Finite Group Databasebeta

← #30640625

Subgroups — #30640625

Order 16256.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550311C211yesmaximal, characteristic
2#22549998C211yesmaximal, characteristic, complemented
3#22549987C211yesmaximal, characteristic, complemented
4#22549698C211yesmaximal, characteristic, complemented
5#22549690C211yesmaximal, characteristic, complemented
6#22549653C211yesmaximal, characteristic, complemented
7#360×C127C211yesmaximal, characteristic, complemented
8C2×D1016C2211yescharacteristic
9#16861216—21no—
10#16861171—21no—
11#16861152C412yescomplemented
12#16861150C412yescomplemented
13#16861149C2211yescharacteristic
14#16861145C2211yescharacteristic
15#16861136—21no—
16#16861129—21no—
17#16861119C2211yescharacteristic
18#135×C127C2211yescharacteristic
19M32×C127C2211yescharacteristic
20#106×C127C2211yescharacteristic
21C2×D508—21no—
22C2×C4×D127—41no—
23D1016—21no—
24D1016C2×C412yes—
25#14523232—21no—
26#14523232—41no—
27C2×C127⋊C8—41no—
28C2×C127⋊C8D411yescharacteristic
29(C2×D127)⋊C4—41no—
30C4×Dic127D411yescharacteristic
31C127⋊C16C2×C412yescomplemented
32#41×C127—21no—
33Q16×C127C2×C411yescharacteristic, complemented
34SD16×C127—21no—
35D8×C127C2×C411yescharacteristic, complemented
36C2×C8×C127C2311yescharacteristic
37C4⋊C4×C127—21no—
38C16×C127—21no—
39C22×D127—41no—
40D508—22no—
41D508—41no—
42C2×Dic127—21no—
43C4×D127—82no—
44C127⋊C8—21no—
45C127⋊C8—41no—
46Q8×C127—21no—
47D4×C127—21no—
48D4×C127—41no—
49C2×C4×C127—41no—
50C2×C4×C127—41no—
51C2×C4×C127C2×D411yescharacteristic
52C8×C127C22×C411yescharacteristic
53C8×C127#4111yescharacteristic
54C22×C127—41no—
55C22×C127AGL(1,ℤ/8)11yescharacteristic
56C2×D127—42no—
57C2×D127—81no—
58C2×D127—81no—
59C4×C127—41no—
60C4×C127—81no—
61C4×C127C4×D411yescharacteristic
62C4×C127AGL(1,ℤ/8)11yescharacteristic
63Dic127—41no—
64C2×C127—21no—
65C2×C127—81no—
66C2×C127#44211yescharacteristic
67D127—82no—
68AGL(1,ℤ/16)—1271nomaximal, Sylow
69C127AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—1271no—
71#444—1271no—
72#442—1271no—
73#365—1271no—
74#361—1271no—
75#360D12711yescharacteristic, complemented
76#347—1271no—
77#135C2×D12711yescharacteristic
78C2×D8—1271no—
79#131—2541no—
80C4×D4—2541no—
81SD32—1272no—
82D16—1272no—
83M32—1271no—
84M32C2×D12711yescharacteristic
85#108—1271no—
86#106C2×D12711yescharacteristic
87#104—2541no—
88#102—2541no—
89#97—1271no—
90#41—21no—
91C2×D4—2541no—
92C22×C4—5081no—
93Q16C4×D12711yescharacteristic, complemented
94SD16—21no—
95D8—1272no—
96D8—2541no—
97D8C4×D12711yescharacteristic, complemented
98M16—5081no—
99M16—2541no—
100C2×C8—1271no—
101C2×C8—5081no—
102C2×C8C22×D12711yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—5081no—
105C42—1271no—
106C16—21no—
107C16—1272no—
108C23—5081no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—5081no—
113D4—2542no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—10162no—
117C2×C4—2541no—
118C2×C4D4×D12711yescharacteristic
119C8—5081no—
120C8—2541no—
121C8C2×C4×D12711yescharacteristic
122C8#1452324711yescharacteristic
123C22—41no—
124C22—10161no—
125C22—10161no—
126C22—5082no—
127C22#1686123311yescharacteristic
128C4—41no—
129C4—81no—
130C4—5081no—
131C4(C22⋊D127)⋊C411yescharacteristic
132C4#1686123611yescharacteristic
133C2—21no—
134C2—81no—
135C2—10162no—
136C2#2254998511yescharacteristic
subgroupG/H
centerC2#22549985
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#360×C127C2
socleC2×C127#442
prime radical#22550311C2
cosocleC2×C8×C127C23
hypercenter#360D127
hypocenterC127AGL(1,ℤ/16)
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22549998C2semidirect
3. #22549987C2semidirect
4. #22549698C2semidirect
5. #22549690C2semidirect
6. #22549653C2semidirect
7. #360×C127C2semidirect
11. #16861152C4semidirect
12. #16861150C4semidirect
31. C127⋊C16C2×C4semidirect
33. Q16×C127C2×C4semidirect
35. D8×C127C2×C4semidirect
69. C127AGL(1,ℤ/16)semidirect
75. #360D127semidirect
93. Q16C4×D127semidirect
97. D8C4×D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225503112
#225499982
#225499872
#225496982
#225496902
#225496532
#360×C1272
AGL(1,ℤ/16)127