Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
172 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 29 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 | #11716940 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#87770 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11716891 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11716793 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11716792 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C5×#709 | — | 2 | 1 | no | — |
| 7 | C2×C5×D20 | — | 2 | 1 | no | — |
| 8 | C2×#6029 | C22 | 1 | 1 | yes | characteristic |
| 9 | #87775 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #87772 | — | 2 | 1 | no | — |
| 11 | #87770 | — | 2 | 1 | no | — |
| 12 | #87770 | C22 | 1 | 2 | yes | — |
| 13 | #87770 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | #87765 | C22 | 1 | 2 | yes | complemented |
| 15 | C22×C5×D5 | — | 4 | 1 | no | — |
| 16 | C2×C4×C52 | D4 | 1 | 1 | yes | characteristic |
| 17 | C5×C22⋊D5 | — | 4 | 1 | no | — |
| 18 | C5×D20 | — | 2 | 2 | no | — |
| 19 | C5×D20 | — | 2 | 1 | no | — |
| 20 | C5×D20 | — | 4 | 1 | no | — |
| 21 | C4×C5×D5 | — | 4 | 1 | no | — |
| 22 | C5×Dic10 | — | 2 | 1 | no | — |
| 23 | #6029 | D4 | 1 | 1 | yes | characteristic |
| 24 | #6029 | C23 | 1 | 1 | yes | characteristic |
| 25 | #4096 | — | 10 | 1 | no | — |
| 26 | C2×#686 | — | 10 | 1 | no | — |
| 27 | #3998 | — | 10 | 1 | no | — |
| 28 | C22×C52 | D8 | 1 | 1 | yes | characteristic |
| 29 | C2×C5×D5 | — | 4 | 1 | no | — |
| 30 | C2×C5×D5 | — | 4 | 2 | no | — |
| 31 | C2×C5×D5 | — | 8 | 2 | no | — |
| 32 | C4×C52 | D8 | 1 | 1 | yes | characteristic |
| 33 | C4×C52 | C2×D4 | 1 | 1 | yes | characteristic |
| 34 | C5×Dic5 | — | 4 | 1 | no | — |
| 35 | C5×#41 | — | 10 | 1 | no | — |
| 36 | C2×C5×D4 | — | 10 | 1 | no | — |
| 37 | #709 | — | 2 | 1 | no | — |
| 38 | C2×D20 | — | 2 | 1 | no | — |
| 39 | #689 | — | 10 | 1 | no | — |
| 40 | #688 | — | 10 | 1 | no | — |
| 41 | #687 | — | 10 | 1 | no | — |
| 42 | #686 | — | 10 | 1 | no | — |
| 43 | #686 | — | 10 | 2 | no | — |
| 44 | #686 | — | 10 | 2 | no | — |
| 45 | C2×#168 | — | 10 | 1 | no | — |
| 46 | C2×#168 | — | 10 | 1 | no | — |
| 47 | C2×#168 | — | 10 | 1 | no | — |
| 48 | #674 | — | 10 | 2 | no | — |
| 49 | #509 | — | 25 | 1 | no | maximal, Sylow |
| 50 | C2×C52 | — | 2 | 1 | no | — |
| 51 | C2×C52 | C2×D8 | 1 | 1 | yes | characteristic |
| 52 | C5×D5 | — | 8 | 1 | no | — |
| 53 | C5×D5 | — | 8 | 2 | no | — |
| 54 | C23×C5 | — | 20 | 1 | no | — |
| 55 | C22×D5 | — | 4 | 1 | no | — |
| 56 | C5×Q8 | — | 10 | 1 | no | — |
| 57 | C5×D4 | — | 10 | 2 | no | — |
| 58 | C5×D4 | — | 10 | 1 | no | — |
| 59 | C5×D4 | — | 20 | 1 | no | — |
| 60 | C5×D4 | — | 20 | 1 | no | — |
| 61 | C2×C4×C5 | — | 2 | 1 | no | — |
| 62 | C2×C4×C5 | — | 2 | 1 | no | — |
| 63 | C2×C4×C5 | — | 2 | 1 | no | — |
| 64 | C2×C4×C5 | — | 20 | 1 | no | — |
| 65 | C22⋊D5 | — | 4 | 1 | no | — |
| 66 | D20 | — | 2 | 2 | no | — |
| 67 | D20 | — | 2 | 1 | no | — |
| 68 | D20 | — | 4 | 1 | no | — |
| 69 | C4×D5 | — | 4 | 1 | no | — |
| 70 | Dic10 | — | 2 | 1 | no | — |
| 71 | #168 | — | 10 | 1 | no | — |
| 72 | #168 | — | 10 | 1 | no | — |
| 73 | #168 | — | 10 | 1 | no | — |
| 74 | #168 | — | 10 | 1 | no | — |
| 75 | #168 | — | 10 | 1 | no | — |
| 76 | #168 | — | 10 | 1 | no | — |
| 77 | #135 | — | 25 | 1 | no | — |
| 78 | C2×D8 | — | 25 | 1 | no | — |
| 79 | SD32 | — | 25 | 2 | no | — |
| 80 | D16 | — | 25 | 2 | no | — |
| 81 | M32 | — | 25 | 1 | no | — |
| 82 | C52 | #509 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 83 | C22×C5 | — | 2 | 1 | no | — |
| 84 | C22×C5 | — | 2 | 1 | no | — |
| 85 | C22×C5 | — | 2 | 1 | no | — |
| 86 | C22×C5 | — | 20 | 1 | no | — |
| 87 | C22×C5 | — | 20 | 2 | no | — |
| 88 | C22×C5 | — | 40 | 2 | no | — |
| 89 | C2×D5 | — | 4 | 1 | no | — |
| 90 | C2×D5 | — | 4 | 2 | no | — |
| 91 | C2×D5 | — | 8 | 2 | no | — |
| 92 | C4×C5 | — | 2 | 1 | no | — |
| 93 | C4×C5 | — | 2 | 1 | no | — |
| 94 | C4×C5 | — | 2 | 1 | no | — |
| 95 | C4×C5 | — | 2 | 1 | no | — |
| 96 | C4×C5 | — | 2 | 1 | no | — |
| 97 | C4×C5 | — | 2 | 1 | no | — |
| 98 | C4×C5 | — | 20 | 1 | no | — |
| 99 | Dic5 | — | 4 | 1 | no | — |
| 100 | #41 | — | 10 | 1 | no | — |
| 101 | C2×D4 | — | 10 | 1 | no | — |
| 102 | Q16 | — | 25 | 1 | no | — |
| 103 | SD16 | — | 50 | 1 | no | — |
| 104 | D8 | — | 25 | 1 | no | — |
| 105 | D8 | — | 25 | 2 | no | — |
| 106 | D8 | — | 50 | 1 | no | — |
| 107 | C2×C8 | — | 25 | 1 | no | — |
| 108 | C16 | — | 25 | 2 | no | — |
| 109 | C2×C5 | — | 2 | 2 | no | — |
| 110 | C2×C5 | — | 2 | 1 | no | — |
| 111 | C2×C5 | — | 2 | 1 | no | — |
| 112 | C2×C5 | — | 2 | 1 | no | — |
| 113 | C2×C5 | — | 4 | 1 | no | — |
| 114 | C2×C5 | — | 4 | 1 | no | — |
| 115 | C2×C5 | — | 40 | 2 | no | — |
| 116 | C2×C5 | — | 40 | 1 | no | — |
| 117 | D5 | — | 8 | 2 | no | — |
| 118 | D5 | — | 8 | 1 | no | — |
| 119 | C23 | — | 20 | 1 | no | — |
| 120 | Q8 | — | 10 | 1 | no | — |
| 121 | D4 | — | 10 | 2 | no | — |
| 122 | D4 | — | 10 | 1 | no | — |
| 123 | D4 | — | 20 | 1 | no | — |
| 124 | D4 | — | 20 | 1 | no | — |
| 125 | C2×C4 | — | 20 | 1 | no | — |
| 126 | C2×C4 | D52⋊C2 | 1 | 1 | yes | characteristic |
| 127 | C8 | — | 25 | 1 | no | — |
| 128 | C8 | — | 25 | 1 | no | — |
| 129 | C5 | — | 2 | 1 | no | — |
| 130 | C5 | — | 2 | 1 | no | — |
| 131 | C5 | — | 2 | 1 | no | — |
| 132 | C22 | — | 20 | 1 | no | — |
| 133 | C22 | — | 20 | 2 | no | — |
| 134 | C22 | — | 40 | 2 | no | — |
| 135 | C22 | #87837 | 1 | 1 | yes | characteristic |
| 136 | C4 | — | 20 | 1 | no | — |
| 137 | C4 | #87837 | 1 | 1 | yes | characteristic |
| 138 | C4 | C2×D52⋊C2 | 1 | 1 | yes | characteristic |
| 139 | C2 | — | 2 | 1 | no | — |
| 140 | C2 | — | 40 | 2 | no | — |
| 141 | C2 | — | 40 | 1 | no | — |
| 142 | C2 | C2×#87837 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11716940 | C2 | semidirect |
| 2. C2×#87770 | C2 | semidirect |
| 3. #11716891 | C2 | semidirect |
| 4. #11716793 | C2 | semidirect |
| 5. #11716792 | C2 | semidirect |
| 9. #87775 | C22 | semidirect |
| 13. #87770 | C22 | semidirect |
| 14. #87765 | C22 | semidirect |
| 82. C52 | #509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11716940 | 2 |
| C2×#87770 | 2 |
| #11716891 | 2 |
| #11716793 | 2 |
| #11716792 | 2 |
| #509 | 25 |