Finite Group DatabasebetaFinite Group DatabasebetaOrder 4352.
1800 conjugacy classes of subgroups.
The subgroup lattice has 222 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #14545330 | C2 | 1 | 3 | yes | maximal |
| 2 | #14545318 | C2 | 1 | 12 | yes | maximal |
| 3 | #14545284 | C2 | 1 | 12 | yes | maximal |
| 4 | #125×#595 | C2 | 1 | 3 | yes | maximal |
| 5 | D17×#564 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #11763760 | C22 | 1 | 6 | yes | — |
| 7 | #11763760 | C22 | 1 | 12 | yes | — |
| 8 | #11763760 | C22 | 1 | 12 | yes | — |
| 9 | #11763758 | C22 | 1 | 12 | yes | — |
| 10 | #11763758 | C22 | 1 | 12 | yes | — |
| 11 | #11763758 | C22 | 1 | 12 | yes | — |
| 12 | #11763758 | C22 | 1 | 12 | yes | — |
| 13 | #11763758 | C22 | 1 | 12 | yes | — |
| 14 | #11763754 | C22 | 1 | 4 | yes | — |
| 15 | #11763753 | C22 | 1 | 3 | yes | — |
| 16 | C4⋊C4×#595 | C22 | 1 | 6 | yes | — |
| 17 | C4⋊C4×#595 | C22 | 1 | 12 | yes | — |
| 18 | C4⋊C4×#595 | C22 | 1 | 12 | yes | — |
| 19 | C4×C4⋊#595 | C22 | 1 | 12 | yes | — |
| 20 | C42×#595 | C22 | 1 | 1 | yes | characteristic |
| 21 | C17⋊#564 | C4 | 1 | 3 | yes | complemented |
| 22 | C17⋊#564 | C4 | 1 | 12 | yes | complemented |
| 23 | #125×D17 | C22 | 1 | 3 | yes | — |
| 24 | #125×D17 | C22 | 1 | 12 | yes | — |
| 25 | C17×#564 | C4 | 1 | 1 | yes | characteristic, complemented |
| 26 | C2×C4⋊#595 | C23 | 1 | 12 | yes | — |
| 27 | C2×C4⋊#595 | C23 | 1 | 12 | yes | — |
| 28 | C2×C4⋊#595 | C23 | 1 | 6 | yes | — |
| 29 | C2×C4×#595 | C23 | 1 | 3 | yes | — |
| 30 | C2×C4×#595 | C23 | 1 | 3 | yes | — |
| 31 | C2×C4×#595 | C23 | 1 | 12 | yes | — |
| 32 | C2×C4×#595 | C23 | 1 | 12 | yes | — |
| 33 | C17×#125 | C2×C4 | 1 | 3 | yes | — |
| 34 | C17×#125 | C2×C4 | 1 | 12 | yes | — |
| 35 | C4⋊C4×D17 | C23 | 1 | 6 | yes | — |
| 36 | C4⋊C4×D17 | C23 | 1 | 12 | yes | — |
| 37 | C4⋊C4×D17 | C23 | 1 | 12 | yes | — |
| 38 | #10587576 | C2×C4 | 1 | 6 | yes | — |
| 39 | #10587576 | C2×C4 | 1 | 12 | yes | — |
| 40 | #10587576 | C2×C4 | 1 | 12 | yes | — |
| 41 | #10587575 | C2×C4 | 1 | 12 | yes | — |
| 42 | #10587575 | C2×C4 | 1 | 12 | yes | — |
| 43 | #10587575 | C2×C4 | 1 | 12 | yes | — |
| 44 | #10587575 | C2×C4 | 1 | 12 | yes | — |
| 45 | #10587575 | C2×C4 | 1 | 12 | yes | — |
| 46 | C42×D17 | C23 | 1 | 1 | yes | characteristic |
| 47 | C42×D17 | C23 | 1 | 4 | yes | — |
| 48 | #10587556 | C2×C4 | 1 | 3 | yes | — |
| 49 | #10587556 | C2×C4 | 1 | 12 | yes | — |
| 50 | (C2×#595)⋊C4 | C23 | 1 | 12 | yes | — |
| 51 | (C2×#595)⋊C4 | C23 | 1 | 12 | yes | — |
| 52 | (C2×#595)⋊C4 | C23 | 1 | 12 | yes | — |
| 53 | (C2×#595)⋊C4 | C23 | 1 | 12 | yes | — |
| 54 | (C2×#595)⋊C4 | C23 | 1 | 12 | yes | — |
| 55 | C22×#595 | C24 | 1 | 12 | yes | — |
| 56 | C22×#595 | C24 | 1 | 1 | yes | characteristic |
| 57 | C22×#595 | C24 | 1 | 3 | yes | — |
| 58 | C2×C4×D17 | C24 | 1 | 3 | yes | — |
| 59 | C2×C4×D17 | C24 | 1 | 12 | yes | — |
| 60 | C4⋊#595 | — | 2 | 24 | no | — |
| 61 | C4⋊#595 | — | 4 | 12 | no | — |
| 62 | C4⋊#595 | — | 4 | 6 | no | — |
| 63 | C4×#595 | — | 2 | 6 | no | — |
| 64 | C4×#595 | — | 4 | 12 | no | — |
| 65 | C4×#595 | — | 4 | 12 | no | — |
| 66 | C4×#595 | — | 4 | 3 | no | — |
| 67 | C4⋊C4×C17 | C22×C4 | 1 | 12 | yes | — |
| 68 | C4⋊C4×C17 | C22×C4 | 1 | 6 | yes | — |
| 69 | C4⋊C4×C17 | C22×C4 | 1 | 12 | yes | — |
| 70 | C42×C17 | C22×C4 | 1 | 1 | yes | characteristic |
| 71 | C42×C17 | C22×C4 | 1 | 4 | yes | — |
| 72 | C4⋊Dic17 | C22×C4 | 1 | 6 | yes | — |
| 73 | C4⋊Dic17 | C22×C4 | 1 | 12 | yes | — |
| 74 | C4⋊Dic17 | C22×C4 | 1 | 12 | yes | — |
| 75 | #63225 | C22×C4 | 1 | 12 | yes | — |
| 76 | #63225 | C22×C4 | 1 | 12 | yes | — |
| 77 | #63225 | C22×C4 | 1 | 12 | yes | — |
| 78 | #63225 | C22×C4 | 1 | 12 | yes | — |
| 79 | #63225 | C22×C4 | 1 | 12 | yes | — |
| 80 | C4×Dic17 | C22×C4 | 1 | 3 | yes | — |
| 81 | C4×Dic17 | C22×C4 | 1 | 12 | yes | — |
| 82 | #34526 | — | 17 | 1 | no | maximal, Sylow |
| 83 | C22×D17 | C25 | 1 | 1 | yes | characteristic |
| 84 | C2×#595 | — | 2 | 6 | no | — |
| 85 | C2×#595 | — | 2 | 3 | no | — |
| 86 | C2×#595 | — | 2 | 12 | no | — |
| 87 | C2×#595 | — | 2 | 12 | no | — |
| 88 | C2×#595 | — | 2 | 12 | no | — |
| 89 | C2×#595 | 2−1+4 | 1 | 6 | yes | — |
| 90 | C2×C4×C17 | C23×C4 | 1 | 3 | yes | — |
| 91 | C2×C4×C17 | C23×C4 | 1 | 12 | yes | — |
| 92 | C2×Dic17 | C23×C4 | 1 | 3 | yes | — |
| 93 | C2×Dic17 | C23×C4 | 1 | 12 | yes | — |
| 94 | C4×D17 | — | 2 | 3 | no | — |
| 95 | C4×D17 | — | 2 | 12 | no | — |
| 96 | C4×D17 | — | 2 | 12 | no | — |
| 97 | C4×D17 | — | 2 | 3 | no | — |
| 98 | C2×#564 | — | 17 | 1 | no | — |
| 99 | #2382 | — | 17 | 3 | no | — |
| 100 | #2370 | — | 17 | 12 | no | — |
| 101 | #2336 | — | 17 | 12 | no | — |
| 102 | C4×#125 | — | 17 | 3 | no | — |
| 103 | C22×C17 | C24×C4 | 1 | 1 | yes | characteristic |
| 104 | C2×D17 | C2×2−1+4 | 1 | 3 | yes | — |
| 105 | C2×D17 | #585 | 1 | 3 | yes | — |
| 106 | #595 | — | 2 | 6 | no | — |
| 107 | #595 | — | 4 | 12 | no | — |
| 108 | #595 | — | 4 | 1 | no | — |
| 109 | C4×C17 | — | 2 | 3 | no | — |
| 110 | C4×C17 | — | 2 | 12 | no | — |
| 111 | Dic17 | — | 2 | 3 | no | — |
| 112 | Dic17 | — | 2 | 12 | no | — |
| 113 | #564 | — | 17 | 12 | no | — |
| 114 | #564 | — | 17 | 3 | no | — |
| 115 | #564 | #595 | 1 | 1 | yes | characteristic, complemented |
| 116 | C2×#125 | — | 17 | 3 | no | — |
| 117 | C2×#125 | — | 17 | 12 | no | — |
| 118 | #389 | — | 17 | 6 | no | — |
| 119 | #389 | — | 17 | 12 | no | — |
| 120 | #389 | — | 17 | 12 | no | — |
| 121 | #387 | — | 17 | 12 | no | — |
| 122 | #387 | — | 17 | 12 | no | — |
| 123 | #387 | — | 17 | 12 | no | — |
| 124 | #387 | — | 17 | 12 | no | — |
| 125 | #387 | — | 17 | 12 | no | — |
| 126 | #383 | — | 17 | 4 | no | — |
| 127 | #382 | — | 17 | 3 | no | — |
| 128 | C4×C4⋊C4 | — | 17 | 6 | no | — |
| 129 | C4×C4⋊C4 | — | 17 | 12 | no | — |
| 130 | C4×C4⋊C4 | — | 17 | 12 | no | — |
| 131 | C4×C4⋊C4 | — | 17 | 12 | no | — |
| 132 | C43 | — | 17 | 1 | no | — |
| 133 | C2×C17 | C4×2−1+4 | 1 | 3 | yes | — |
| 134 | D17 | #3553 | 1 | 3 | yes | — |
| 135 | D17 | #3556 | 1 | 1 | yes | characteristic |
| 136 | #125 | — | 17 | 12 | no | — |
| 137 | #125 | — | 17 | 12 | no | — |
| 138 | #125 | — | 17 | 12 | no | — |
| 139 | #125 | — | 17 | 12 | no | — |
| 140 | #125 | — | 17 | 12 | no | — |
| 141 | #125 | — | 17 | 6 | no | — |
| 142 | #125 | — | 17 | 12 | no | — |
| 143 | #125 | — | 17 | 12 | no | — |
| 144 | #125 | — | 17 | 3 | no | — |
| 145 | #125 | — | 17 | 12 | no | — |
| 146 | #125 | C2×#595 | 1 | 3 | yes | — |
| 147 | #125 | C2×#595 | 1 | 12 | yes | — |
| 148 | C2×C4⋊C4 | — | 17 | 6 | no | — |
| 149 | C2×C4⋊C4 | — | 17 | 12 | no | — |
| 150 | C2×C4⋊C4 | — | 17 | 12 | no | — |
| 151 | C2×C4⋊C4 | — | 17 | 12 | no | — |
| 152 | C2×C4⋊C4 | — | 17 | 6 | no | — |
| 153 | C2×C4⋊C4 | — | 17 | 12 | no | — |
| 154 | C2×C42 | — | 17 | 1 | no | — |
| 155 | C2×C42 | — | 17 | 4 | no | — |
| 156 | C2×C42 | — | 17 | 3 | no | — |
| 157 | C2×C42 | — | 17 | 3 | no | — |
| 158 | C2×C42 | — | 17 | 12 | no | — |
| 159 | C2×C42 | — | 17 | 12 | no | — |
| 160 | #95 | — | 17 | 12 | no | — |
| 161 | #95 | — | 17 | 12 | no | — |
| 162 | #95 | — | 17 | 12 | no | — |
| 163 | #95 | — | 17 | 12 | no | — |
| 164 | #95 | — | 17 | 12 | no | — |
| 165 | C17 | #34526 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 166 | C22×C4 | — | 17 | 3 | no | — |
| 167 | C22×C4 | — | 17 | 3 | no | — |
| 168 | C22×C4 | — | 17 | 12 | no | — |
| 169 | C22×C4 | — | 17 | 1 | no | — |
| 170 | C22×C4 | — | 17 | 12 | no | — |
| 171 | C4⋊C4 | — | 17 | 12 | no | — |
| 172 | C4⋊C4 | — | 17 | 12 | no | — |
| 173 | C4⋊C4 | — | 17 | 12 | no | — |
| 174 | C4⋊C4 | — | 17 | 12 | no | — |
| 175 | C4⋊C4 | — | 17 | 12 | no | — |
| 176 | C4⋊C4 | — | 17 | 6 | no | — |
| 177 | C4⋊C4 | — | 17 | 12 | no | — |
| 178 | C4⋊C4 | — | 17 | 12 | no | — |
| 179 | C4⋊C4 | — | 34 | 24 | no | — |
| 180 | C4⋊C4 | — | 68 | 12 | no | — |
| 181 | C4⋊C4 | — | 68 | 6 | no | — |
| 182 | C4⋊C4 | C22×#595 | 1 | 6 | yes | — |
| 183 | C4⋊C4 | C22×#595 | 1 | 12 | yes | — |
| 184 | C4⋊C4 | C22×#595 | 1 | 12 | yes | — |
| 185 | C42 | — | 17 | 3 | no | — |
| 186 | C42 | — | 17 | 12 | no | — |
| 187 | C42 | — | 34 | 6 | no | — |
| 188 | C42 | — | 68 | 12 | no | — |
| 189 | C42 | — | 68 | 12 | no | — |
| 190 | C42 | — | 68 | 3 | no | — |
| 191 | C42 | C22×#595 | 1 | 1 | yes | characteristic |
| 192 | C42 | C22×#595 | 1 | 4 | yes | — |
| 193 | C23 | — | 17 | 1 | no | — |
| 194 | C2×C4 | — | 17 | 6 | no | — |
| 195 | C2×C4 | — | 17 | 3 | no | — |
| 196 | C2×C4 | — | 17 | 12 | no | — |
| 197 | C2×C4 | — | 34 | 3 | no | — |
| 198 | C2×C4 | — | 34 | 12 | no | — |
| 199 | C2×C4 | — | 34 | 3 | no | — |
| 200 | C2×C4 | — | 34 | 12 | no | — |
| 201 | C2×C4 | — | 34 | 6 | no | — |
| 202 | C2×C4 | — | 34 | 3 | no | — |
| 203 | C2×C4 | — | 34 | 12 | no | — |
| 204 | C2×C4 | — | 34 | 12 | no | — |
| 205 | C2×C4 | — | 34 | 12 | no | — |
| 206 | C2×C4 | C23×#595 | 1 | 3 | yes | — |
| 207 | C2×C4 | C23×#595 | 1 | 12 | yes | — |
| 208 | C22 | — | 17 | 3 | no | — |
| 209 | C22 | — | 17 | 3 | no | — |
| 210 | C22 | C24×#595 | 1 | 1 | yes | characteristic |
| 211 | C4 | — | 2 | 3 | no | — |
| 212 | C4 | — | 2 | 12 | no | — |
| 213 | C4 | — | 34 | 6 | no | — |
| 214 | C4 | — | 34 | 3 | no | — |
| 215 | C4 | — | 34 | 12 | no | — |
| 216 | C4 | — | 68 | 12 | no | — |
| 217 | C4 | — | 68 | 1 | no | — |
| 218 | C2 | — | 17 | 3 | no | — |
| 219 | C2 | — | 17 | 1 | no | — |
| 220 | C2 | 2−1+4×#595 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×#595 |
| commutator, socle | C22×C17 | C24×C4 |
| Fitting | C17×#564 | C4 |
| prime radical, cosocle | C22×D17 | C25 |
| hypercenter | #564 | #595 |
| hypocenter | C17 | #34526 |
| LCS | G ▷ C22×C17 ▷ C17 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 21. C17⋊#564 | C4 | semidirect |
| 22. C17⋊#564 | C4 | semidirect |
| 25. C17×#564 | C4 | semidirect |
| 115. #564 | #595 | semidirect |
| 165. C17 | #34526 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14545330 | 2 |
| #14545318 | 2 |
| #14545284 | 2 |
| #125×#595 | 2 |
| D17×#564 | 2 |
| #34526 | 17 |