← C67×#33297 #30712548
Subgroups — C67×#33297 #30712548
Order 17152.
750 conjugacy classes of subgroups.
The subgroup lattice has 218 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C67×#2868 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C67×#2859 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C67×#1993 | C2 | 1 | 4 | yes | maximal |
| 4 | C67×#1992 | C2 | 1 | 4 | yes | maximal |
| 5 | C67×#1834 | C2 | 1 | 4 | yes | maximal |
| 6 | C4×#97×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C2×C4⋊Q8×C67 | — | 2 | 2 | no | — |
| 8 | C2×C4⋊Q8×C67 | C22 | 1 | 2 | yes | — |
| 9 | C2×#125×C67 | C22 | 1 | 2 | yes | — |
| 10 | C2×#97×C67 | C22 | 1 | 2 | yes | — |
| 11 | C2×#97×C67 | C22 | 1 | 2 | yes | — |
| 12 | C2×C4×C8×C67 | C22 | 1 | 2 | yes | — |
| 13 | #389×C67 | — | 2 | 2 | no | — |
| 14 | #389×C67 | — | 2 | 2 | no | — |
| 15 | #384×C67 | — | 2 | 2 | no | — |
| 16 | #384×C67 | C22 | 1 | 2 | yes | — |
| 17 | #384×C67 | C22 | 1 | 2 | yes | — |
| 18 | #384×C67 | C22 | 1 | 2 | yes | — |
| 19 | #382×C67 | C22 | 1 | 2 | yes | — |
| 20 | C43×C67 | C22 | 1 | 1 | yes | characteristic |
| 21 | #340×C67 | C22 | 1 | 16 | yes | — |
| 22 | C22×Q8×C67 | — | 2 | 2 | no | — |
| 23 | C22×C8×C67 | C23 | 1 | 4 | yes | — |
| 24 | C4⋊Q8×C67 | — | 2 | 8 | no | — |
| 25 | C4⋊Q8×C67 | — | 4 | 8 | no | — |
| 26 | #125×C67 | — | 2 | 8 | no | — |
| 27 | C2×C4⋊C4×C67 | — | 2 | 4 | no | — |
| 28 | C2×C4⋊C4×C67 | — | 2 | 4 | no | — |
| 29 | C2×C4⋊C4×C67 | — | 2 | 4 | no | — |
| 30 | C2×C4⋊C4×C67 | — | 2 | 2 | no | — |
| 31 | C2×C4⋊C4×C67 | — | 2 | 2 | no | — |
| 32 | C2×C4⋊C4×C67 | — | 2 | 2 | no | — |
| 33 | C2×C4⋊C4×C67 | C23 | 1 | 4 | yes | — |
| 34 | C2×C4⋊C4×C67 | C23 | 1 | 4 | yes | — |
| 35 | C2×C42×C67 | D4 | 1 | 2 | yes | — |
| 36 | C2×C42×C67 | D4 | 1 | 2 | yes | — |
| 37 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 38 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 40 | #97×C67 | — | 2 | 4 | no | — |
| 41 | #97×C67 | — | 2 | 4 | no | — |
| 42 | C4×C8×C67 | Q8 | 1 | 8 | yes | complemented |
| 43 | #95×C67 | — | 2 | 2 | no | — |
| 44 | #95×C67 | — | 2 | 2 | no | — |
| 45 | C2×Q8×C67 | — | 4 | 8 | no | — |
| 46 | C2×Q8×C67 | — | 4 | 4 | no | — |
| 47 | C22×C4×C67 | — | 2 | 4 | no | — |
| 48 | C22×C4×C67 | — | 2 | 4 | no | — |
| 49 | C22×C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 50 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 51 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | C22×C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 53 | C22×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 54 | C2×C8×C67 | — | 2 | 4 | no | — |
| 55 | C2×C8×C67 | — | 2 | 4 | no | — |
| 56 | C2×C8×C67 | C2×Q8 | 1 | 8 | yes | — |
| 57 | C4⋊C4×C67 | — | 2 | 8 | no | — |
| 58 | C4⋊C4×C67 | — | 2 | 8 | no | — |
| 59 | C4⋊C4×C67 | — | 4 | 8 | no | — |
| 60 | C4⋊C4×C67 | — | 4 | 8 | no | — |
| 61 | C4⋊C4×C67 | — | 4 | 4 | no | — |
| 62 | C4⋊C4×C67 | — | 4 | 4 | no | — |
| 63 | C4⋊C4×C67 | — | 4 | 8 | no | — |
| 64 | C4⋊C4×C67 | — | 4 | 4 | no | — |
| 65 | C42×C67 | — | 2 | 4 | no | — |
| 66 | C42×C67 | — | 2 | 4 | no | — |
| 67 | C42×C67 | C2×Q8 | 1 | 4 | yes | — |
| 68 | C42×C67 | #41 | 1 | 4 | yes | — |
| 69 | C42×C67 | #41 | 1 | 4 | yes | — |
| 70 | C23×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 71 | Q8×C67 | — | 8 | 8 | no | — |
| 72 | C2×C4×C67 | — | 2 | 2 | no | — |
| 73 | C2×C4×C67 | — | 2 | 2 | no | — |
| 74 | C2×C4×C67 | — | 2 | 2 | no | — |
| 75 | C2×C4×C67 | — | 2 | 2 | no | — |
| 76 | C2×C4×C67 | — | 4 | 4 | no | — |
| 77 | C2×C4×C67 | — | 4 | 4 | no | — |
| 78 | C2×C4×C67 | — | 4 | 8 | no | — |
| 79 | C2×C4×C67 | — | 4 | 4 | no | — |
| 80 | C2×C4×C67 | — | 4 | 4 | no | — |
| 81 | C2×C4×C67 | #124 | 1 | 2 | yes | — |
| 82 | C2×C4×C67 | #124 | 1 | 4 | yes | — |
| 83 | C2×C4×C67 | #124 | 1 | 2 | yes | — |
| 84 | C2×C4×C67 | C4⋊Q8 | 1 | 4 | yes | — |
| 85 | C2×C4×C67 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 86 | C2×C4×C67 | #137 | 1 | 4 | yes | — |
| 87 | C2×C4×C67 | C22×Q8 | 1 | 2 | yes | — |
| 88 | C2×C4×C67 | C2×#41 | 1 | 2 | yes | — |
| 89 | C2×C4×C67 | C2×#41 | 1 | 2 | yes | — |
| 90 | C8×C67 | — | 2 | 8 | no | — |
| 91 | C22×C67 | C2×#124 | 1 | 1 | yes | characteristic |
| 92 | C22×C67 | C2×#124 | 1 | 1 | yes | characteristic |
| 93 | C22×C67 | C2×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 94 | C22×C67 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 95 | C22×C67 | C2×#137 | 1 | 2 | yes | — |
| 96 | C4×C67 | — | 2 | 2 | no | — |
| 97 | C4×C67 | — | 2 | 2 | no | — |
| 98 | C4×C67 | — | 2 | 4 | no | — |
| 99 | C4×C67 | — | 8 | 4 | no | — |
| 100 | C4×C67 | — | 8 | 4 | no | — |
| 101 | C4×C67 | #501 | 1 | 8 | yes | — |
| 102 | C4×C67 | #533 | 1 | 4 | yes | — |
| 103 | #33297 | C67 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 104 | C2×C67 | #2862 | 1 | 1 | yes | characteristic |
| 105 | C2×C67 | #3138 | 1 | 1 | yes | characteristic |
| 106 | C2×C67 | #3139 | 1 | 1 | yes | characteristic |
| 107 | C2×C67 | #3141 | 1 | 2 | yes | — |
| 108 | C2×C67 | C2×#501 | 1 | 2 | yes | — |
| 109 | #2868 | C2×C67 | 1 | 1 | yes | characteristic |
| 110 | #2859 | C2×C67 | 1 | 1 | yes | characteristic |
| 111 | #1993 | C2×C67 | 1 | 4 | yes | — |
| 112 | #1992 | C2×C67 | 1 | 4 | yes | — |
| 113 | #1834 | C2×C67 | 1 | 4 | yes | — |
| 114 | C4×#97 | C2×C67 | 1 | 1 | yes | characteristic |
| 115 | C67 | #33297 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 116 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 117 | C2×C4⋊Q8 | C22×C67 | 1 | 2 | yes | — |
| 118 | C2×#125 | C22×C67 | 1 | 2 | yes | — |
| 119 | C2×#97 | C22×C67 | 1 | 2 | yes | — |
| 120 | C2×#97 | C22×C67 | 1 | 2 | yes | — |
| 121 | C2×C4×C8 | C22×C67 | 1 | 2 | yes | — |
| 122 | #389 | — | 2 | 2 | no | — |
| 123 | #389 | — | 2 | 2 | no | — |
| 124 | #384 | — | 2 | 2 | no | — |
| 125 | #384 | C22×C67 | 1 | 2 | yes | — |
| 126 | #384 | C22×C67 | 1 | 2 | yes | — |
| 127 | #384 | C22×C67 | 1 | 2 | yes | — |
| 128 | #382 | C22×C67 | 1 | 2 | yes | — |
| 129 | C43 | C22×C67 | 1 | 1 | yes | characteristic |
| 130 | #340 | C22×C67 | 1 | 16 | yes | — |
| 131 | C22×Q8 | — | 2 | 2 | no | — |
| 132 | C22×C8 | C23×C67 | 1 | 4 | yes | — |
| 133 | C4⋊Q8 | — | 2 | 8 | no | — |
| 134 | C4⋊Q8 | — | 4 | 8 | no | — |
| 135 | #125 | — | 2 | 8 | no | — |
| 136 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 137 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 138 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 139 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 140 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 141 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 142 | C2×C4⋊C4 | C23×C67 | 1 | 4 | yes | — |
| 143 | C2×C4⋊C4 | C23×C67 | 1 | 4 | yes | — |
| 144 | C2×C42 | D4×C67 | 1 | 2 | yes | — |
| 145 | C2×C42 | D4×C67 | 1 | 2 | yes | — |
| 146 | C2×C42 | C23×C67 | 1 | 1 | yes | characteristic |
| 147 | C2×C42 | C23×C67 | 1 | 1 | yes | characteristic |
| 148 | C2×C42 | C23×C67 | 1 | 1 | yes | characteristic |
| 149 | #97 | — | 2 | 4 | no | — |
| 150 | #97 | — | 2 | 4 | no | — |
| 151 | C4×C8 | Q8×C67 | 1 | 8 | yes | complemented |
| 152 | #95 | — | 2 | 2 | no | — |
| 153 | #95 | — | 2 | 2 | no | — |
| 154 | C2×Q8 | — | 4 | 8 | no | — |
| 155 | C2×Q8 | — | 4 | 4 | no | — |
| 156 | C22×C4 | — | 2 | 4 | no | — |
| 157 | C22×C4 | — | 2 | 4 | no | — |
| 158 | C22×C4 | C2×D4×C67 | 1 | 1 | yes | characteristic |
| 159 | C22×C4 | C2×D4×C67 | 1 | 1 | yes | characteristic |
| 160 | C22×C4 | C2×D4×C67 | 1 | 2 | yes | — |
| 161 | C22×C4 | C2×D4×C67 | 1 | 2 | yes | — |
| 162 | C22×C4 | C24×C67 | 1 | 1 | yes | characteristic |
| 163 | C2×C8 | — | 2 | 4 | no | — |
| 164 | C2×C8 | — | 2 | 4 | no | — |
| 165 | C2×C8 | C2×Q8×C67 | 1 | 8 | yes | — |
| 166 | C4⋊C4 | — | 2 | 8 | no | — |
| 167 | C4⋊C4 | — | 2 | 8 | no | — |
| 168 | C4⋊C4 | — | 4 | 8 | no | — |
| 169 | C4⋊C4 | — | 4 | 8 | no | — |
| 170 | C4⋊C4 | — | 4 | 4 | no | — |
| 171 | C4⋊C4 | — | 4 | 8 | no | — |
| 172 | C4⋊C4 | — | 4 | 4 | no | — |
| 173 | C4⋊C4 | — | 4 | 4 | no | — |
| 174 | C42 | — | 2 | 4 | no | — |
| 175 | C42 | — | 2 | 4 | no | — |
| 176 | C42 | C2×Q8×C67 | 1 | 4 | yes | — |
| 177 | C42 | #41×C67 | 1 | 4 | yes | — |
| 178 | C42 | #41×C67 | 1 | 4 | yes | — |
| 179 | C23 | C22×D4×C67 | 1 | 1 | yes | characteristic |
| 180 | Q8 | — | 8 | 8 | no | — |
| 181 | C2×C4 | — | 2 | 2 | no | — |
| 182 | C2×C4 | — | 2 | 2 | no | — |
| 183 | C2×C4 | — | 2 | 2 | no | — |
| 184 | C2×C4 | — | 2 | 2 | no | — |
| 185 | C2×C4 | — | 4 | 4 | no | — |
| 186 | C2×C4 | — | 4 | 4 | no | — |
| 187 | C2×C4 | — | 4 | 8 | no | — |
| 188 | C2×C4 | — | 4 | 4 | no | — |
| 189 | C2×C4 | — | 4 | 4 | no | — |
| 190 | C2×C4 | #124×C67 | 1 | 2 | yes | — |
| 191 | C2×C4 | #124×C67 | 1 | 4 | yes | — |
| 192 | C2×C4 | #124×C67 | 1 | 2 | yes | — |
| 193 | C2×C4 | C4⋊Q8×C67 | 1 | 4 | yes | — |
| 194 | C2×C4 | AGL(1,ℤ/8)×C67 | 1 | 4 | yes | — |
| 195 | C2×C4 | #137×C67 | 1 | 4 | yes | — |
| 196 | C2×C4 | C22×Q8×C67 | 1 | 2 | yes | — |
| 197 | C2×C4 | C2×#41×C67 | 1 | 2 | yes | — |
| 198 | C2×C4 | C2×#41×C67 | 1 | 2 | yes | — |
| 199 | C8 | — | 2 | 8 | no | — |
| 200 | C22 | C2×#124×C67 | 1 | 1 | yes | characteristic |
| 201 | C22 | C2×#124×C67 | 1 | 1 | yes | characteristic |
| 202 | C22 | C2×C4⋊Q8×C67 | 1 | 1 | yes | characteristic |
| 203 | C22 | C2×AGL(1,ℤ/8)×C67 | 1 | 2 | yes | — |
| 204 | C22 | C2×#137×C67 | 1 | 2 | yes | — |
| 205 | C4 | — | 2 | 2 | no | — |
| 206 | C4 | — | 2 | 2 | no | — |
| 207 | C4 | — | 2 | 4 | no | — |
| 208 | C4 | — | 8 | 4 | no | — |
| 209 | C4 | — | 8 | 4 | no | — |
| 210 | C4 | #501×C67 | 1 | 8 | yes | — |
| 211 | C4 | #533×C67 | 1 | 4 | yes | — |
| 212 | C2 | C67×#2862 | 1 | 1 | yes | characteristic |
| 213 | C2 | C67×#3138 | 1 | 1 | yes | characteristic |
| 214 | C2 | C67×#3139 | 1 | 1 | yes | characteristic |
| 215 | C2 | C67×#3141 | 1 | 2 | yes | — |
| 216 | C2 | C2×#501×C67 | 1 | 2 | yes | — |
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