Finite Group Databasebeta

← C2×#20292770 #28345660

Subgroups — C2×#20292770 #28345660

Order 13696.

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×#13133818C211yesmaximal, characteristic, complemented
2C22×#13133806C211yesmaximal, characteristic, complemented
3C22×#13133803C211yesmaximal, characteristic, complemented
4#20292770C218yesmaximal, direct product factor
5C2×#16812032C211yesmaximal, characteristic, complemented
6C2×#16812029C211yesmaximal, characteristic, complemented
7C2×C22⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
8C2×#121×C107C211yesmaximal, characteristic, complemented
9C2×#13133818—24no—
10C2×#13133818—24no—
11C2×#13133818—22no—
12C2×#13133818C2214yescomplemented
13C22×Dic214C2211yescharacteristic
14C22×D428C2211yescharacteristic
15C22×C22⋊D107—21no—
16C22×C4×D107—21no—
17C2×#13133806—24no—
18C2×#13133806C2214yescomplemented
19C2×#13133803—24no—
20C2×#13133803C2214yescomplemented
21C22×C107⋊C8C2211yescharacteristic
22C22×C107⋊C8C2211yescharacteristic
23#16812032C2214yescomplemented
24#16812029C2214yescomplemented
25C22⋊(C107⋊C8)C2214yescomplemented
26C22×D4×C107—21no—
27C22×D4×C107C2211yescharacteristic
28C23×C4×C107C2211yescharacteristic
29#121×C107—22no—
30#121×C107C2214yescomplemented
31C2×C4⋊C4×C107C2211yescharacteristic
32C2×C22⋊C4×C107—21no—
33C23×D107—21no—
34#13133818—216no—
35#13133818—28no—
36#13133818—28no—
37C2×Dic214—22no—
38C2×Dic214—21no—
39C2×Dic214D414yescomplemented
40C2×Dic214C2312yes—
41C2×D428—22no—
42C2×D428—21no—
43C2×D428D414yescomplemented
44C2×D428C2312yes—
45C2×C22⋊D107—28no—
46C2×C22⋊D107—24no—
47C22×Dic107—21no—
48C2×C4×D107—28no—
49C2×C4×D107—24no—
50#13133806—28no—
51#13133803—28no—
52C2×C107⋊C8—22no—
53C2×C107⋊C8—22no—
54C2×C107⋊C8C2312yes—
55C2×C107⋊C8C2312yes—
56C24×C107—21no—
57C24×C107D411yescharacteristic
58C2×D4×C107—24no—
59C2×D4×C107—22no—
60C2×D4×C107—22no—
61C2×D4×C107—21no—
62C2×D4×C107—44no—
63C2×D4×C107C2312yes—
64C22×C4×C107—22no—
65C22×C4×C107—22no—
66C22×C4×C107—21no—
67C22×C4×C107D411yescharacteristic
68C22×C4×C107D412yes—
69C22×C4×C107C2312yes—
70C22×C4×C107C2311yescharacteristic
71C4⋊C4×C107—21no—
72C4⋊C4×C107C2312yes—
73C22⋊C4×C107—24no—
74C23×C107—22no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—42no—
79C23×C107—42no—
80C23×C107—42no—
81C23×C107C2×D412yes—
82C23×C107C2×D411yescharacteristic
83C22×D107—24no—
84C22×D107—22no—
85C22×D107—42no—
86C22×D107—42no—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—48no—
90D4×C107—42no—
91C2×C4×C107—22no—
92C2×C4×C107—24no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—42no—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2×D412yes—
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107C2411yescharacteristic
102C22⋊D107—216no—
103C2×Dic107—24no—
104C2×Dic107—22no—
105D428—24no—
106D428—22no—
107D428C2×D414yescomplemented
108C4×D107—216no—
109Dic214—24no—
110Dic214—22no—
111Dic214C2×D414yescomplemented
112C107⋊C8—22no—
113C107⋊C8—22no—
114C22×C107—22no—
115C22×C107—22no—
116C22×C107—22no—
117C22×C107—44no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107—84no—
122C22×C107#12012yes—
123C22×C107#13512yes—
124C22×C107AGL(1,ℤ/8)12yes—
125C22×C107C22×D411yescharacteristic
126C2×D107—24no—
127C2×D107—42no—
128C2×D107—42no—
129C2×D107—48no—
130C4×C107—22no—
131C4×C107—42no—
132C4×C107#12012yes—
133C4×C107C22×D411yescharacteristic
134C4×C107C22×D411yescharacteristic
135Dic107—24no—
136C2×C107—42no—
137C2×C107—82no—
138C2×C107#44914yescomplemented
139C2×C107C2×#12011yescharacteristic
140C2×C107C2×#13511yescharacteristic
141C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
142D107—44no—
143C2×#449—1071nomaximal, Sylow
144C107C2×#44911yescharacteristic, complemented, Sylow
145C22×#41—1071no—
146C22×SD16—1071no—
147C22×D8—1071no—
148C2×#121D10711yescharacteristic, complemented
149#449—1078no—
150C2×#103—1071no—
151C2×#102—1071no—
152C2×C22⋊C8—1071no—
153C2×#41—1074no—
154C2×#41—2144no—
155C2×#41—2144no—
156C2×#41—2142no—
157C22×Q8—1071no—
158C22×D4—21no—
159C22×D4—1071no—
160C22×D4—2141no—
161C22×D4C2×D10711yescharacteristic
162C23×C4—2141no—
163C23×C4C2×D10711yescharacteristic
164C2×SD16—1074no—
165C2×SD16—2144no—
166C2×D8—1074no—
167C2×D8—2144no—
168C22×C8—1071no—
169C22×C8—1071no—
170#121—22no—
171#121C2×D10714yescomplemented
172C2×C4⋊C4C2×D10711yescharacteristic
173C2×C22⋊C4—21no—
174#103—1074no—
175#102—1074no—
176C22⋊C8—1074no—
177C24—21no—
178C24—2141no—
179C24C22⋊D10711yescharacteristic
180#41—2148no—
181#41—2148no—
182#41—21416no—
183C2×Q8—1074no—
184C2×Q8—1072no—
185C2×Q8—2142no—
186C2×Q8—2141no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—44no—
192C2×D4—1074no—
193C2×D4—1072no—
194C2×D4—2148no—
195C2×D4—2144no—
196C2×D4—2142no—
197C2×D4—2141no—
198C2×D4C22×D10712yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—2148no—
203C22×C4—2141no—
204C22×C4—2144no—
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—21no—
216C4⋊C4C22×D10712yes—
217C22⋊C4—24no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—4282no—
226C23—4282no—
227C23—2144no—
228C23—2142no—
229C23C2×C22⋊D10712yes—
230C23C2×C22⋊D10711yescharacteristic
231Q8—1074no—
232Q8—2144no—
233Q8—2142no—
234D4—22no—
235D4—22no—
236D4—48no—
237D4—42no—
238D4—1074no—
239D4—21416no—
240D4—2144no—
241D4—2142no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—21416no—
249C2×C4—2144no—
250C2×C4—2142no—
251C2×C4D4×D10712yes—
252C2×C4D4×D10712yes—
253C2×C4C2×C22⋊D10712yes—
254C2×C4C2×C22⋊D10711yescharacteristic
255C2×C4C23×D10711yescharacteristic
256C8—2142no—
257C8—2142no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—44no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—4282no—
267C22—4282no—
268C22—4288no—
269C22—2144no—
270C22#1681207712yes—
271C22#1681212912yes—
272C22#1681213512yes—
273C22C22×C22⋊D10711yescharacteristic
274C4—22no—
275C4—42no—
276C4—2144no—
277C4#1681207712yes—
278C4C2×D4×D10711yescharacteristic
279C4C2×D4×D10711yescharacteristic
280C2—42no—
281C2—82no—
282C2—4284no—
283C2#2029277014yesdirect product factor
284C2C2×#1681207711yescharacteristic
285C2C2×#1681212911yescharacteristic
286C2C2×#1681213511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#121×C107C2
socleC23×C107C2×D4
hypercenterC2×#121D107
hypocenterC107C2×#449
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C22×#13133806C2semidirect
3. C22×#13133803C2semidirect
4. #20292770C2direct (N × H)
5. C2×#16812032C2semidirect
6. C2×#16812029C2semidirect
7. C2×C22⋊(C107⋊C8)C2semidirect
8. C2×#121×C107C2semidirect
12. C2×#13133818C22semidirect
18. C2×#13133806C22semidirect
20. C2×#13133803C22semidirect
23. #16812032C22semidirect
24. #16812029C22semidirect
25. C22⋊(C107⋊C8)C22semidirect
30. #121×C107C22semidirect
39. C2×Dic214D4semidirect
43. C2×D428D4semidirect
107. D428C2×D4semidirect
111. Dic214C2×D4semidirect
138. C2×C107#449semidirect
144. C107C2×#449semidirect
148. C2×#121D107semidirect
171. #121C2×D107semidirect
283. C2#20292770direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338182
C22×#131338062
C22×#131338032
#202927702
C2×#168120322
C2×#168120292
C2×C22⋊(C107⋊C8)2
C2×#121×C1072
C2×#449107