C2×C67 | — | 2 | 2 | no | — |
| 91 | C2×C67 | — | 2 | 1 | no | — |
| 92 | C2×C67 | — | 2 | 12 | no | — |
| 93 | C2×C67 | C22×#95 | 1 | 1 | yes | characteristic |
| 94 | C2×#568 | C2×C67 | 1 | 1 | yes | characteristic |
| 95 | #3428 | C2×C67 | 1 | 2 | yes | — |
| 96 | C2×H(3,ℤ/4) | C2×C67 | 1 | 12 | yes | complemented |
| 97 | C67 | #20338 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 98 | C22×#41 | C22×C67 | 1 | 1 | yes | characteristic |
| 99 | #568 | — | 2 | 8 | no | — |
| 100 | C2×#131 | C4×C67 | 1 | 8 | yes | complemented |
| 101 | C22×M16 | C22×C67 | 1 | 6 | yes | — |
| 102 | #520 | — | 2 | 8 | no | — |
| 103 | #519 | — | 2 | 2 | no | — |
| 104 | #518 | — | 2 | 6 | no | — |
| 105 | C4×#41 | C4×C67 | 1 | 16 | yes | complemented |
| 106 | C2×#117 | C22×C67 | 1 | 12 | yes | — |
| 107 | H(3,ℤ/4) | C22×C67 | 1 | 16 | yes | — |
| 108 | C2×#41 | C2×C4×C67 | 1 | 8 | yes | — |
| 109 | C2×#41 | C2×C4×C67 | 1 | 4 | yes | — |
| 110 | C2×#41 | D4×C67 | 1 | 8 | yes | — |
| 111 | C2×#41 | Q8×C67 | 1 | 4 | yes | — |
| 112 | C22×Q8 | D4×C67 | 1 | 1 | yes | characteristic |
| 113 | C22×D4 | D4×C67 | 1 | 3 | yes | — |
| 114 | C23×C4 | C23×C67 | 1 | 3 | yes | — |
| 115 | #131 | — | 2 | 32 | no | — |
| 116 | C2×M16 | — | 2 | 24 | no | — |
| 117 | C2×M16 | C2×C4×C67 | 1 | 12 | yes | complemented |
| 118 | C2×M16 | C23×C67 | 1 | 4 | yes | — |
| 119 | C22×C8 | C2×C4×C67 | 1 | 12 | yes | complemented |
| 120 | C4×Q8 | — | 2 | 16 | no | — |
| 121 | C4×D4 | — | 2 | 48 | no | — |
| 122 | #117 | — | 2 | 12 | no | — |
| 123 | #117 | — | 2 | 12 | no | — |
| 124 | #117 | C2×C4×C67 | 1 | 24 | yes | complemented |
| 125 | #117 | C23×C67 | 1 | 8 | yes | — |
| 126 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 127 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 128 | C2×C42 | C2×C4×C67 | 1 | 24 | yes | complemented |
| 129 | C24 | C2×D4×C67 | 1 | 3 | yes | — |
| 130 | #41 | C42×C67 | 1 | 16 | yes | — |
| 131 | #41 | C22⋊C4×C67 | 1 | 16 | yes | — |
| 132 | #41 | C4⋊C4×C67 | 1 | 32 | yes | — |
| 133 | C2×Q8 | C22⋊C4×C67 | 1 | 8 | yes | — |
| 134 | C2×Q8 | C4⋊C4×C67 | 1 | 4 | yes | — |
| 135 | C2×D4 | C22⋊C4×C67 | 1 | 24 | yes | — |
| 136 | C2×D4 | C4⋊C4×C67 | 1 | 12 | yes | — |
| 137 | C22×C4 | — | 2 | 24 | no | — |
| 138 | C22×C4 | C22×C4×C67 | 1 | 6 | yes | — |
| 139 | C22×C4 | C22×C4×C67 | 1 | 12 | yes | — |
| 140 | C22×C4 | C2×D4×C67 | 1 | 3 | yes | — |
| 141 | C22×C4 | C2×D4×C67 | 1 | 12 | yes | — |
| 142 | C22×C4 | C2×Q8×C67 | 1 | 6 | yes | — |
| 143 | C22×C4 | C24×C67 | 1 | 1 | yes | characteristic |
| 144 | M16 | — | 2 | 24 | no | — |
| 145 | M16 | — | 2 | 8 | no | — |
| 146 | C2×C8 | — | 2 | 24 | no | — |
| 147 | C2×C8 | C22×C4×C67 | 1 | 8 | yes | — |
| 148 | C4⋊C4 | — | 2 | 24 | no | — |
| 149 | C4⋊C4 | — | 2 | 8 | no | — |
| 150 | C22⋊C4 | — | 2 | 24 | no | — |
| 151 | C22⋊C4 | — | 2 | 8 | no | — |
| 152 | C42 | — | 2 | 24 | no | — |
| 153 | C42 | C22×C4×C67 | 1 | 16 | yes | — |
| 154 | C23 | — | 8 | 3 | no | — |
| 155 | C23 | C2×C22⋊C4×C67 | 1 | 12 | yes | — |
| 156 | C23 | C2×C4⋊C4×C67 | 1 | 6 | yes | — |
| 157 | C23 | C22×D4×C67 | 1 | 1 | yes | characteristic |
| 158 | Q8 | #95×C67 | 1 | 16 | yes | — |
| 159 | D4 | #95×C67 | 1 | 48 | yes | — |
| 160 | C2×C4 | — | 2 | 16 | no | — |
| 161 | C2×C4 | — | 8 | 24 | no | — |
| 162 | C2×C4 | C2×C42×C67 | 1 | 12 | yes | — |
| 163 | C2×C4 | C2×C22⋊C4×C67 | 1 | 12 | yes | — |
| 164 | C2×C4 | C2×C22⋊C4×C67 | 1 | 12 | yes | — |
| 165 | C2×C4 | C2×C4⋊C4×C67 | 1 | 24 | yes | — |
| 166 | C2×C4 | C2×C4⋊C4×C67 | 1 | 6 | yes | — |
| 167 | C2×C4 | C23×C4×C67 | 1 | 1 | yes | characteristic |
| 168 | C2×C4 | C23×C4×C67 | 1 | 2 | yes | — |
| 169 | C2×C4 | C22×D4×C67 | 1 | 2 | yes | — |
| 170 | C2×C4 | C22×Q8×C67 | 1 | 1 | yes | characteristic |
| 171 | C8 | — | 2 | 8 | no | — |
| 172 | C22 | — | 4 | 12 | no | — |
| 173 | C22 | — | 4 | 1 | no | — |
| 174 | C22 | — | 4 | 6 | no | — |
| 175 | C22 | C2×#95×C67 | 1 | 12 | yes | — |
| 176 | C22 | C22×C22⋊C4×C67 | 1 | 2 | yes | — |
| 177 | C22 | C22×C4⋊C4×C67 | 1 | 1 | yes | characteristic |
| 178 | C4 | — | 4 | 16 | no | — |
| 179 | C4 | C2×#95×C67 | 1 | 12 | yes | — |
| 180 | C4 | C22×C42×C67 | 1 | 1 | yes | characteristic |
| 181 | C4 | C22×C22⋊C4×C67 | 1 | 1 | yes | characteristic |
| 182 | C4 | C22×C4⋊C4×C67 | 1 | 2 | yes | — |
| 183 | C2 | — | 2 | 2 | no | — |
| 184 | C2 | — | 2 | 1 | no | — |
| 185 | C2 | — | 2 | 12 | no | — |
| 186 | C2 | C22×#95×C67 | 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