Finite Group Databasebeta

← C2×#20292856 #28346021

Subgroups — C2×#20292856 #28346021

Order 13696.

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

normal subgroups only

This group has 564 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×#13133807C211yesmaximal, characteristic, complemented
2C2×#16812088C211yesmaximal, characteristic, complemented
3C2×#16812083C211yesmaximal, characteristic
4#20292856C218yesmaximal, direct product factor
5C2×#16812041C211yesmaximal, characteristic, complemented
6C2×#16812034C211yesmaximal, characteristic, complemented
7C2×#16812031C211yesmaximal, characteristic, complemented
8C23×C8×C107C211yesmaximal, characteristic, complemented
9C22×Dic214C2211yescharacteristic
10C22×D428C2211yescharacteristic
11C22×C22⋊D107—21no—
12C2×#13133807—24no—
13C2×#13133807C2214yescomplemented
14#16812088—22no—
15#16812088C2214yescomplemented
16#16812083—22no—
17#16812083C2214yes—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4—21no—
21C2×C22⋊Dic107—21no—
22C2×(C2×D107)⋊C4—21no—
23#16812041C2214yescomplemented
24#16812034C2214yescomplemented
25#16812031C2214yescomplemented
26C23×C4×C107C2211yescharacteristic
27C22×C8×C107—24no—
28C22×C8×C107C2211yescharacteristic
29C22×C8×C107C2211yescharacteristic
30C22×C8×C107C2214yescomplemented
31C23×D107—21no—
32C2×Dic214—22no—
33C2×Dic214—22no—
34C2×Dic214—21no—
35C2×Dic214C2312yes—
36C2×D428—22no—
37C2×D428—22no—
38C2×D428—21no—
39C2×D428C2312yes—
40C2×C22⋊D107—24no—
41C2×C22⋊D107—44no—
42C22×Dic107—21no—
43C22×Dic107—21no—
44C22×Dic107—21no—
45#13133807—28no—
46C4⋊Dic107—21no—
47C4⋊Dic107—21no—
48C4⋊Dic107C2312yes—
49C4⋊Dic107C2312yes—
50Dic107⋊C4—24no—
51C22⋊Dic107—24no—
52(C2×D107)⋊C4—24no—
53C24×C107D411yescharacteristic
54C22×C4×C107—22no—
55C22×C4×C107—22no—
56C22×C4×C107D411yescharacteristic
57C22×C4×C107D412yes—
58C22×C4×C107C2312yes—
59C22×C4×C107C2311yescharacteristic
60C2×C8×C107—22no—
61C2×C8×C107—28no—
62C2×C8×C107D414yescomplemented
63C2×C8×C107C2312yes—
64C2×C8×C107C2312yes—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107SD1614yescomplemented
68C23×C107C2×D412yes—
69C23×C107C2×D411yescharacteristic
70C22×D107—22no—
71C22×D107—42no—
72C22×D107—42no—
73C22×D107—42no—
74C2×C4×C107—22no—
75C2×C4×C107—24no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2×D412yes—
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107#4112yes—
82C2×C4×C107C2411yescharacteristic
83C22⋊D107—48no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—42no—
88C2×Dic107—42no—
89C2×Dic107—42no—
90D428—22no—
91D428—22no—
92D428—42no—
93Dic214—22no—
94Dic214—22no—
95Dic214—42no—
96C8×C107—22no—
97C8×C107C2×D414yescomplemented
98C22×C107—22no—
99C22×C107—22no—
100C22×C107—28no—
101C22×C107#12112yes—
102C22×C107C2×SD1612yes—
103C22×C107C2×SD1614yescomplemented
104C22×C107#13512yes—
105C22×C107C22×D411yescharacteristic
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—42no—
109C2×D107—84no—
110C4×C107—22no—
111C4×C107#12112yes—
112C4×C107C22×D411yescharacteristic
113C4×C107C2×#4111yescharacteristic
114Dic107—42no—
115Dic107—42no—
116Dic107—42no—
117C2×C107—24no—
118C2×C107#46514yescomplemented
119C2×C107C2×#12111yescharacteristic
120C2×C107C22×SD1611yescharacteristic
121C2×C107C2×#13511yescharacteristic
122D107—82no—
123C2×#465—1071nomaximal, Sylow
124C107C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16—1071no—
126C23×C8D10711yescharacteristic, complemented
127C2×C22⋊Q8—1071no—
128C2×#121—1071no—
129#465—1078no—
130C2×#106—1071no—
131C2×#103—1071no—
132C2×#102—1071no—
133C22×Q8—1071no—
134C22×D4—1071no—
135C22×D4—2141no—
136C23×C4C2×D10711yescharacteristic
137C2×SD16—1074no—
138C2×SD16—2144no—
139C22×C8—24no—
140C22×C8C2×D10711yescharacteristic
141C22×C8C2×D10711yescharacteristic
142C22×C8C2×D10714yescomplemented
143C22⋊Q8—1074no—
144C22⋊Q8—2142no—
145#121—1074no—
146#121—2142no—
147C2×C4⋊C4—1071no—
148C2×C4⋊C4—1071no—
149C2×C4⋊C4—2141no—
150C2×C22⋊C4—2141no—
151C2×C22⋊C4—2141no—
152#106—1074no—
153#103—1074no—
154#102—1074no—
155C24—2141no—
156C24D42811yescharacteristic
157C2×Q8—1072no—
158C2×Q8—2142no—
159C2×Q8—2142no—
160C2×Q8—2141no—
161C2×D4—1072no—
162C2×D4—4284no—
163C2×D4—2144no—
164C2×D4—2142no—
165C2×D4—2141no—
166C2×D4—2142no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—2141no—
170C22×C4—2141no—
171C22×C4—2141no—
172C22×C4D42812yes—
173C22×C4D42811yescharacteristic
174C22×C4C22×D10712yes—
175C22×C4C22×D10711yescharacteristic
176SD16—2148no—
177C2×C8—22no—
178C2×C8—28no—
179C2×C8C22⋊D10714yescomplemented
180C2×C8C22×D10712yes—
181C2×C8C22×D10712yes—
182C4⋊C4—1072no—
183C4⋊C4—1072no—
184C4⋊C4—2141no—
185C4⋊C4—2141no—
186C4⋊C4—2144no—
187C22⋊C4—2144no—
188C22⋊C4—2144no—
189C23—22no—
190C23—22no—
191C23—4282no—
192C23—4282no—
193C23—4282no—
194C23—2142no—
195C23#1313380714yescomplemented
196C23C2×D42812yes—
197C23C2×D42811yescharacteristic
198Q8—4282no—
199Q8—2142no—
200Q8—2142no—
201D4—4288no—
202D4—4282no—
203D4—2142no—
204D4—2142no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—4282no—
210C2×C4—4282no—
211C2×C4—4282no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4C2×C22⋊D10712yes—
216C2×C4C2×D42812yes—
217C2×C4C2×D42811yescharacteristic
218C2×C4#1313381812yes—
219C2×C4C23×D10711yescharacteristic
220C8—22no—
221C8C2×C22⋊D10714yescomplemented
222C22—22no—
223C22—28no—
224C22—22no—
225C22—8564no—
226C22—4282no—
227C22—4282no—
228C22—4282no—
229C22#1681208312yes—
230C22C2×#1313380712yes—
231C22C2×#1313380714yescomplemented
232C22#1681213112yes—
233C22C22×D42811yescharacteristic
234C4—22no—
235C4—4282no—
236C4—4282no—
237C4—4282no—
238C4#1681208312yes—
239C4C22×C22⋊D10711yescharacteristic
240C4C2×#1313381811yescharacteristic
241C2—24no—
242C2—8562no—
243C2#2029285614yesdirect product factor
244C2C2×#1681208311yescharacteristic
245C2C22×#1313380711yescharacteristic
246C2C2×#1681213111yescharacteristic
subgroupG/H
centerC23C2×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC23×C8×C107C2
socleC23×C107C2×D4
prime radicalC2×#16812083C2
hypercenterC23×C8D107
hypocenterC107C2×#465
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. C22×#13133807C2semidirect
2. C2×#16812088C2semidirect
4. #20292856C2direct (N × H)
5. C2×#16812041C2semidirect
6. C2×#16812034C2semidirect
7. C2×#16812031C2semidirect
8. C23×C8×C107C2semidirect
13. C2×#13133807C22semidirect
15. #16812088C22semidirect
23. #16812041C22semidirect
24. #16812034C22semidirect
25. #16812031C22semidirect
30. C22×C8×C107C22semidirect
62. C2×C8×C107D4semidirect
67. C23×C107SD16semidirect
97. C8×C107C2×D4semidirect
103. C22×C107C2×SD16semidirect
118. C2×C107#465semidirect
124. C107C2×#465semidirect
126. C23×C8D107semidirect
142. C22×C8C2×D107semidirect
179. C2×C8C22⋊D107semidirect
195. C23#13133807semidirect
221. C8C2×C22⋊D107semidirect
231. C22C2×#13133807semidirect
243. C2#20292856direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338072
C2×#168120882
C2×#168120832
#202928562
C2×#168120412
C2×#168120342
C2×#168120312
C23×C8×C1072
C2×#465107