Finite Group Databasebeta

← C2×#20292789 #28345670

Subgroups — C2×#20292789 #28345670

Order 13696.

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

normal subgroups only

This group has 760 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×#13133808C211yesmaximal, characteristic, complemented
2C22×#13133806C211yesmaximal, characteristic, complemented
3C2×#16812088C211yesmaximal, characteristic, complemented
4#20292789C218yesmaximal, direct product factor
5C2×#16812033C211yesmaximal, characteristic, complemented
6C2×#16812030C211yesmaximal, characteristic, complemented
7C2×C22⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
8C22×#41×C107C211yesmaximal, characteristic
9C22×Dic214C2211yescharacteristic
10C2×#13133808—24no—
11C2×#13133808C2214yescomplemented
12C2×#13133806—24no—
13C2×#13133806C2214yescomplemented
14C22×C107⋊C8C2211yescharacteristic
15C22×C107⋊C8C2211yescharacteristic
16#16812088—22no—
17#16812088C2214yescomplemented
18C2×C4⋊Dic107C2211yescharacteristic
19C2×Dic107⋊C4—21no—
20C2×C22⋊Dic107—21no—
21#16812033C2214yescomplemented
22#16812030C2214yescomplemented
23C22⋊(C107⋊C8)C2214yescomplemented
24C2×#41×C107—24no—
25C2×#41×C107—24no—
26C2×#41×C107—22no—
27C2×#41×C107C2214yes—
28C22×Q8×C107C2211yescharacteristic
29C22×D4×C107—21no—
30C22×D4×C107C2211yescharacteristic
31C23×C4×C107—21no—
32C23×C4×C107C2211yescharacteristic
33C2×Dic214—22no—
34C2×Dic214—22no—
35C2×Dic214—21no—
36C2×Dic214C2312yes—
37C22×Dic107—21no—
38C22×Dic107—21no—
39#13133808—28no—
40#13133806—28no—
41C2×C107⋊C8—22no—
42C2×C107⋊C8—22no—
43C2×C107⋊C8C2312yes—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107—21no—
46C4⋊Dic107C2312yes—
47Dic107⋊C4—24no—
48C22⋊Dic107—24no—
49C24×C107—21no—
50C24×C107D411yescharacteristic
51#41×C107—216no—
52#41×C107—28no—
53#41×C107—28no—
54C2×Q8×C107—22no—
55C2×Q8×C107—21no—
56C2×Q8×C107D414yes—
57C2×Q8×C107C2312yes—
58C2×D4×C107—28no—
59C2×D4×C107—24no—
60C2×D4×C107—21no—
61C2×D4×C107—22no—
62C2×D4×C107D414yes—
63C2×D4×C107C2312yes—
64C22×C4×C107—28no—
65C22×C4×C107—21no—
66C22×C4×C107—24no—
67C22×C4×C107—22no—
68C22×C4×C107—22no—
69C22×C4×C107D411yescharacteristic
70C22×C4×C107D412yes—
71C22×C4×C107C2312yes—
72C22×C4×C107C2311yescharacteristic
73C23×C107—22no—
74C23×C107—24no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—42no—
79C23×C107—42no—
80C23×C107C2×D412yes—
81C23×C107C2×D411yescharacteristic
82Q8×C107—24no—
83Q8×C107—22no—
84Q8×C107C2×D414yes—
85D4×C107—216no—
86D4×C107—24no—
87D4×C107—22no—
88D4×C107C2×D414yes—
89C2×C4×C107—216no—
90C2×C4×C107—22no—
91C2×C4×C107—24no—
92C2×C4×C107—22no—
93C2×C4×C107—24no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2×D412yes—
100C2×C4×C107C2411yescharacteristic
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—42no—
104C2×Dic107—42no—
105Dic214—22no—
106Dic214—22no—
107Dic214—42no—
108C107⋊C8—22no—
109C107⋊C8—22no—
110C22×C107—22no—
111C22×C107—24no—
112C22×C107—22no—
113C22×C107—22no—
114C22×C107—42no—
115C22×C107—42no—
116C22×C107—44no—
117C22×C107—48no—
118C22×C107#12012yes—
119C22×C107#13512yes—
120C22×C107#13712yes—
121C22×C107C22×D411yescharacteristic
122C4×C107—22no—
123C4×C107—24no—
124C4×C107#12012yes—
125C4×C107C22×D411yescharacteristic
126C4×C107C22×D411yescharacteristic
127Dic107—42no—
128Dic107—42no—
129C2×C107—42no—
130C2×C107—44no—
131C2×C107#45214yescomplemented
132C2×C107C2×#12011yescharacteristic
133C2×C107C2×#13511yescharacteristic
134C2×C107C2×#13711yescharacteristic
135C2×#452—1071nomaximal, Sylow
136C107C2×#45211yescharacteristic, complemented, Sylow
137C22×#41D10711yescharacteristic
138C22×Q16—1071no—
139C22×SD16—1071no—
140C2×C22⋊Q8—1071no—
141#452—1078no—
142C2×#103—1071no—
143C2×#102—1071no—
144C2×C22⋊C8—1071no—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—22no—
148C2×#41C2×D10714yes—
149C22×Q8—1071no—
150C22×Q8C2×D10711yescharacteristic
151C22×D4—21no—
152C22×D4C2×D10711yescharacteristic
153C23×C4—21no—
154C23×C4C2×D10711yescharacteristic
155C2×Q16—1074no—
156C2×Q16—2144no—
157C2×SD16—1074no—
158C2×SD16—2144no—
159C22×C8—1071no—
160C22×C8—1071no—
161C22⋊Q8—1074no—
162C22⋊Q8—2142no—
163C2×C4⋊C4—1071no—
164C2×C4⋊C4—2141no—
165C2×C22⋊C4—2141no—
166#103—1074no—
167#102—1074no—
168C22⋊C8—1074no—
169C24—21no—
170C24C22⋊D10711yescharacteristic
171#41—216no—
172#41—28no—
173#41—28no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—1072no—
177C2×Q8—2142no—
178C2×Q8—2142no—
179C2×Q8—2141no—
180C2×Q8C22⋊D10714yes—
181C2×Q8C22×D10712yes—
182C2×D4—28no—
183C2×D4—24no—
184C2×D4—21no—
185C2×D4—22no—
186C2×D4C22⋊D10714yes—
187C2×D4C22×D10712yes—
188C22×C4—28no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—2141no—
194C22×C4—2141no—
195C22×C4C22⋊D10712yes—
196C22×C4C22⋊D10711yescharacteristic
197C22×C4C22×D10712yes—
198C22×C4C22×D10711yescharacteristic
199Q16—2148no—
200SD16—2148no—
201C2×C8—1072no—
202C2×C8—1072no—
203C2×C8—2142no—
204C2×C8—2142no—
205C4⋊C4—1072no—
206C4⋊C4—2141no—
207C4⋊C4—2144no—
208C22⋊C4—2144no—
209C23—22no—
210C23—24no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—42no—
215C23—42no—
216C23C2×C22⋊D10712yes—
217C23C2×C22⋊D10711yescharacteristic
218Q8—24no—
219Q8—22no—
220Q8—4282no—
221Q8—2142no—
222Q8—2142no—
223Q8C2×C22⋊D10714yes—
224D4—216no—
225D4—24no—
226D4—22no—
227D4C2×C22⋊D10714yes—
228C2×C4—216no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—4282no—
236C2×C4—4282no—
237C2×C4—2142no—
238C2×C4—2142no—
239C2×C4C2×C22⋊D10712yes—
240C2×C4C2×C22⋊D10711yescharacteristic
241C2×C4C2×C22⋊D10712yes—
242C2×C4C2×C22⋊D10712yes—
243C2×C4C23×D10711yescharacteristic
244C8—2142no—
245C8—2142no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—24no—
250C22—42no—
251C22—42no—
252C22—44no—
253C22—48no—
254C22#1681207812yes—
255C22#1681212912yes—
256C22#1681213912yes—
257C22C22×C22⋊D10711yescharacteristic
258C4—22no—
259C4—24no—
260C4—4282no—
261C4—4282no—
262C4#1681207812yes—
263C4C22×C22⋊D10711yescharacteristic
264C4C22×C22⋊D10711yescharacteristic
265C2—42no—
266C2—44no—
267C2#2029278914yesdirect product factor
268C2C2×#1681207811yescharacteristic
269C2C2×#1681212911yescharacteristic
270C2C2×#1681213911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalC22×#41×C107C2
socleC23×C107C2×D4
hypercenterC22×#41D107
hypocenterC107C2×#452
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×#13133808C2semidirect
2. C22×#13133806C2semidirect
3. C2×#16812088C2semidirect
4. #20292789C2direct (N × H)
5. C2×#16812033C2semidirect
6. C2×#16812030C2semidirect
7. C2×C22⋊(C107⋊C8)C2semidirect
11. C2×#13133808C22semidirect
13. C2×#13133806C22semidirect
17. #16812088C22semidirect
21. #16812033C22semidirect
22. #16812030C22semidirect
23. C22⋊(C107⋊C8)C22semidirect
131. C2×C107#452semidirect
136. C107C2×#452semidirect
267. C2#20292789direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338082
C22×#131338062
C2×#168120882
#202927892
C2×#168120332
C2×#168120302
C2×C22⋊(C107⋊C8)2
C22×#41×C1072
C2×#452107