Finite Group Databasebeta

← C2×#22550009 #30647905

Subgroups — C2×#22550009 #30647905

Order 16256.

1048 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 181 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×D508C211yesmaximal, characteristic, complemented
2C22×#14523233C212yesmaximal, complemented
3#22550009C218yesmaximal, direct product factor
4C2×#16861129C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6C2×#121×C127C211yesmaximal, characteristic, complemented
7C24×D127—21no—
8C22×D508—28no—
9C22×D508—21no—
10C22×D508—22no—
11C22×D508C2212yes—
12C22×D508C2214yescomplemented
13C2×#14523233—28no—
14C2×#14523233C2218yescomplemented
15C22×C127⋊C8C2212yescomplemented
16#16861129C2218yescomplemented
17C22⋊(C127⋊C8)C2214yescomplemented
18C22×D4×C127—21no—
19C22×D4×C127C2211yescharacteristic, complemented
20C23×C4×C127C2211yescharacteristic
21#121×C127—22no—
22#121×C127C2214yescomplemented
23C2×C4⋊C4×C127C2211yescharacteristic, complemented
24C2×C22⋊C4×C127—21no—
25C23×D127—28no—
26C23×D127—22no—
27C23×D127—24no—
28C23×D127—44no—
29C23×D127—44no—
30C2×D508—28no—
31C2×D508—216no—
32C2×D508—216no—
33C2×D508—24no—
34C2×D508—22no—
35C2×D508—24no—
36C2×D508D418yescomplemented
37C2×D508C2314yes—
38#14523233—216no—
39C2×C127⋊C8—24no—
40C2×C127⋊C8C2314yescomplemented
41C24×C127—21no—
42C24×C127D411yescharacteristic
43C2×D4×C127—24no—
44C2×D4×C127—24no—
45C2×D4×C127—21no—
46C2×D4×C127—44no—
47C2×D4×C127C2312yescomplemented
48C22×C4×C127—24no—
49C22×C4×C127—21no—
50C22×C4×C127D411yescharacteristic
51C22×C4×C127D412yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—21no—
55C4⋊C4×C127C2312yescomplemented
56C22⋊C4×C127—24no—
57C22×D127—216no—
58C22×D127—28no—
59C22×D127—24no—
60C22×D127—48no—
61C22×D127—432no—
62C22×D127—44no—
63C22×D127—48no—
64C22×D127—44no—
65D508—216no—
66D508—28no—
67D508—24no—
68D508C2×D418yescomplemented
69C127⋊C8—24no—
70C23×C127—22no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127—42no—
74C23×C127—42no—
75C23×C127—42no—
76C23×C127D814yescomplemented
77C23×C127C2×D412yes—
78C23×C127C2×D411yescharacteristic
79D4×C127—24no—
80D4×C127—48no—
81D4×C127—42no—
82C2×C4×C127—22no—
83C2×C4×C127—24no—
84C2×C4×C127—24no—
85C2×C4×C127—22no—
86C2×C4×C127—42no—
87C2×C4×C127C2×D414yes—
88C2×C4×C127C2×D412yes—
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2411yescharacteristic
91C22×C127—22no—
92C22×C127—22no—
93C22×C127—28no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—42no—
97C22×C127—84no—
98C22×C127#12012yes—
99C22×C127C2×D812yes—
100C22×C127C2×D814yescomplemented
101C22×C127AGL(1,ℤ/8)12yes—
102C22×C127C22×D411yescharacteristic
103C2×D127—28no—
104C2×D127—432no—
105C2×D127—44no—
106C2×D127—44no—
107C2×D127—416no—
108C4×C127—22no—
109C4×C127—42no—
110C4×C127#12012yes—
111C4×C127C22×D412yes—
112C2×C127—24no—
113C2×C127—82no—
114C2×C127#44714yescomplemented
115C2×C127C2×#12011yescharacteristic
116C2×C127C22×D811yescharacteristic
117C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
118D127—48no—
119C2×#447—1271nomaximal, Sylow
120C127C2×#44711yescharacteristic, complemented, Sylow
121C23×D4—1271no—
122C22×D8—1272no—
123C2×#121D12711yescharacteristic, complemented
124#447—1278no—
125C2×#102—1272no—
126C2×C22⋊C8—1271no—
127C25—2541no—
128C22×D4—21no—
129C22×D4—1272no—
130C22×D4—1274no—
131C22×D4—2548no—
132C22×D4—2542no—
133C22×D4—2541no—
134C22×D4C2×D12711yescharacteristic, complemented
135C23×C4C2×D12711yescharacteristic
136C2×D8—1278no—
137C2×D8—2548no—
138C22×C8—1272no—
139#121—22no—
140#121C2×D12714yescomplemented
141C2×C4⋊C4C2×D12711yescharacteristic, complemented
142C2×C22⋊C4—21no—
143#102—1278no—
144C22⋊C8—1274no—
145C24—21no—
146C24—5084no—
147C24—5084no—
148C24—2548no—
149C24—2544no—
150C24—2542no—
151C24C22⋊D12711yescharacteristic
152C2×D4—24no—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—44no—
156C2×D4—1278no—
157C2×D4—1274no—
158C2×D4—25416no—
159C2×D4—2548no—
160C2×D4—25416no—
161C2×D4—2544no—
162C2×D4—2544no—
163C2×D4—2542no—
164C2×D4C22×D12712yescomplemented
165C22×C4—24no—
166C22×C4—21no—
167C22×C4C22⋊D12712yes—
168C22×C4C22⋊D12711yescharacteristic
169C22×C4C22×D12712yes—
170C22×C4C22×D12711yescharacteristic
171D8—25416no—
172C2×C8—1274no—
173C2×C8—2544no—
174C4⋊C4—21no—
175C4⋊C4C22×D12712yescomplemented
176C22⋊C4—24no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—5088no—
184C23—50832no—
185C23—5084no—
186C23—5088no—
187C23—5084no—
188C23—25416no—
189C23—2544no—
190C23—2548no—
191C23#1452323314yescomplemented
192C23C2×C22⋊D12712yes—
193C23C2×C22⋊D12711yescharacteristic
194D4—24no—
195D4—48no—
196D4—42no—
197D4—1278no—
198D4—25416no—
199D4—2544no—
200D4—2548no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4D4×D12714yes—
207C2×C4C2×C22⋊D12712yes—
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C23×D12711yescharacteristic
210C8—2544no—
211C22—22no—
212C22—22no—
213C22—28no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—84no—
218C22—50832no—
219C22—5084no—
220C22—5084no—
221C22—50816no—
222C22—2548no—
223C22#1686117712yes—
224C22C2×#1452323312yes—
225C22C2×#1452323314yescomplemented
226C22#1686123412yes—
227C22C22×C22⋊D12711yescharacteristic
228C4—22no—
229C4—42no—
230C4#1686117712yes—
231C4C2×D4×D12712yes—
232C2—24no—
233C2—82no—
234C2—5088no—
235C2#2255000914yesdirect product factor
236C2C2×#1686117711yescharacteristic
237C2C22×#1452323311yescharacteristic
238C2C2×#1686123411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#121×C127C2
socleC23×C127C2×D4
hypercenterC2×#121D127
hypocenterC127C2×#447
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×D508C2semidirect
2. C22×#14523233C2semidirect
3. #22550009C2direct (N × H)
4. C2×#16861129C2semidirect
5. C2×C22⋊(C127⋊C8)C2semidirect
6. C2×#121×C127C2semidirect
12. C22×D508C22semidirect
14. C2×#14523233C22semidirect
15. C22×C127⋊C8C22semidirect
16. #16861129C22semidirect
17. C22⋊(C127⋊C8)C22semidirect
19. C22×D4×C127C22semidirect
22. #121×C127C22semidirect
23. C2×C4⋊C4×C127C22semidirect
36. C2×D508D4semidirect
40. C2×C127⋊C8C23semidirect
47. C2×D4×C127C23semidirect
55. C4⋊C4×C127C23semidirect
68. D508C2×D4semidirect
76. C23×C127D8semidirect
100. C22×C127C2×D8semidirect
114. C2×C127#447semidirect
120. C127C2×#447semidirect
123. C2×#121D127semidirect
134. C22×D4C2×D127semidirect
140. #121C2×D127semidirect
141. C2×C4⋊C4C2×D127semidirect
164. C2×D4C22×D127semidirect
175. C4⋊C4C22×D127semidirect
191. C23#14523233semidirect
225. C22C2×#14523233semidirect
235. C2#22550009direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D5082
C22×#145232332
#225500092
C2×#168611292
C2×C22⋊(C127⋊C8)2
C2×#121×C1272
C2×#447127