Finite Group Databasebeta

← C2×#22550438 #30652033

Subgroups — C2×#22550438 #30652033

Order 16256.

1984 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C23×C22⋊D127C212yesmaximal, complemented
2#22550438C2116yesmaximal, direct product factor
3C2×#16861193C214yesmaximal, complemented
4C2×#16861183C214yesmaximal, complemented
5C2×#16861179C214yesmaximal, complemented
6C22×C22⋊C4×C127C211yesmaximal, characteristic, complemented
7C24×D127C2212yescomplemented
8C22×D508C2214yescomplemented
9C22×C22⋊D127—216no—
10C22×C22⋊D127C2214yes—
11C22×C22⋊D127C2214yescomplemented
12C22×C22⋊D127C22116yescomplemented
13C23×Dic127C2212yescomplemented
14#16861193C22132yescomplemented
15#16861183C22132yescomplemented
16#16861179C22132yescomplemented
17C2×C4⋊Dic127C2214yescomplemented
18C2×(C2×D127)⋊C4C2218yescomplemented
19C25×C127C2211yescharacteristic
20C23×C4×C127C2212yescomplemented
21C2×C22⋊C4×C127C2214yescomplemented
22C2×C22⋊C4×C127C2218yescomplemented
23C23×D127—28no—
24C23×D127—48no—
25C23×D127C2314yes—
26C23×D127C2318yescomplemented
27C2×D508—48no—
28C2×D508C23116yescomplemented
29C2×C22⋊D127—216no—
30C2×C22⋊D127—216no—
31C2×C22⋊D127—264no—
32C2×C22⋊D127C23116yes—
33C2×C22⋊D127C23116yescomplemented
34C22×Dic127—28no—
35C22×Dic127C2314yes—
36C22×Dic127C2318yescomplemented
37C4⋊Dic127C23116yescomplemented
38(C2×D127)⋊C4C23132yescomplemented
39C24×C127—44no—
40C24×C127D418yescomplemented
41C24×C127C2312yes—
42C24×C127C2311yescharacteristic
43C24×C127C2314yes—
44C22×C4×C127—44no—
45C22×C4×C127C2314yes—
46C22×C4×C127C2318yescomplemented
47C22⋊C4×C127C23116yescomplemented
48C22×D127—28no—
49C22×D127—216no—
50C22×D127—216no—
51C22×D127—416no—
52C22×D127—432no—
53C22×D127C2418yes—
54D508—416no—
55C22⋊D127—232no—
56C22⋊D127—232no—
57C2×Dic127—28no—
58C2×Dic127—216no—
59C2×Dic127C2418yes—
60C23×C127—24no—
61C23×C127—28no—
62C23×C127—416no—
63C23×C127—48no—
64C23×C127C2×D418yes—
65C23×C127C2×D414yes—
66C23×C127C2×D4116yescomplemented
67C23×C127C2414yes—
68C23×C127C2412yes—
69C23×C127C2411yescharacteristic
70C2×C4×C127—28no—
71C2×C4×C127—48no—
72C2×C4×C127C2418yes—
73C22×C127—28no—
74C22×C127—216no—
75C22×C127—24no—
76C22×C127—28no—
77C22×C127—416no—
78C22×C127C22×D418yes—
79C22×C127C22×D414yes—
80C22×C127C22×D412yes—
81C22×C127H(5,𝔽2)14yes—
82C22×C127C2511yescharacteristic
83C2×D127—28no—
84C2×D127—216no—
85C2×D127—432no—
86C2×D127—416no—
87C4×C127—28no—
88Dic127—28no—
89C2×C127—28no—
90C2×C127—24no—
91C2×C127#53414yescomplemented
92C2×C127C23×D411yescharacteristic
93C2×C127C2×H(5,𝔽2)12yes—
94D127—48no—
95C2×#534—1271nomaximal, Sylow
96C127C2×#53411yescharacteristic, complemented, Sylow
97C23×D4—1272no—
98#534—12716no—
99C2×#123—1274no—
100C2×#121—1274no—
101C2×#120—1274no—
102C22×C22⋊C4D12711yescharacteristic, complemented
103C25—1272no—
104C25C2×D12711yescharacteristic
105C22×D4—1274no—
106C22×D4—1274no—
107C22×D4—12716no—
108C22×D4—1274no—
109C22×D4—25416no—
110C23×C4—1272no—
111C23×C4C2×D12712yescomplemented
112#123—12732no—
113#121—12732no—
114#120—12732no—
115C2×C4⋊C4—1274no—
116C2×C22⋊C4—1278no—
117C2×C22⋊C4C2×D12714yescomplemented
118C2×C22⋊C4C2×D12718yescomplemented
119C24—44no—
120C24—1278no—
121C24—1274no—
122C24—5088no—
123C24—2548no—
124C24D50818yescomplemented
125C24C22×D12712yes—
126C24C22×D12711yescharacteristic
127C24C22×D12714yes—
128C2×D4—12716no—
129C2×D4—12716no—
130C2×D4—12716no—
131C2×D4—5088no—
132C2×D4—25416no—
133C2×D4—25416no—
134C2×D4—25464no—
135C22×C4—44no—
136C22×C4—1278no—
137C22×C4—1274no—
138C22×C4—2548no—
139C22×C4C22×D12714yes—
140C22×C4C22×D12718yescomplemented
141C4⋊C4—12716no—
142C22⋊C4—12732no—
143C22⋊C4C22×D127116yescomplemented
144C23—24no—
145C23—28no—
146C23—416no—
147C23—48no—
148C23—1278no—
149C23—50832no—
150C23—50816no—
151C23—25416no—
152C23—2548no—
153C23—25416no—
154C23C2×D50818yes—
155C23C2×D50814yes—
156C23C2×D508116yescomplemented
157C23C23×D12714yes—
158C23C23×D12712yes—
159C23C23×D12711yescharacteristic
160D4—50816no—
161D4—25432no—
162D4—25432no—
163C2×C4—28no—
164C2×C4—48no—
165C2×C4—1278no—
166C2×C4—25416no—
167C2×C4—2548no—
168C2×C4C23×D12718yes—
169C22—216no—
170C22—28no—
171C22—24no—
172C22—28no—
173C22—416no—
174C22—50832no—
175C22—50816no—
176C22—2548no—
177C22—25416no—
178C22C22×D50818yes—
179C22C22×D50814yes—
180C22C22×D50812yes—
181C22#1686125414yes—
182C22C24×D12711yescharacteristic
183C4—28no—
184C4—2548no—
185C2—28no—
186C2—24no—
187C2—5088no—
188C2#2255043814yesdirect product factor
189C2C23×D50811yescharacteristic
190C2C2×#1686125412yes—
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC22×C22⋊C4×C127C2
socleC23×C127C24
hypercenterC22×C22⋊C4D127
hypocenterC127C2×#534
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D127C2semidirect
2. #22550438C2direct (N × H)
3. C2×#16861193C2semidirect
4. C2×#16861183C2semidirect
5. C2×#16861179C2semidirect
6. C22×C22⋊C4×C127C2semidirect
7. C24×D127C22semidirect
8. C22×D508C22semidirect
11. C22×C22⋊D127C22semidirect
12. C22×C22⋊D127C22semidirect
13. C23×Dic127C22semidirect
14. #16861193C22semidirect
15. #16861183C22semidirect
16. #16861179C22semidirect
17. C2×C4⋊Dic127C22semidirect
18. C2×(C2×D127)⋊C4C22semidirect
20. C23×C4×C127C22semidirect
21. C2×C22⋊C4×C127C22semidirect
22. C2×C22⋊C4×C127C22semidirect
26. C23×D127C23semidirect
28. C2×D508C23semidirect
33. C2×C22⋊D127C23semidirect
36. C22×Dic127C23semidirect
37. C4⋊Dic127C23semidirect
38. (C2×D127)⋊C4C23semidirect
40. C24×C127D4semidirect
46. C22×C4×C127C23semidirect
47. C22⋊C4×C127C23semidirect
66. C23×C127C2×D4semidirect
91. C2×C127#534semidirect
96. C127C2×#534semidirect
102. C22×C22⋊C4D127semidirect
111. C23×C4C2×D127semidirect
117. C2×C22⋊C4C2×D127semidirect
118. C2×C22⋊C4C2×D127semidirect
124. C24D508semidirect
140. C22×C4C22×D127semidirect
143. C22⋊C4C22×D127semidirect
156. C23C2×D508semidirect
188. C2#22550438direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1272
#225504382
C2×#168611932
C2×#168611832
C2×#168611792
C22×C22⋊C4×C1272
C2×#534127