Finite Group Databasebeta

← C2×#20292772 #28345662

Subgroups — C2×#20292772 #28345662

Order 13696.

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

normal subgroups only

This group has 812 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
1C22×#13133805C211yesmaximal, characteristic, complemented
2C22×#13133803C211yesmaximal, characteristic, complemented
3C2×#16812083C211yesmaximal, characteristic, complemented
4#20292772C218yesmaximal, direct product factor
5C2×#16812033C211yesmaximal, characteristic, complemented
6C2×#16812030C211yesmaximal, characteristic, complemented
7C2×C22⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
8C22×#41×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107—21no—
11C2×#13133805—24no—
12C2×#13133805C2214yescomplemented
13C2×#13133803—24no—
14C2×#13133803C2214yescomplemented
15C22×C107⋊C8C2211yescharacteristic
16C22×C107⋊C8C2211yescharacteristic
17#16812083—22no—
18#16812083C2214yescomplemented
19C2×C4⋊Dic107C2211yescharacteristic
20C2×(C2×D107)⋊C4—21no—
21#16812033C2214yescomplemented
22#16812030C2214yescomplemented
23C22⋊(C107⋊C8)C2214yescomplemented
24C2×#41×C107—24no—
25C2×#41×C107—24no—
26C2×#41×C107—22no—
27C2×#41×C107C2214yescomplemented
28C22×Q8×C107C2211yescharacteristic
29C22×D4×C107—21no—
30C22×D4×C107C2211yescharacteristic
31C23×C4×C107—21no—
32C23×C4×C107C2211yescharacteristic
33C23×D107—21no—
34C2×D428—22no—
35C2×D428—22no—
36C2×D428—21no—
37C2×D428C2312yes—
38C2×C22⋊D107—24no—
39C2×C22⋊D107—44no—
40C22×Dic107—21no—
41#13133805—28no—
42#13133803—28no—
43C2×C107⋊C8—22no—
44C2×C107⋊C8—22no—
45C2×C107⋊C8C2312yes—
46C2×C107⋊C8C2312yes—
47C4⋊Dic107—21no—
48C4⋊Dic107C2312yes—
49(C2×D107)⋊C4—24no—
50C24×C107—21no—
51C24×C107D411yescharacteristic
52#41×C107—216no—
53#41×C107—28no—
54#41×C107—28no—
55C2×Q8×C107—22no—
56C2×Q8×C107—21no—
57C2×Q8×C107D414yescomplemented
58C2×Q8×C107C2312yes—
59C2×D4×C107—28no—
60C2×D4×C107—24no—
61C2×D4×C107—21no—
62C2×D4×C107—22no—
63C2×D4×C107D414yescomplemented
64C2×D4×C107C2312yes—
65C22×C4×C107—28no—
66C22×C4×C107—21no—
67C22×C4×C107—24no—
68C22×C4×C107—22no—
69C22×C4×C107—22no—
70C22×C4×C107D411yescharacteristic
71C22×C4×C107D412yes—
72C22×C4×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74C23×C107—22no—
75C23×C107—24no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—42no—
80C23×C107—42no—
81C23×C107C2×D412yes—
82C23×C107C2×D411yescharacteristic
83C22×D107—22no—
84C22×D107—42no—
85C22×D107—42no—
86C22×D107—42no—
87Q8×C107—24no—
88Q8×C107—22no—
89Q8×C107C2×D414yescomplemented
90D4×C107—216no—
91D4×C107—24no—
92D4×C107—22no—
93D4×C107C2×D414yescomplemented
94C2×C4×C107—216no—
95C2×C4×C107—22no—
96C2×C4×C107—24no—
97C2×C4×C107—22no—
98C2×C4×C107—24no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2×D412yes—
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2411yescharacteristic
106C22⋊D107—48no—
107C2×Dic107—22no—
108C2×Dic107—42no—
109D428—22no—
110D428—22no—
111D428—42no—
112C107⋊C8—22no—
113C107⋊C8—22no—
114C22×C107—22no—
115C22×C107—24no—
116C22×C107—22no—
117C22×C107—22no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—44no—
121C22×C107—48no—
122C22×C107#12012yes—
123C22×C107#13512yes—
124C22×C107AGL(1,ℤ/8)12yes—
125C22×C107C22×D411yescharacteristic
126C2×D107—42no—
127C2×D107—42no—
128C2×D107—42no—
129C2×D107—84no—
130C4×C107—22no—
131C4×C107—24no—
132C4×C107#12012yes—
133C4×C107C22×D411yescharacteristic
134C4×C107C22×D411yescharacteristic
135Dic107—42no—
136C2×C107—42no—
137C2×C107—44no—
138C2×C107#44914yescomplemented
139C2×C107C2×#12011yescharacteristic
140C2×C107C2×#13511yescharacteristic
141C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
142D107—82no—
143C2×#449—1071nomaximal, Sylow
144C107C2×#44911yescharacteristic, complemented, Sylow
145C22×#41D10711yescharacteristic, complemented
146C22×SD16—1071no—
147C22×D8—1071no—
148C2×#121—1071no—
149#449—1078no—
150C2×#103—1071no—
151C2×#102—1071no—
152C2×C22⋊C8—1071no—
153C2×#41—24no—
154C2×#41—24no—
155C2×#41—22no—
156C2×#41C2×D10714yescomplemented
157C22×Q8C2×D10711yescharacteristic
158C22×D4—21no—
159C22×D4—1071no—
160C22×D4—2141no—
161C22×D4C2×D10711yescharacteristic
162C23×C4—21no—
163C23×C4C2×D10711yescharacteristic
164C2×SD16—1074no—
165C2×SD16—2144no—
166C2×D8—1074no—
167C2×D8—2144no—
168C22×C8—1071no—
169C22×C8—1071no—
170#121—1074no—
171#121—2142no—
172C2×C4⋊C4—1071no—
173C2×C22⋊C4—2141no—
174#103—1074no—
175#102—1074no—
176C22⋊C8—1074no—
177C24—21no—
178C24—2141no—
179C24C22⋊D10711yescharacteristic
180#41—216no—
181#41—28no—
182#41—28no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8C22⋊D10714yescomplemented
186C2×Q8C22×D10712yes—
187C2×D4—28no—
188C2×D4—24no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—1072no—
192C2×D4—4284no—
193C2×D4—2144no—
194C2×D4—2141no—
195C2×D4—2142no—
196C2×D4—2142no—
197C2×D4C22⋊D10714yescomplemented
198C2×D4C22×D10712yes—
199C22×C4—28no—
200C22×C4—21no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—2141no—
205C22×C4C22⋊D10712yes—
206C22×C4C22⋊D10711yescharacteristic
207C22×C4C22×D10712yes—
208C22×C4C22×D10711yescharacteristic
209SD16—2148no—
210D8—2148no—
211C2×C8—1072no—
212C2×C8—1072no—
213C2×C8—2142no—
214C2×C8—2142no—
215C4⋊C4—1072no—
216C4⋊C4—2141no—
217C22⋊C4—2144no—
218C23—22no—
219C23—24no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—4282no—
226C23—4282no—
227C23—4282no—
228C23—2142no—
229C23C2×C22⋊D10712yes—
230C23C2×C22⋊D10711yescharacteristic
231Q8—24no—
232Q8—22no—
233Q8C2×C22⋊D10714yescomplemented
234D4—216no—
235D4—24no—
236D4—22no—
237D4—4288no—
238D4—4282no—
239D4—2142no—
240D4—2142no—
241D4C2×C22⋊D10714yescomplemented
242C2×C4—216no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—4282no—
250C2×C4—2142no—
251C2×C4C2×C22⋊D10712yes—
252C2×C4C2×C22⋊D10711yescharacteristic
253C2×C4C2×C22⋊D10712yes—
254C2×C4C2×C22⋊D10712yes—
255C2×C4C23×D10711yescharacteristic
256C8—2142no—
257C8—2142no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—24no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—48no—
266C22—8564no—
267C22—4282no—
268C22—4282no—
269C22—4282no—
270C22#1681207812yes—
271C22#1681212912yes—
272C22#1681213412yes—
273C22C22×C22⋊D10711yescharacteristic
274C4—22no—
275C4—24no—
276C4—4282no—
277C4#1681207812yes—
278C4C22×C22⋊D10711yescharacteristic
279C4C22×C22⋊D10711yescharacteristic
280C2—42no—
281C2—44no—
282C2—8562no—
283C2#2029277214yesdirect product factor
284C2C2×#1681207811yescharacteristic
285C2C2×#1681212911yescharacteristic
286C2C2×#1681213411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×#41×C107C2
socleC23×C107C2×D4
hypercenterC22×#41D107
hypocenterC107C2×#449
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#13133805C2semidirect
2. C22×#13133803C2semidirect
3. C2×#16812083C2semidirect
4. #20292772C2direct (N × H)
5. C2×#16812033C2semidirect
6. C2×#16812030C2semidirect
7. C2×C22⋊(C107⋊C8)C2semidirect
8. C22×#41×C107C2semidirect
12. C2×#13133805C22semidirect
14. C2×#13133803C22semidirect
18. #16812083C22semidirect
21. #16812033C22semidirect
22. #16812030C22semidirect
23. C22⋊(C107⋊C8)C22semidirect
27. C2×#41×C107C22semidirect
57. C2×Q8×C107D4semidirect
63. C2×D4×C107D4semidirect
89. Q8×C107C2×D4semidirect
93. D4×C107C2×D4semidirect
138. C2×C107#449semidirect
144. C107C2×#449semidirect
145. C22×#41D107semidirect
156. C2×#41C2×D107semidirect
185. C2×Q8C22⋊D107semidirect
197. C2×D4C22⋊D107semidirect
233. Q8C2×C22⋊D107semidirect
241. D4C2×C22⋊D107semidirect
283. C2#20292772direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338052
C22×#131338032
C2×#168120832
#202927722
C2×#168120332
C2×#168120302
C2×C22⋊(C107⋊C8)2
C22×#41×C1072
C2×#449107