Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
425 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #11717474 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #11717452 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#87839 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11717397 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #11717395 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #11717386 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11717384 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #11717381 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #87902 | C22 | 1 | 1 | yes | characteristic |
| 10 | C5×#711 | — | 2 | 2 | no | — |
| 11 | C2×#6039 | C22 | 1 | 4 | yes | — |
| 12 | #87880 | C22 | 1 | 4 | yes | — |
| 13 | #87879 | — | 2 | 4 | no | — |
| 14 | #87873 | C22 | 1 | 2 | yes | — |
| 15 | #87872 | C22 | 1 | 2 | yes | — |
| 16 | D5×Dic10 | — | 2 | 2 | no | — |
| 17 | #87867 | C22 | 1 | 2 | yes | — |
| 18 | C2×#6034 | C22 | 1 | 2 | yes | — |
| 19 | #87863 | C22 | 1 | 1 | yes | characteristic |
| 20 | #87862 | C22 | 1 | 1 | yes | characteristic |
| 21 | #87839 | C22 | 1 | 1 | yes | characteristic |
| 22 | #87839 | C22 | 1 | 4 | yes | — |
| 23 | #87839 | C22 | 1 | 2 | yes | — |
| 24 | #87839 | C22 | 1 | 2 | yes | — |
| 25 | #87838 | C22 | 1 | 2 | yes | — |
| 26 | #87838 | C22 | 1 | 4 | yes | — |
| 27 | #87837 | C22 | 1 | 1 | yes | characteristic |
| 28 | C52×D4 | D4 | 1 | 1 | yes | characteristic |
| 29 | C22⋊(C5⋊D5) | D4 | 1 | 2 | yes | — |
| 30 | C2×#1039 | C23 | 1 | 2 | yes | — |
| 31 | C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 32 | #6045 | D4 | 1 | 1 | yes | characteristic |
| 33 | C5×C22⋊D5 | — | 2 | 4 | no | — |
| 34 | C2×C5×Dic5 | — | 2 | 4 | no | — |
| 35 | C4×C5×D5 | — | 2 | 2 | no | — |
| 36 | C5×Dic10 | — | 2 | 2 | no | — |
| 37 | #6039 | — | 2 | 4 | no | — |
| 38 | #6039 | C23 | 1 | 4 | yes | — |
| 39 | #6039 | C23 | 1 | 2 | yes | — |
| 40 | #6038 | — | 2 | 2 | no | — |
| 41 | #6037 | C23 | 1 | 2 | yes | — |
| 42 | #6036 | — | 2 | 2 | no | — |
| 43 | D5×Dic5 | — | 2 | 4 | no | — |
| 44 | #6034 | C23 | 1 | 1 | yes | characteristic |
| 45 | #6034 | C23 | 1 | 1 | yes | characteristic |
| 46 | #6034 | C23 | 1 | 2 | yes | — |
| 47 | #4150 | — | 10 | 2 | no | — |
| 48 | #4131 | — | 10 | 1 | no | — |
| 49 | C22×C52 | C2×D4 | 1 | 2 | yes | — |
| 50 | C2×C5⋊D5 | C2×D4 | 1 | 1 | yes | characteristic |
| 51 | C2×C5×D5 | — | 2 | 2 | no | — |
| 52 | C4×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 53 | #1039 | C2×D4 | 1 | 2 | yes | — |
| 54 | #1039 | C24 | 1 | 1 | yes | characteristic |
| 55 | C5×Dic5 | — | 2 | 4 | no | — |
| 56 | C5×Dic5 | — | 2 | 2 | no | — |
| 57 | C5×#41 | — | 10 | 2 | no | — |
| 58 | Q8×D5 | — | 10 | 2 | no | — |
| 59 | Q8×D5 | — | 10 | 2 | no | — |
| 60 | #711 | — | 2 | 2 | no | — |
| 61 | #711 | — | 10 | 4 | no | — |
| 62 | #711 | — | 10 | 1 | no | — |
| 63 | #711 | — | 10 | 2 | no | — |
| 64 | #711 | — | 10 | 4 | no | — |
| 65 | #709 | — | 10 | 2 | no | — |
| 66 | #709 | — | 10 | 4 | no | — |
| 67 | C2×Dic10 | — | 10 | 4 | no | — |
| 68 | C2×Dic10 | — | 10 | 2 | no | — |
| 69 | #704 | — | 10 | 2 | no | — |
| 70 | C2×#170 | — | 10 | 2 | no | — |
| 71 | #700 | — | 10 | 1 | no | — |
| 72 | D5⋊C8 | — | 10 | 1 | no | — |
| 73 | #578 | — | 25 | 1 | no | maximal, Sylow |
| 74 | C2×C52 | — | 2 | 2 | no | — |
| 75 | C2×C52 | C22×D4 | 1 | 1 | yes | characteristic |
| 76 | C5⋊D5 | — | 2 | 1 | no | — |
| 77 | C5×D5 | — | 4 | 2 | no | — |
| 78 | C5×Q8 | — | 10 | 2 | no | — |
| 79 | C5×D4 | — | 2 | 2 | no | — |
| 80 | C5×D4 | — | 2 | 1 | no | — |
| 81 | C5×D4 | — | 10 | 4 | no | — |
| 82 | C2×C4×C5 | — | 10 | 2 | no | — |
| 83 | C2×C4×C5 | — | 10 | 4 | no | — |
| 84 | C22⋊D5 | — | 2 | 4 | no | — |
| 85 | C22⋊D5 | — | 10 | 2 | no | — |
| 86 | C22⋊D5 | — | 10 | 4 | no | — |
| 87 | C22⋊D5 | — | 10 | 2 | no | — |
| 88 | C22⋊D5 | — | 10 | 2 | no | — |
| 89 | C2×Dic5 | — | 2 | 4 | no | — |
| 90 | C2×Dic5 | — | 10 | 2 | no | — |
| 91 | C2×Dic5 | — | 10 | 4 | no | — |
| 92 | C2×Dic5 | — | 10 | 4 | no | — |
| 93 | D20 | — | 10 | 2 | no | — |
| 94 | C4×D5 | — | 2 | 2 | no | — |
| 95 | C4×D5 | — | 10 | 1 | no | — |
| 96 | C4×D5 | — | 10 | 2 | no | — |
| 97 | C4×D5 | — | 10 | 4 | no | — |
| 98 | C4×D5 | — | 10 | 4 | no | — |
| 99 | Dic10 | — | 2 | 2 | no | — |
| 100 | Dic10 | — | 10 | 2 | no | — |
| 101 | Dic10 | — | 10 | 4 | no | — |
| 102 | Dic10 | — | 10 | 2 | no | — |
| 103 | Dic10 | — | 10 | 1 | no | — |
| 104 | Dic10 | — | 10 | 4 | no | — |
| 105 | Dic10 | — | 10 | 4 | no | — |
| 106 | #170 | — | 10 | 1 | no | — |
| 107 | #170 | — | 10 | 1 | no | — |
| 108 | #170 | — | 10 | 2 | no | — |
| 109 | 2−1+4 | — | 25 | 2 | no | — |
| 110 | #137 | — | 25 | 2 | no | — |
| 111 | #137 | — | 25 | 4 | no | — |
| 112 | #135 | — | 25 | 1 | no | — |
| 113 | #135 | — | 25 | 2 | no | — |
| 114 | C2×Q16 | — | 25 | 1 | no | — |
| 115 | C2×Q16 | — | 25 | 2 | no | — |
| 116 | #131 | — | 25 | 1 | no | — |
| 117 | C52 | #578 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 118 | C22×C5 | — | 2 | 4 | no | — |
| 119 | C22×C5 | — | 2 | 2 | no | — |
| 120 | C22×C5 | — | 10 | 2 | no | — |
| 121 | C2×D5 | — | 2 | 2 | no | — |
| 122 | C2×D5 | — | 10 | 2 | no | — |
| 123 | C2×D5 | — | 10 | 1 | no | — |
| 124 | C4×C5 | — | 2 | 2 | no | — |
| 125 | C4×C5 | — | 2 | 1 | no | — |
| 126 | C4×C5 | — | 10 | 2 | no | — |
| 127 | C4×C5 | — | 10 | 4 | no | — |
| 128 | Dic5 | — | 2 | 4 | no | — |
| 129 | Dic5 | — | 2 | 2 | no | — |
| 130 | Dic5 | — | 10 | 4 | no | — |
| 131 | Dic5 | — | 10 | 2 | no | — |
| 132 | Dic5 | — | 10 | 1 | no | — |
| 133 | Dic5 | — | 10 | 2 | no | — |
| 134 | #41 | — | 10 | 2 | no | — |
| 135 | #41 | — | 25 | 2 | no | — |
| 136 | #41 | — | 25 | 1 | no | — |
| 137 | #41 | — | 25 | 4 | no | — |
| 138 | #41 | — | 50 | 4 | no | — |
| 139 | C2×Q8 | — | 25 | 4 | no | — |
| 140 | C2×Q8 | — | 25 | 2 | no | — |
| 141 | C2×Q8 | — | 50 | 2 | no | — |
| 142 | Q16 | — | 25 | 1 | no | — |
| 143 | Q16 | — | 25 | 2 | no | — |
| 144 | Q16 | — | 25 | 4 | no | — |
| 145 | Q16 | — | 25 | 2 | no | — |
| 146 | SD16 | — | 25 | 2 | no | — |
| 147 | SD16 | — | 25 | 4 | no | — |
| 148 | D8 | — | 25 | 1 | no | — |
| 149 | M16 | — | 25 | 1 | no | — |
| 150 | M16 | — | 25 | 2 | no | — |
| 151 | C2×C8 | — | 25 | 1 | no | — |
| 152 | C2×C8 | — | 25 | 2 | no | — |
| 153 | C2×C5 | — | 2 | 2 | no | — |
| 154 | C2×C5 | — | 2 | 1 | no | — |
| 155 | C2×C5 | — | 4 | 4 | no | — |
| 156 | C2×C5 | — | 4 | 2 | no | — |
| 157 | C2×C5 | — | 20 | 2 | no | — |
| 158 | D5 | — | 4 | 2 | no | — |
| 159 | D5 | — | 20 | 2 | no | — |
| 160 | D5 | — | 20 | 1 | no | — |
| 161 | Q8 | — | 10 | 2 | no | — |
| 162 | Q8 | — | 25 | 2 | no | — |
| 163 | Q8 | — | 25 | 4 | no | — |
| 164 | Q8 | — | 25 | 1 | no | — |
| 165 | Q8 | — | 50 | 4 | no | — |
| 166 | D4 | — | 10 | 4 | no | — |
| 167 | D4 | — | 25 | 2 | no | — |
| 168 | D4 | — | 25 | 2 | no | — |
| 169 | D4 | — | 50 | 2 | no | — |
| 170 | D4 | D52⋊C2 | 1 | 1 | yes | characteristic |
| 171 | C2×C4 | — | 10 | 4 | no | — |
| 172 | C2×C4 | — | 10 | 2 | no | — |
| 173 | C2×C4 | — | 25 | 2 | no | — |
| 174 | C2×C4 | — | 25 | 1 | no | — |
| 175 | C2×C4 | — | 50 | 2 | no | — |
| 176 | C2×C4 | — | 50 | 4 | no | — |
| 177 | C8 | — | 25 | 1 | no | — |
| 178 | C8 | — | 25 | 1 | no | — |
| 179 | C8 | — | 25 | 2 | no | — |
| 180 | C5 | — | 2 | 2 | no | — |
| 181 | C5 | — | 2 | 1 | no | — |
| 182 | C22 | — | 10 | 2 | no | — |
| 183 | C22 | — | 25 | 1 | no | — |
| 184 | C22 | C2×D52⋊C2 | 1 | 2 | yes | — |
| 185 | C4 | — | 10 | 2 | no | — |
| 186 | C4 | — | 10 | 4 | no | — |
| 187 | C4 | — | 25 | 1 | no | — |
| 188 | C4 | — | 25 | 2 | no | — |
| 189 | C4 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 190 | C2 | — | 2 | 2 | no | — |
| 191 | C2 | — | 20 | 2 | no | — |
| 192 | C2 | — | 50 | 1 | no | — |
| 193 | C2 | C22×D52⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D52⋊C2 |
| commutator, cosocle | #1039 | C24 |
| Fitting | C52×D4 | D4 |
| socle | C2×C52 | C22×D4 |
| hypercenter | D4 | D52⋊C2 |
| hypocenter | C52 | #578 |
| LCS | G ▷ #1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11717474 | C2 | semidirect |
| 2. #11717452 | C2 | semidirect |
| 3. C2×#87839 | C2 | semidirect |
| 4. #11717397 | C2 | semidirect |
| 5. #11717395 | C2 | semidirect |
| 6. #11717386 | C2 | semidirect |
| 7. #11717384 | C2 | semidirect |
| 8. #11717381 | C2 | semidirect |
| 117. C52 | #578 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717474 | 2 |
| #11717452 | 2 |
| C2×#87839 | 2 |
| #11717397 | 2 |
| #11717395 | 2 |
| #11717386 | 2 |
| #11717384 | 2 |
| #11717381 | 2 |
| #578 | 25 |