Finite Group Databasebeta

← #30640631

Subgroups — #30640631

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#22550312C211yesmaximal, characteristic
2#22549997C211yesmaximal, characteristic, complemented
3#22549988C211yesmaximal, characteristic, complemented
4#22549700C211yesmaximal, characteristic, complemented
5#22549687C211yesmaximal, characteristic, complemented
6#22549653C211yesmaximal, characteristic, complemented
7#361×C127C211yesmaximal, characteristic, complemented
8#16861231C2211yescharacteristic
9#16861217—21no—
10D4×Dic127—21no—
11#16861153C412yescomplemented
12#16861150C412yescomplemented
13#16861149C2211yescharacteristic
14#16861141C2211yescharacteristic
15#16861135—21no—
16#16861130—21no—
17#16861119C2211yescharacteristic
18C2×D8×C127C2211yescharacteristic
19M32×C127C2211yescharacteristic
20#108×C127C2211yescharacteristic
21#14523248—21no—
22C22×Dic127—41no—
23Dic508C2×C411yescharacteristic, complemented
24#14523237—21no—
25D1016C2×C411yescharacteristic, complemented
26#14523231—41no—
27C2×C127⋊C8D411yescharacteristic
28C8×D127—41no—
29C4⋊Dic127—21no—
30C22⋊Dic127—41no—
31C4×Dic127D411yescharacteristic
32C127⋊C16C2×C412yescomplemented
33C2×D4×C127—21no—
34D8×C127—21no—
35D8×C127C2×C412yes—
36M16×C127—21no—
37C2×C8×C127C2311yescharacteristic
38C16×C127—21no—
39Dic254—21no—
40D508—21no—
41C22⋊D127—41no—
42C2×Dic127—21no—
43C2×Dic127—41no—
44C2×Dic127—82no—
45C4×D127—41no—
46C127⋊C8—21no—
47C23×C127—41no—
48D4×C127—22no—
49D4×C127—41no—
50C2×C4×C127C2×D411yescharacteristic
51C8×C127—41no—
52C8×C127C22×C411yescharacteristic
53C8×C127#4111yescharacteristic
54C22×C127—42no—
55C22×C127—81no—
56C22×C127—81no—
57C22×C127AGL(1,ℤ/8)11yescharacteristic
58C2×D127—41no—
59C4×C127C4×D411yescharacteristic
60C4×C127AGL(1,ℤ/8)11yescharacteristic
61Dic127—41no—
62Dic127—41no—
63Dic127—81no—
64C2×C127—21no—
65C2×C127—82no—
66C2×C127#44211yescharacteristic
67D127—81no—
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#361D12711yescharacteristic, complemented
75#360—1271no—
76#347—1271no—
77#135—1271no—
78C2×D8C2×D12711yescharacteristic
79#131—2541no—
80C4×D4—2541no—
81SD32—1272no—
82D16—1272no—
83M32—1271no—
84M32C2×D12711yescharacteristic
85#108C2×D12711yescharacteristic
86#106—1271no—
87#104—2541no—
88#102—2541no—
89#97—1271no—
90#41—2541no—
91C2×D4—21no—
92C22×C4—5081no—
93Q16—1271no—
94SD16—2541no—
95D8—21no—
96D8—1271no—
97D8C4×D12712yes—
98M16—21no—
99M16—5081no—
100C2×C8—1271no—
101C2×C8—5081no—
102C2×C8C22×D12711yescharacteristic
103C4⋊C4—2541no—
104C22⋊C4—5081no—
105C42—1271no—
106C16—21no—
107C16—1272no—
108C23—41no—
109Q8—2541no—
110D4—22no—
111D4—41no—
112D4—5081no—
113D4—2541no—
114C2×C4—10162no—
115C2×C4—5081no—
116C2×C4—5081no—
117C2×C4—2541no—
118C2×C4D4×D12711yescharacteristic
119C8—41no—
120C8—2541no—
121C8C2×C4×D12711yescharacteristic
122C8#1452324711yescharacteristic
123C22—42no—
124C22—81no—
125C22—81no—
126C22—5081no—
127C22#1686123611yescharacteristic
128C4—10161no—
129C4—5081no—
130C4—5081no—
131C4(C22⋊D127)⋊C411yescharacteristic
132C4#1686123311yescharacteristic
133C2—21no—
134C2—82no—
135C2—10161no—
136C2#2254998511yescharacteristic
subgroupG/H
centerC2#22549985
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#361×C127C2
socleC2×C127#442
prime radical#22550312C2
cosocleC2×C8×C127C23
hypercenter#361D127
hypocenterC127AGL(1,ℤ/16)
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #22549997C2semidirect
3. #22549988C2semidirect
4. #22549700C2semidirect
5. #22549687C2semidirect
6. #22549653C2semidirect
7. #361×C127C2semidirect
11. #16861153C4semidirect
12. #16861150C4semidirect
23. Dic508C2×C4semidirect
25. D1016C2×C4semidirect
32. C127⋊C16C2×C4semidirect
69. C127AGL(1,ℤ/16)semidirect
74. #361D127semidirect

Maximal subgroups

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

subgroupindex
#225503122
#225499972
#225499882
#225497002
#225496872
#225496532
#361×C1272
AGL(1,ℤ/16)127