Finite Group Databasebeta

← #30640629

Subgroups — #30640629

Order 16256.

154 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 59 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#22549998C211yesmaximal, characteristic, complemented
2#22549988C211yesmaximal, characteristic, complemented
3#22549699C211yesmaximal, characteristic, complemented
4#22549689C211yesmaximal, characteristic, complemented
5#22549686C211yesmaximal, characteristic, complemented
6#22549653C211yesmaximal, characteristic, complemented
7#509×C127C211yesmaximal, characteristic
8#16861216—21no—
9D4×Dic127—21no—
10#16861149C2211yescharacteristic
11#16861145C2211yescharacteristic
12#16861141C2211yescharacteristic
13#16861136—21no—
14#16861130—21no—
15#16861119C2211yescharacteristic
16#135×C127C2211yescharacteristic
17C2×D8×C127C2211yescharacteristic
18SD32×C127C412yescomplemented
19D16×C127C412yescomplemented
20M32×C127C2211yescharacteristic
21C22×Dic127—41no—
22#14523232—21no—
23#14523232—41no—
24C2×C127⋊C8—41no—
25C2×C127⋊C8D411yescharacteristic
26C4⋊Dic127—21no—
27C22⋊Dic127—41no—
28C4×Dic127D411yescharacteristic
29C127⋊C16—21no—
30#41×C127—21no—
31C2×D4×C127—21no—
32Q16×C127C2×C411yescharacteristic, complemented
33SD16×C127—21no—
34D8×C127—21no—
35D8×C127C2×C412yes—
36D8×C127C2×C411yescharacteristic, complemented
37C2×C8×C127C2311yescharacteristic
38C16×C127C2×C412yescomplemented
39C2×Dic127—21no—
40C2×Dic127—41no—
41C2×Dic127—82no—
42C127⋊C8—21no—
43C127⋊C8—41no—
44C23×C127—41no—
45Q8×C127—21no—
46D4×C127—22no—
47D4×C127—21no—
48D4×C127—41no—
49D4×C127—41no—
50C2×C4×C127—41no—
51C2×C4×C127C2×D411yescharacteristic
52C8×C127C22×C411yescharacteristic
53C8×C127#4111yescharacteristic
54C22×C127—41no—
55C22×C127—42no—
56C22×C127—81no—
57C22×C127—81no—
58C22×C127AGL(1,ℤ/8)11yescharacteristic
59C4×C127—41no—
60C4×C127C4×D411yescharacteristic
61C4×C127AGL(1,ℤ/8)11yescharacteristic
62Dic127—41no—
63Dic127—81no—
64C2×C127—21no—
65C2×C127—81no—
66C2×C127—82no—
67C2×C127#44211yescharacteristic
68AGL(1,ℤ/16)—1271nomaximal, Sylow
69C127AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509D12711yescharacteristic
71#444—1271no—
72#442—1271no—
73#365—1271no—
74#361—1271no—
75#360—1271no—
76#347—1271no—
77#135C2×D12711yescharacteristic
78C2×D8C2×D12711yescharacteristic
79#131—2541no—
80C4×D4—2541no—
81SD32Dic12712yescomplemented
82D16Dic12712yescomplemented
83M32—1271no—
84M32C2×D12711yescharacteristic
85#108—1271no—
86#106—1271no—
87#104—2541no—
88#102—2541no—
89#97—1271no—
90#41—21no—
91C2×D4—21no—
92C22×C4—5081no—
93Q16C2×Dic12711yescharacteristic, complemented
94SD16—21no—
95D8—21no—
96D8C2×Dic12712yes—
97D8C2×Dic12711yescharacteristic, complemented
98M16—5081no—
99M16—2541no—
100C2×C8—1271no—
101C2×C8—5081no—
102C2×C8C22×D12711yescharacteristic
103C4⋊C4—2541no—
104C22⋊C4—5081no—
105C42—1271no—
106C16—2541no—
107C16C2×Dic12712yescomplemented
108C23—41no—
109Q8—21no—
110D4—22no—
111D4—21no—
112D4—41no—
113D4—41no—
114C2×C4—41no—
115C2×C4—10162no—
116C2×C4—5081no—
117C2×C4—2541no—
118C2×C4D4×D12711yescharacteristic
119C8—5081no—
120C8—2541no—
121C8C22×Dic12711yescharacteristic
122C8#1452324711yescharacteristic
123C22—41no—
124C22—42no—
125C22—81no—
126C22—81no—
127C22#1686123311yescharacteristic
128C4—41no—
129C4—10161no—
130C4—5081no—
131C4D4×Dic12711yescharacteristic
132C4#1686123311yescharacteristic
133C2—21no—
134C2—82no—
135C2—81no—
136C2#2254998811yescharacteristic
subgroupG/H
centerC2#22549988
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting, prime radical#509×C127C2
socleC2×C127#442
cosocleC2×C8×C127C23
hypercenter#509D127
hypocenterC127AGL(1,ℤ/16)
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22549998C2semidirect
2. #22549988C2semidirect
3. #22549699C2semidirect
4. #22549689C2semidirect
5. #22549686C2semidirect
6. #22549653C2semidirect
18. SD32×C127C4semidirect
19. D16×C127C4semidirect
32. Q16×C127C2×C4semidirect
36. D8×C127C2×C4semidirect
38. C16×C127C2×C4semidirect
69. C127AGL(1,ℤ/16)semidirect
81. SD32Dic127semidirect
82. D16Dic127semidirect
93. Q16C2×Dic127semidirect
97. D8C2×Dic127semidirect
107. C16C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225499982
#225499882
#225496992
#225496892
#225496862
#225496532
#509×C1272
AGL(1,ℤ/16)127