Finite Group Databasebeta

← #30640622

Subgroups — #30640622

Order 16256.

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

normal subgroups only

This group has 142 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
2#22549983C211yesmaximal, characteristic, complemented
3#22549699C211yesmaximal, characteristic, complemented
4#22549692C211yesmaximal, characteristic, complemented
5#22549686C211yesmaximal, characteristic
6#22549653C211yesmaximal, characteristic, complemented
7#510×C127C211yesmaximal, characteristic
8#16861216—21no—
9Q8×Dic127—21no—
10#16861149C2211yescharacteristic
11#16861145C2211yescharacteristic
12#16861141C2211yescharacteristic
13#16861136—21no—
14#16861133—21no—
15#16861119C2211yescharacteristic
16#135×C127C2211yescharacteristic
17C2×Q16×C127C2211yescharacteristic
18Q32×C127C412yescomplemented
19SD32×C127C412yescomplemented
20M32×C127C2211yescharacteristic
21#14523232—21no—
22#14523232—41no—
23C2×C127⋊C8—41no—
24C2×C127⋊C8D411yescharacteristic
25C4⋊Dic127—21no—
26C4⋊Dic127—41no—
27C4×Dic127—41no—
28C4×Dic127D411yescharacteristic
29C127⋊C16—21no—
30#41×C127—21no—
31C2×Q8×C127—21no—
32Q16×C127—21no—
33Q16×C127C2×C412yes—
34Q16×C127C2×C411yescharacteristic
35SD16×C127—21no—
36D8×C127C2×C411yescharacteristic
37C2×C8×C127C2311yescharacteristic
38C16×C127C2×C412yes—
39C2×Dic127—21no—
40C2×Dic127—41no—
41C127⋊C8—21no—
42C127⋊C8—41no—
43Q8×C127—21no—
44Q8×C127—22no—
45Q8×C127—41no—
46D4×C127—21no—
47D4×C127—41no—
48C2×C4×C127—41no—
49C2×C4×C127—41no—
50C2×C4×C127C2×D411yescharacteristic
51C8×C127C22×C411yescharacteristic
52C8×C127#4111yescharacteristic
53C22×C127—41no—
54C22×C127AGL(1,ℤ/8)11yescharacteristic
55C4×C127—41no—
56C4×C127—42no—
57C4×C127C4×D411yescharacteristic
58C4×C127AGL(1,ℤ/8)11yescharacteristic
59Dic127—41no—
60Dic127—81no—
61C2×C127—21no—
62C2×C127—81no—
63C2×C127#44211yescharacteristic
64#2180—1271nomaximal, Sylow
65C127#218011yescharacteristic, complemented, Sylow
66#510D12711yescharacteristic
67#444—1271no—
68#441—1271no—
69#365—1271no—
70#362—1271no—
71#360—1271no—
72#347—1271no—
73#135C2×D12711yescharacteristic
74C2×Q16C2×D12711yescharacteristic
75#131—2541no—
76C4×Q8—2541no—
77Q32Dic12712yescomplemented
78SD32Dic12712yescomplemented
79M32—1271no—
80M32C2×D12711yescharacteristic
81#108—1271no—
82#106—1271no—
83#104—2541no—
84#103—2541no—
85#97—1271no—
86#41—21no—
87C2×Q8—21no—
88Q16—21no—
89Q16C2×Dic12712yes—
90Q16C2×Dic12711yescharacteristic
91SD16—21no—
92D8C2×Dic12711yescharacteristic
93M16—5081no—
94M16—2541no—
95C2×C8—1271no—
96C2×C8—5081no—
97C2×C8C22×D12711yescharacteristic
98C4⋊C4—5081no—
99C4⋊C4—2541no—
100C42—1271no—
101C42—5081no—
102C16—2541no—
103C16C2×Dic12712yes—
104Q8—21no—
105Q8—22no—
106Q8—41no—
107D4—21no—
108D4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—5081no—
112C2×C4—2541no—
113C2×C4D4×D12711yescharacteristic
114C8—5081no—
115C8—2541no—
116C8C22×Dic12711yescharacteristic
117C8#1452324711yescharacteristic
118C22—41no—
119C22#1686123311yescharacteristic
120C4—41no—
121C4—42no—
122C4—10161no—
123C4—5081no—
124C4D4×Dic12711yescharacteristic
125C4#1686123311yescharacteristic
126C2—21no—
127C2—81no—
128C2#2254998811yescharacteristic
subgroupG/H
centerC2#22549988
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#510×C127C2
socleC2×C127#442
prime radical#135×C127C22
cosocleC2×C8×C127C23
hypercenter#510D127
hypocenterC127#2180
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #22549983C2semidirect
3. #22549699C2semidirect
4. #22549692C2semidirect
6. #22549653C2semidirect
18. Q32×C127C4semidirect
19. SD32×C127C4semidirect
65. C127#2180semidirect
77. Q32Dic127semidirect
78. SD32Dic127semidirect

Maximal subgroups

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

subgroupindex
#225499982
#225499832
#225496992
#225496922
#225496862
#225496532
#510×C1272
#2180127