Finite Group Databasebeta

← C2×#22550024 #30647934

Subgroups — C2×#22550024 #30647934

Order 16256.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14523249C211yesmaximal, characteristic, complemented
2C22×#14523235C211yesmaximal, characteristic, complemented
3C22×D1016C211yesmaximal, characteristic, complemented
4C2×#16861184C211yesmaximal, characteristic, complemented
5#22550024C218yesmaximal, direct product factor
6C2×#16861129C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
8C2×#103×C127C211yesmaximal, characteristic, complemented
9C2×#14523249—24no—
10C2×#14523249—22no—
11C2×#14523249—24no—
12C2×#14523249C2214yescomplemented
13C22×D508—21no—
14C22×D508—21no—
15C22×D508C2211yescharacteristic
16C22×D508C2211yescharacteristic
17C22×C4×D127—21no—
18C22×C4×D127C2211yescharacteristic
19C2×#14523235—24no—
20C2×#14523235C2214yescomplemented
21C2×D1016—24no—
22C2×D1016C2214yescomplemented
23C22×C127⋊C8C2211yescharacteristic
24#16861184—22no—
25#16861184C2214yescomplemented
26C2×(C2×D127)⋊C4—21no—
27#16861129C2214yescomplemented
28(C2×D127)⋊C8C2214yescomplemented
29C22×Q8×C127C2211yescharacteristic
30C22×C8×C127C2211yescharacteristic
31C2×C4⋊C4×C127C2211yescharacteristic
32#103×C127C2214yescomplemented
33C23×D127—21no—
34C23×D127—21no—
35C23×D127D411yescharacteristic
36#14523249—216no—
37#14523249—28no—
38#14523249—28no—
39C2×D508—28no—
40C2×D508—24no—
41C2×D508—24no—
42C2×D508—21no—
43C2×D508—22no—
44C2×D508—22no—
45C2×D508—22no—
46C2×D508—21no—
47C2×D508—44no—
48C2×D508D414yescomplemented
49C2×D508C2312yes—
50C2×D508C2312yes—
51C22×Dic127D411yescharacteristic
52C2×C4×D127—28no—
53C2×C4×D127—24no—
54C2×C4×D127—22no—
55C2×C4×D127—22no—
56C2×C4×D127D412yes—
57C2×C4×D127C2312yes—
58#14523235—28no—
59D1016—28no—
60C2×C127⋊C8—22no—
61C2×C127⋊C8C2312yes—
62(C2×D127)⋊C4—24no—
63C2×Q8×C127—22no—
64C2×Q8×C127—21no—
65C2×Q8×C127D414yescomplemented
66C2×Q8×C127C2312yes—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70C2×C8×C127—22no—
71C2×C8×C127C2312yes—
72C4⋊C4×C127—21no—
73C4⋊C4×C127C2312yes—
74C22×D127—22no—
75C22×D127—24no—
76C22×D127—22no—
77C22×D127—22no—
78C22×D127—22no—
79C22×D127—22no—
80C22×D127—42no—
81C22×D127—42no—
82C22×D127—42no—
83C22×D127—42no—
84C22×D127—42no—
85C22×D127C2×D412yes—
86D508—216no—
87D508—24no—
88D508—22no—
89D508—22no—
90D508—22no—
91D508—48no—
92D508—42no—
93D508C2×D414yescomplemented
94C2×Dic127—22no—
95C2×Dic127C2×D412yes—
96C4×D127—216no—
97C4×D127—24no—
98C4×D127—22no—
99C4×D127—22no—
100C127⋊C8—22no—
101C23×C127C2×D411yescharacteristic
102Q8×C127—24no—
103Q8×C127—22no—
104Q8×C127C2×D414yescomplemented
105C2×C4×C127—24no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—42no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D412yes—
111C2×C4×C127C2×D412yes—
112C2×C4×C127C2411yescharacteristic
113C8×C127—22no—
114C22×C127#12012yes—
115C22×C127#13512yes—
116C22×C127AGL(1,ℤ/8)12yes—
117C22×C127C22×D411yescharacteristic
118C2×D127—22no—
119C2×D127—24no—
120C2×D127—22no—
121C2×D127—22no—
122C2×D127—42no—
123C2×D127—42no—
124C2×D127—44no—
125C2×D127—48no—
126C2×D127—42no—
127C2×D127—42no—
128C2×D127—42no—
129C2×D127—84no—
130C4×C127—24no—
131C4×C127—42no—
132C4×C127#12012yes—
133C4×C127C22×D411yescharacteristic
134C4×C127C22×D411yescharacteristic
135Dic127—22no—
136C2×C127#44914yescomplemented
137C2×C127C2×#12011yescharacteristic
138C2×C127C2×#13511yescharacteristic
139C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
140D127—42no—
141D127—44no—
142D127—82no—
143C2×#449—1271nomaximal, Sylow
144C127C2×#44911yescharacteristic, complemented, Sylow
145C22×#41—1271no—
146C22×SD16—1271no—
147C22×D8—1271no—
148C2×#121—1271no—
149#449—1278no—
150C2×#103D12711yescharacteristic, complemented
151C2×#102—1271no—
152C2×C22⋊C8—1271no—
153C2×#41—1274no—
154C2×#41—2542no—
155C2×#41—2544no—
156C2×#41—2544no—
157C22×Q8C2×D12711yescharacteristic
158C22×D4—1271no—
159C22×D4—1271no—
160C22×D4—2541no—
161C22×D4—2541no—
162C23×C4—1271no—
163C23×C4—2541no—
164C2×SD16—1274no—
165C2×SD16—2544no—
166C2×D8—1274no—
167C2×D8—2544no—
168C22×C8—1271no—
169C22×C8C2×D12711yescharacteristic
170#121—1274no—
171#121—2542no—
172C2×C4⋊C4C2×D12711yescharacteristic
173C2×C22⋊C4—2541no—
174#103C2×D12714yescomplemented
175#102—1274no—
176C22⋊C8—1274no—
177C24—1271no—
178C24—2541no—
179C24—2541no—
180#41—2548no—
181#41—25416no—
182#41—2548no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8D50814yescomplemented
186C2×Q8C22×D12712yes—
187C2×D4—1274no—
188C2×D4—1272no—
189C2×D4—1272no—
190C2×D4—5084no—
191C2×D4—2544no—
192C2×D4—2544no—
193C2×D4—2548no—
194C2×D4—2541no—
195C2×D4—2542no—
196C2×D4—2541no—
197C2×D4—2542no—
198C2×D4—2542no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1271no—
202C22×C4—1272no—
203C22×C4—1272no—
204C22×C4—2544no—
205C22×C4—2548no—
206C22×C4—2542no—
207C22×C4—2542no—
208C22×C4C22×D12711yescharacteristic
209SD16—2548no—
210D8—2548no—
211C2×C8—22no—
212C2×C8—1272no—
213C2×C8—2542no—
214C2×C8C22×D12712yes—
215C4⋊C4—21no—
216C4⋊C4C22×D12712yes—
217C22⋊C4—2544no—
218C23—1272no—
219C23—5082no—
220C23—5082no—
221C23—5082no—
222C23—5082no—
223C23—5082no—
224C23—2542no—
225C23—2542no—
226C23—2542no—
227C23—2542no—
228C23—2544no—
229C23—2542no—
230C23D4×D12711yescharacteristic
231Q8—22no—
232Q8—24no—
233Q8C2×D50814yescomplemented
234D4—1274no—
235D4—5088no—
236D4—5082no—
237D4—25416no—
238D4—2542no—
239D4—2542no—
240D4—2542no—
241D4—2544no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—1272no—
247C2×C4—25416no—
248C2×C4—2542no—
249C2×C4—2542no—
250C2×C4—2542no—
251C2×C4—2544no—
252C2×C4D4×D12712yes—
253C2×C4D4×D12711yescharacteristic
254C2×C4C2×D50812yes—
255C2×C4C23×D12711yescharacteristic
256C8—22no—
257C8—2542no—
258C22—10164no—
259C22—5082no—
260C22—5082no—
261C22—5082no—
262C22—5088no—
263C22—5084no—
264C22—5082no—
265C22—5082no—
266C22—2544no—
267C22—2542no—
268C22—2542no—
269C22—2542no—
270C22#1686117912yes—
271C22#1686123212yes—
272C22#1686123612yes—
273C22C2×D4×D12711yescharacteristic
274C4—24no—
275C4—42no—
276C4—2542no—
277C4#1686117912yes—
278C4C2×D4×D12711yescharacteristic
279C4C22×D50811yescharacteristic
280C2—10162no—
281C2—5084no—
282C2—5082no—
283C2#2255002414yesdirect product factor
284C2C2×#1686117911yescharacteristic
285C2C2×#1686123211yescharacteristic
286C2C2×#1686123611yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
hypercenterC2×#103D127
hypocenterC127C2×#449
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#14523249C2semidirect
2. C22×#14523235C2semidirect
3. C22×D1016C2semidirect
4. C2×#16861184C2semidirect
5. #22550024C2direct (N × H)
6. C2×#16861129C2semidirect
7. C2×(C2×D127)⋊C8C2semidirect
8. C2×#103×C127C2semidirect
12. C2×#14523249C22semidirect
20. C2×#14523235C22semidirect
22. C2×D1016C22semidirect
25. #16861184C22semidirect
27. #16861129C22semidirect
28. (C2×D127)⋊C8C22semidirect
32. #103×C127C22semidirect
48. C2×D508D4semidirect
65. C2×Q8×C127D4semidirect
93. D508C2×D4semidirect
104. Q8×C127C2×D4semidirect
136. C2×C127#449semidirect
144. C127C2×#449semidirect
150. C2×#103D127semidirect
174. #103C2×D127semidirect
185. C2×Q8D508semidirect
233. Q8C2×D508semidirect
283. C2#22550024direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232492
C22×#145232352
C22×D10162
C2×#168611842
#225500242
C2×#168611292
C2×(C2×D127)⋊C82
C2×#103×C1272
C2×#449127