Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
138 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 15 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 | C5×Q8⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10602735 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | #4147⋊C3 | — | 4 | 1 | no | maximal |
| 4 | #2205⋊C3 | — | 5 | 1 | no | maximal, Hall |
| 5 | C5×Q82 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | #65384 | — | 3 | 1 | no | — |
| 7 | #64768 | — | 3 | 1 | no | — |
| 8 | C2×C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 9 | C2×C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 10 | C2×C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 11 | D5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 12 | C5⋊(#41⋊C3) | — | 4 | 1 | no | — |
| 13 | Q8⋊SL(2,𝔽3) | D5 | 1 | 1 | yes | characteristic, complemented |
| 14 | #4147 | A4 | 1 | 1 | yes | characteristic |
| 15 | C5×C4⋊Q8 | — | 3 | 1 | no | — |
| 16 | C5×C4⋊Q8 | — | 6 | 1 | no | — |
| 17 | C4×C5×Q8 | — | 6 | 1 | no | — |
| 18 | #4088 | — | 6 | 1 | no | — |
| 19 | #3968 | — | 3 | 1 | no | — |
| 20 | #3937 | — | 6 | 1 | no | — |
| 21 | #2205 | — | 5 | 1 | no | Sylow |
| 22 | C3×C22⋊D5 | — | 16 | 1 | no | — |
| 23 | C5×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 24 | C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 25 | C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 26 | 2−1+4⋊C3 | — | 20 | 1 | no | — |
| 27 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 28 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 29 | C2×C5×Q8 | — | 3 | 1 | no | — |
| 30 | C2×C5×Q8 | C2×A4 | 1 | 1 | yes | characteristic |
| 31 | #713 | — | 4 | 1 | no | — |
| 32 | Q8×D5 | — | 4 | 1 | no | — |
| 33 | #709 | — | 6 | 1 | no | — |
| 34 | C5×C4⋊C4 | — | 6 | 2 | no | — |
| 35 | C5×C4⋊C4 | — | 6 | 1 | no | — |
| 36 | C42×C5 | — | 3 | 1 | no | — |
| 37 | C42×C5 | — | 6 | 1 | no | — |
| 38 | #689 | — | 12 | 1 | no | — |
| 39 | #688 | — | 12 | 1 | no | — |
| 40 | #681 | — | 6 | 1 | no | — |
| 41 | Q82 | C3×D5 | 1 | 1 | yes | characteristic, complemented |
| 42 | #456 | — | 15 | 1 | no | — |
| 43 | #356 | — | 15 | 1 | no | — |
| 44 | C22×C3×C5 | — | 16 | 1 | no | — |
| 45 | C2×C3×D5 | — | 16 | 1 | no | — |
| 46 | C3×Dic5 | — | 16 | 1 | no | — |
| 47 | #41⋊C3 | — | 20 | 1 | no | — |
| 48 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 49 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 50 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 51 | C2×SL(2,𝔽3) | — | 20 | 1 | no | — |
| 52 | C5×Q8 | — | 2 | 1 | no | — |
| 53 | C5×Q8 | — | 4 | 1 | no | — |
| 54 | C5×Q8 | — | 6 | 1 | no | — |
| 55 | C5×Q8 | — | 8 | 1 | no | — |
| 56 | C5×Q8 | — | 12 | 1 | no | — |
| 57 | C2×C4×C5 | — | 3 | 1 | no | — |
| 58 | C2×C4×C5 | — | 6 | 1 | no | — |
| 59 | C2×C4×C5 | — | 6 | 1 | no | — |
| 60 | C22⋊D5 | — | 4 | 1 | no | — |
| 61 | D20 | — | 6 | 1 | no | — |
| 62 | C4×D5 | — | 12 | 1 | no | — |
| 63 | Dic10 | — | 6 | 1 | no | — |
| 64 | #168 | — | 12 | 1 | no | — |
| 65 | 2−1+4 | — | 5 | 1 | no | — |
| 66 | #137 | — | 30 | 1 | no | — |
| 67 | C4⋊Q8 | — | 3 | 1 | no | — |
| 68 | C4⋊Q8 | — | 6 | 1 | no | — |
| 69 | C4×Q8 | — | 6 | 1 | no | — |
| 70 | #104 | — | 30 | 1 | no | — |
| 71 | #101 | — | 15 | 1 | no | — |
| 72 | C2×C3×C5 | — | 16 | 1 | no | — |
| 73 | C2×C3×C5 | — | 32 | 1 | no | — |
| 74 | C3×D5 | — | 32 | 1 | no | — |
| 75 | C3×D4 | — | 80 | 1 | no | — |
| 76 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 77 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 78 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 79 | C22×C5 | #120⋊C3 | 1 | 1 | yes | characteristic |
| 80 | C2×D5 | — | 4 | 1 | no | — |
| 81 | C4×C5 | — | 6 | 1 | no | — |
| 82 | C4×C5 | — | 6 | 1 | no | — |
| 83 | C4×C5 | — | 6 | 1 | no | — |
| 84 | C4×C5 | — | 12 | 1 | no | — |
| 85 | Dic5 | — | 4 | 1 | no | — |
| 86 | #41 | — | 20 | 1 | no | — |
| 87 | #41 | — | 30 | 1 | no | — |
| 88 | C2×Q8 | — | 2 | 1 | no | — |
| 89 | C2×Q8 | — | 2 | 1 | no | — |
| 90 | C2×Q8 | — | 3 | 1 | no | — |
| 91 | C2×Q8 | — | 20 | 1 | no | — |
| 92 | C2×Q8 | D5×A4 | 1 | 1 | yes | characteristic |
| 93 | Q16 | — | 60 | 1 | no | — |
| 94 | SD16 | — | 60 | 1 | no | — |
| 95 | M16 | — | 30 | 1 | no | — |
| 96 | C4⋊C4 | — | 6 | 2 | no | — |
| 97 | C4⋊C4 | — | 6 | 1 | no | — |
| 98 | C42 | — | 3 | 1 | no | — |
| 99 | C42 | — | 6 | 1 | no | — |
| 100 | C3×C5 | — | 16 | 1 | no | Hall |
| 101 | C22×C3 | — | 16 | 1 | no | — |
| 102 | C22×C3 | — | 80 | 1 | no | — |
| 103 | C3×C4 | — | 80 | 1 | no | — |
| 104 | C2×C5 | — | 2 | 1 | no | — |
| 105 | C2×C5 | B(4,𝔽2)⋊C3 | 1 | 1 | yes | characteristic |
| 106 | D5 | — | 8 | 1 | no | — |
| 107 | Q8 | — | 2 | 1 | no | — |
| 108 | Q8 | — | 4 | 1 | no | — |
| 109 | Q8 | — | 6 | 1 | no | — |
| 110 | Q8 | — | 8 | 1 | no | — |
| 111 | Q8 | — | 12 | 1 | no | — |
| 112 | Q8 | — | 30 | 1 | no | — |
| 113 | D4 | — | 20 | 1 | no | — |
| 114 | D4 | — | 30 | 1 | no | — |
| 115 | C2×C4 | — | 3 | 1 | no | — |
| 116 | C2×C4 | — | 6 | 1 | no | — |
| 117 | C2×C4 | — | 6 | 1 | no | — |
| 118 | C2×C4 | — | 60 | 1 | no | — |
| 119 | C8 | — | 60 | 1 | no | — |
| 120 | C2×C3 | — | 16 | 1 | no | — |
| 121 | C2×C3 | — | 32 | 1 | no | — |
| 122 | C2×C3 | — | 160 | 1 | no | — |
| 123 | C5 | #2205⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 124 | C22 | — | 20 | 1 | no | — |
| 125 | C22 | C5⋊(#120⋊C3) | 1 | 1 | yes | characteristic |
| 126 | C4 | — | 6 | 1 | no | — |
| 127 | C4 | — | 6 | 1 | no | — |
| 128 | C4 | — | 6 | 1 | no | — |
| 129 | C4 | — | 12 | 1 | no | — |
| 130 | C4 | — | 20 | 1 | no | — |
| 131 | C3 | — | 16 | 1 | no | Sylow |
| 132 | C2 | — | 2 | 1 | no | — |
| 133 | C2 | — | 40 | 1 | no | — |
| 134 | C2 | C5⋊(B(4,𝔽2)⋊C3) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C5⋊(B(4,𝔽2)⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C5×Q82 | C2×C3 |
| Frattini | C2×Q8 | D5×A4 |
| socle | C2×C5 | B(4,𝔽2)⋊C3 |
| hypercenter | C22 | C5⋊(#120⋊C3) |
| LCS | G ▷ C5×Q82 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×Q8⋊SL(2,𝔽3) | C2 | semidirect |
| 2. #10602735 | C3 | semidirect |
| 5. C5×Q82 | C2×C3 | semidirect |
| 13. Q8⋊SL(2,𝔽3) | D5 | semidirect |
| 41. Q82 | C3×D5 | semidirect |
| 123. C5 | #2205⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×Q8⋊SL(2,𝔽3) | 2 |
| #10602735 | 3 |
| #4147⋊C3 | 4 |
| #2205⋊C3 | 5 |