Finite Group Databasebeta

← C2×#22550511 #30652169

Subgroups — C2×#22550511 #30652169

Order 16256.

2536 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 2536 conjugacy classes of subgroups — too many to draw. Only its 581 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
1C22×D4×D127C214yesmaximal, complemented
2#22550511C2116yesmaximal, direct product factor
3C2×#16861180C214yesmaximal, complemented
4C2×#16861177C214yesmaximal, complemented
5C2×C4×D508C212yesmaximal, complemented
6C2×#127×C127C211yesmaximal, characteristic, complemented
7C24×D127C2214yescomplemented
8C22×D508C2212yes—
9C22×C22⋊D127C2218yescomplemented
10C2×D4×D127—232no—
11C2×D4×D127C22132yescomplemented
12C22×C4×D127C2214yescomplemented
13#16861180C22132yescomplemented
14#16861177C22132yescomplemented
15C4×D508C22116yescomplemented
16C2×C4⋊Dic127C2212yescomplemented
17C2×C22⋊Dic127C2214yescomplemented
18C2×(C2×D127)⋊C4C2214yescomplemented
19C22×D4×C127C2214yescomplemented
20C22×D4×C127C2212yescomplemented
21#127×C127C2218yescomplemented
22C2×C42×C127C2211yescharacteristic, complemented
23C23×D127—232no—
24C23×D127—48no—
25C23×D127C2314yes—
26C23×D127C23116yescomplemented
27C2×D508D4116yescomplemented
28C2×D508C2318yes—
29C2×C22⋊D127—232no—
30C2×C22⋊D127C23132yescomplemented
31D4×D127—2128no—
32C22×Dic127C2314yes—
33C2×C4×D127—216no—
34C2×C4×D127C23116yescomplemented
35C4⋊Dic127C2318yescomplemented
36C22⋊Dic127C23116yescomplemented
37(C2×D127)⋊C4C23116yescomplemented
38C24×C127C2314yes—
39C2×D4×C127—216no—
40C2×D4×C127—44no—
41C2×D4×C127C23116yescomplemented
42C2×D4×C127C2318yescomplemented
43C22×C4×C127C2311yescharacteristic
44C22×C4×C127C2312yes—
45C42×C127C2314yescomplemented
46C22×D127—28no—
47C22×D127—28no—
48C22×D127—264no—
49C22×D127—416no—
50C22×D127—464no—
51C22×D127C2×D4116yescomplemented
52C22×D127C2418yes—
53D508C2×D4132yescomplemented
54C22⋊D127—264no—
55C2×Dic127—28no—
56C2×Dic127C2418yes—
57C4×D127—232no—
58C23×C127—216no—
59C23×C127—28no—
60C23×C127C2418yes—
61C23×C127C2411yescharacteristic
62D4×C127—232no—
63D4×C127—48no—
64C2×C4×C127—22no—
65C2×C4×C127C2×D418yescomplemented
66C2×C4×C127C2412yes—
67C2×C4×C127C2414yes—
68C22×C127—216no—
69C22×C127—28no—
70C22×C127—416no—
71C22×C127C22×D414yes—
72C22×C127H(5,𝔽2)12yes—
73C22×C127C2511yescharacteristic
74C2×D127—232no—
75C2×D127—28no—
76C2×D127—28no—
77C2×D127—464no—
78C2×D127C22×D4116yescomplemented
79C4×C127—22no—
80C4×C127C22×D418yescomplemented
81Dic127—28no—
82C2×C127—48no—
83C2×C127D4214yescomplemented
84C2×C127C23×D412yes—
85C2×C127C2×H(5,𝔽2)11yescharacteristic
86D127—216no—
87C2×D42—1271nomaximal, Sylow
88C127C2×D4211yescharacteristic, complemented, Sylow
89C23×D4—1274no—
90D42—12716no—
91C2×#127D12711yescharacteristic, complemented
92C2×#121—1274no—
93C2×#120—1274no—
94C2×C4×D4—1272no—
95C25—1274no—
96C22×D4—1278no—
97C22×D4—12732no—
98C22×D4—1272no—
99C22×D4—25432no—
100C22×D4C2×D12714yescomplemented
101C22×D4C2×D12712yescomplemented
102C23×C4—1274no—
103#127C2×D12718yescomplemented
104#121—12732no—
105#120—12732no—
106C4×D4—12716no—
107C2×C4⋊C4—1272no—
108C2×C22⋊C4—1274no—
109C2×C22⋊C4—1274no—
110C2×C42C2×D12711yescharacteristic, complemented
111C24—1274no—
112C24—12716no—
113C24—5088no—
114C24—25432no—
115C24C22×D12714yes—
116C2×D4—216no—
117C2×D4—44no—
118C2×D4—12732no—
119C2×D4—12716no—
120C2×D4—1278no—
121C2×D4—254128no—
122C2×D4—25432no—
123C2×D4C22×D127116yescomplemented
124C2×D4C22×D12718yescomplemented
125C22×C4—1274no—
126C22×C4—12716no—
127C22×C4—25416no—
128C22×C4C22×D12711yescharacteristic
129C22×C4C22×D12712yes—
130C4⋊C4—1278no—
131C22⋊C4—12716no—
132C22⋊C4—12716no—
133C42C22×D12714yescomplemented
134C23—216no—
135C23—28no—
136C23—12716no—
137C23—1278no—
138C23—50816no—
139C23—50864no—
140C23—25464no—
141C23—2548no—
142C23—2548no—
143C23C23×D12718yes—
144C23C23×D12711yescharacteristic
145D4—232no—
146D4—48no—
147D4—12732no—
148D4—25464no—
149C2×C4—22no—
150C2×C4—1278no—
151C2×C4—2548no—
152C2×C4—25432no—
153C2×C4D4×D12718yescomplemented
154C2×C4C23×D12712yes—
155C2×C4C23×D12714yes—
156C22—216no—
157C22—28no—
158C22—416no—
159C22—12716no—
160C22—50864no—
161C22—2548no—
162C22—2548no—
163C22—25432no—
164C22C2×D4×D12714yes—
165C22#1686125412yes—
166C22C24×D12711yescharacteristic
167C4—22no—
168C4—2548no—
169C4C2×D4×D12718yescomplemented
170C2—48no—
171C2—25416no—
172C2#2255051114yesdirect product factor
173C2C22×D4×D12712yes—
174C2C2×#1686125411yescharacteristic
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC2×#127×C127C2
socleC23×C127C24
hypercenterC2×#127D127
hypocenterC127C2×D42
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D4×D127C2semidirect
2. #22550511C2direct (N × H)
3. C2×#16861180C2semidirect
4. C2×#16861177C2semidirect
5. C2×C4×D508C2semidirect
6. C2×#127×C127C2semidirect
7. C24×D127C22semidirect
9. C22×C22⋊D127C22semidirect
11. C2×D4×D127C22semidirect
12. C22×C4×D127C22semidirect
13. #16861180C22semidirect
14. #16861177C22semidirect
15. C4×D508C22semidirect
16. C2×C4⋊Dic127C22semidirect
17. C2×C22⋊Dic127C22semidirect
18. C2×(C2×D127)⋊C4C22semidirect
19. C22×D4×C127C22semidirect
20. C22×D4×C127C22semidirect
21. #127×C127C22semidirect
22. C2×C42×C127C22semidirect
26. C23×D127C23semidirect
27. C2×D508D4semidirect
30. C2×C22⋊D127C23semidirect
34. C2×C4×D127C23semidirect
35. C4⋊Dic127C23semidirect
36. C22⋊Dic127C23semidirect
37. (C2×D127)⋊C4C23semidirect
41. C2×D4×C127C23semidirect
42. C2×D4×C127C23semidirect
45. C42×C127C23semidirect
51. C22×D127C2×D4semidirect
53. D508C2×D4semidirect
65. C2×C4×C127C2×D4semidirect
78. C2×D127C22×D4semidirect
80. C4×C127C22×D4semidirect
83. C2×C127D42semidirect
88. C127C2×D42semidirect
91. C2×#127D127semidirect
100. C22×D4C2×D127semidirect
101. C22×D4C2×D127semidirect
103. #127C2×D127semidirect
110. C2×C42C2×D127semidirect
123. C2×D4C22×D127semidirect
124. C2×D4C22×D127semidirect
133. C42C22×D127semidirect
153. C2×C4D4×D127semidirect
169. C4C2×D4×D127semidirect
172. C2#22550511direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1272
#225505112
C2×#168611802
C2×#168611772
C2×C4×D5082
C2×#127×C1272
C2×D42127