Finite Group Databasebeta

← #30640628

Subgroups — #30640628

Order 16256.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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#22549989C211yesmaximal, characteristic, complemented
3#22549700C211yesmaximal, characteristic, complemented
4#22549688C211yesmaximal, characteristic, complemented
5#22549686C211yesmaximal, characteristic, complemented
6#22549654C211yesmaximal, characteristic, complemented
7#444×C127C211yesmaximal, characteristic, complemented
8C2×D1016C2211yescharacteristic
9C4×D508—21no—
10#16861152C412yescomplemented
11#16861150C412yescomplemented
12#16861149C2211yescharacteristic
13#16861149C2211yescharacteristic
14#16861141C2211yescharacteristic
15#16861131—21no—
16#135×C127C2211yescharacteristic
17#131×C127—21no—
18#108×C127C2211yescharacteristic
19#104×C127—21no—
20#97×C127C2211yescharacteristic
21C2×D508—21no—
22C2×C4×D127—41no—
23D1016—21no—
24D1016C2×C412yes—
25C4⋊Dic127—21no—
26(C2×D127)⋊C4—41no—
27C127⋊C16—21no—
28C127⋊C16C2×C412yescomplemented
29#41×C127—21no—
30Q16×C127C2×C411yescharacteristic, complemented
31SD16×C127—21no—
32D8×C127C2×C411yescharacteristic, complemented
33M16×C127—21no—
34M16×C127—41no—
35C2×C8×C127—41no—
36C2×C8×C127D411yescharacteristic
37C2×C8×C127C2311yescharacteristic
38C42×C127D411yescharacteristic
39C22×D127—41no—
40D508—22no—
41D508—41no—
42C2×Dic127—41no—
43C4×D127—82no—
44Q8×C127—21no—
45D4×C127—21no—
46D4×C127—41no—
47C2×C4×C127—21no—
48C2×C4×C127—41no—
49C2×C4×C127C2×D411yescharacteristic
50C8×C127—21no—
51C8×C127—41no—
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—41no—
61C4×C127C4×D411yescharacteristic
62C4×C127AGL(1,ℤ/8)11yescharacteristic
63Dic127—81no—
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#444D12711yescharacteristic, complemented
72#442—1271no—
73#365—1271no—
74#361—1271no—
75#360—1271no—
76#347—1271no—
77#135C2×D12711yescharacteristic
78C2×D8—1271no—
79#131—21no—
80C4×D4—2541no—
81SD32—1272no—
82D16—1272no—
83M32—1271no—
84M32—1271no—
85#108C2×D12711yescharacteristic
86#106—1271no—
87#104—21no—
88#102—2541no—
89#97C2×D12711yescharacteristic
90#41—21no—
91C2×D4—2541no—
92C22×C4—5081no—
93Q16C4×D12711yescharacteristic, complemented
94SD16—21no—
95D8—1272no—
96D8—2541no—
97D8C4×D12711yescharacteristic, complemented
98M16—21no—
99M16—41no—
100C2×C8—41no—
101C2×C8C22⋊D12711yescharacteristic
102C2×C8C22×D12711yescharacteristic
103C4⋊C4—2541no—
104C22⋊C4—5081no—
105C42C22⋊D12711yescharacteristic
106C16—1272no—
107C16—2541no—
108C23—5081no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—5081no—
113D4—2542no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—10162no—
117C2×C4—5081no—
118C2×C4C2×C22⋊D12711yescharacteristic
119C8—21no—
120C8—41no—
121C8C2×C4×D12711yescharacteristic
122C8#1452324811yescharacteristic
123C22—41no—
124C22—10161no—
125C22—10161no—
126C22—5082no—
127C22#1686123411yescharacteristic
128C4—41no—
129C4—41no—
130C4—10161no—
131C4C4×C22⋊D12711yescharacteristic
132C4#1686123511yescharacteristic
133C2—21no—
134C2—81no—
135C2—10162no—
136C2#2254998611yescharacteristic
subgroupG/H
centerC2#22549986
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#444×C127C2
socleC2×C127#442
prime radical#22550311C2
cosocleC2×C8×C127C23
hypercenter#444D127
hypocenterC127AGL(1,ℤ/16)
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
2. #22549989C2semidirect
3. #22549700C2semidirect
4. #22549688C2semidirect
5. #22549686C2semidirect
6. #22549654C2semidirect
7. #444×C127C2semidirect
10. #16861152C4semidirect
11. #16861150C4semidirect
28. C127⋊C16C2×C4semidirect
30. Q16×C127C2×C4semidirect
32. D8×C127C2×C4semidirect
69. C127AGL(1,ℤ/16)semidirect
71. #444D127semidirect
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
#225499892
#225497002
#225496882
#225496862
#225496542
#444×C1272
AGL(1,ℤ/16)127