Finite Group Databasebeta

← C2×#20292854 #28346019

Subgroups — C2×#20292854 #28346019

Order 13696.

564 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13133806C211yesmaximal, characteristic, complemented
2C23×C107⋊C8C211yesmaximal, characteristic, complemented
3C2×#16812088C211yesmaximal, characteristic, complemented
4#20292854C218yesmaximal, direct product factor
5C2×#16812040C211yesmaximal, characteristic, complemented
6C2×#16812032C211yesmaximal, characteristic, complemented
7C2×#16812030C211yesmaximal, characteristic, complemented
8C2×#121×C107C211yesmaximal, characteristic
9C22×Dic214C2211yescharacteristic
10C2×#13133806—24no—
11C2×#13133806C2214yescomplemented
12C22×C107⋊C8—24no—
13C22×C107⋊C8C2211yescharacteristic
14C22×C107⋊C8C2211yescharacteristic
15C22×C107⋊C8C2214yescomplemented
16#16812088—22no—
17#16812088C2214yescomplemented
18C2×C4⋊Dic107C2211yescharacteristic
19C2×Dic107⋊C4—21no—
20C2×C22⋊Dic107—21no—
21#16812040C2214yescomplemented
22#16812032C2214yescomplemented
23#16812030C2214yescomplemented
24C22×D4×C107—21no—
25C22×D4×C107C2211yescharacteristic
26C23×C4×C107C2211yescharacteristic
27#121×C107—22no—
28#121×C107C2214yes—
29C2×C4⋊C4×C107C2211yescharacteristic
30C2×C22⋊C4×C107—21no—
31C2×Dic214—22no—
32C2×Dic214—22no—
33C2×Dic214—21no—
34C2×Dic214C2312yes—
35C22×Dic107—21no—
36C22×Dic107—21no—
37#13133806—28no—
38C2×C107⋊C8—22no—
39C2×C107⋊C8—28no—
40C2×C107⋊C8D414yescomplemented
41C2×C107⋊C8C2312yes—
42C2×C107⋊C8C2312yes—
43C4⋊Dic107—21no—
44C4⋊Dic107C2312yes—
45Dic107⋊C4—24no—
46C22⋊Dic107—24no—
47C24×C107—21no—
48C24×C107D411yescharacteristic
49C2×D4×C107—24no—
50C2×D4×C107—22no—
51C2×D4×C107—21no—
52C2×D4×C107—22no—
53C2×D4×C107—44no—
54C2×D4×C107C2312yes—
55C22×C4×C107—22no—
56C22×C4×C107—22no—
57C22×C4×C107—21no—
58C22×C4×C107D411yescharacteristic
59C22×C4×C107D412yes—
60C22×C4×C107C2312yes—
61C22×C4×C107C2311yescharacteristic
62C4⋊C4×C107—21no—
63C4⋊C4×C107C2312yes—
64C22⋊C4×C107—24no—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107—22no—
68C23×C107—42no—
69C23×C107—42no—
70C23×C107—42no—
71C23×C107SD1614yescomplemented
72C23×C107C2×D412yes—
73C23×C107C2×D411yescharacteristic
74D4×C107—22no—
75D4×C107—22no—
76D4×C107—48no—
77D4×C107—42no—
78C2×C4×C107—22no—
79C2×C4×C107—24no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—42no—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107#4112yes—
88C2×C4×C107C2411yescharacteristic
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—42no—
92C2×Dic107—42no—
93Dic214—22no—
94Dic214—22no—
95Dic214—42no—
96C107⋊C8—22no—
97C107⋊C8C2×D414yescomplemented
98C22×C107—22no—
99C22×C107—22no—
100C22×C107—28no—
101C22×C107—42no—
102C22×C107—42no—
103C22×C107—42no—
104C22×C107—84no—
105C22×C107#12112yes—
106C22×C107C2×SD1612yes—
107C22×C107C2×SD1614yescomplemented
108C22×C107#13512yes—
109C22×C107C22×D411yescharacteristic
110C4×C107—22no—
111C4×C107—42no—
112C4×C107#12112yes—
113C4×C107C22×D411yescharacteristic
114C4×C107C2×#4111yescharacteristic
115Dic107—42no—
116Dic107—42no—
117C2×C107—24no—
118C2×C107—82no—
119C2×C107#46514yescomplemented
120C2×C107C2×#12111yescharacteristic
121C2×C107C22×SD1611yescharacteristic
122C2×C107C2×#13511yescharacteristic
123C2×#465—1071nomaximal, Sylow
124C107C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16—1071no—
126C23×C8—1071no—
127C2×C22⋊Q8—1071no—
128C2×#121D10711yescharacteristic
129#465—1078no—
130C2×#106—1071no—
131C2×#103—1071no—
132C2×#102—1071no—
133C22×Q8—1071no—
134C22×D4—21no—
135C22×D4C2×D10711yescharacteristic
136C23×C4C2×D10711yescharacteristic
137C2×SD16—1074no—
138C2×SD16—2144no—
139C22×C8—1071no—
140C22×C8—1071no—
141C22×C8—1074no—
142C22×C8—2144no—
143C22⋊Q8—1074no—
144C22⋊Q8—2142no—
145#121—22no—
146#121C2×D10714yes—
147C2×C4⋊C4—1071no—
148C2×C4⋊C4—2141no—
149C2×C4⋊C4C2×D10711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2141no—
152#106—1074no—
153#103—1074no—
154#102—1074no—
155C24—21no—
156C24C22⋊D10711yescharacteristic
157C2×Q8—1072no—
158C2×Q8—2142no—
159C2×Q8—2142no—
160C2×Q8—2141no—
161C2×D4—24no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—22no—
165C2×D4—44no—
166C2×D4C22×D10712yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—2141no—
171C22×C4—2141no—
172C22×C4C22⋊D10712yes—
173C22×C4C22⋊D10711yescharacteristic
174C22×C4C22×D10712yes—
175C22×C4C22×D10711yescharacteristic
176SD16—2148no—
177C2×C8—1074no—
178C2×C8—1072no—
179C2×C8—1072no—
180C2×C8—2148no—
181C2×C8—2142no—
182C4⋊C4—21no—
183C4⋊C4—1072no—
184C4⋊C4—2141no—
185C4⋊C4—2144no—
186C4⋊C4C22×D10712yes—
187C22⋊C4—24no—
188C22⋊C4—2144no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23#1313380614yescomplemented
196C23C2×C22⋊D10712yes—
197C23C2×C22⋊D10711yescharacteristic
198Q8—4282no—
199Q8—2142no—
200Q8—2142no—
201D4—22no—
202D4—22no—
203D4—48no—
204D4—42no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—4282no—
212C2×C4—4282no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4D4×D10712yes—
216C2×C4C2×C22⋊D10712yes—
217C2×C4C2×C22⋊D10711yescharacteristic
218C2×C4#1313381712yes—
219C2×C4C23×D10711yescharacteristic
220C8—1074no—
221C8—2142no—
222C22—22no—
223C22—28no—
224C22—22no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—84no—
229C22#1681208112yes—
230C22C2×#1313380612yes—
231C22C2×#1313380614yescomplemented
232C22#1681212912yes—
233C22C22×C22⋊D10711yescharacteristic
234C4—22no—
235C4—42no—
236C4—4282no—
237C4—4282no—
238C4#1681208112yes—
239C4C2×D4×D10711yescharacteristic
240C4C2×#1313381711yescharacteristic
241C2—24no—
242C2—82no—
243C2#2029285414yesdirect product factor
244C2C2×#1681208111yescharacteristic
245C2C22×#1313380611yescharacteristic
246C2C2×#1681212911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalC2×#121×C107C2
socleC23×C107C2×D4
hypercenterC2×#121D107
hypocenterC107C2×#465
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×#13133806C2semidirect
2. C23×C107⋊C8C2semidirect
3. C2×#16812088C2semidirect
4. #20292854C2direct (N × H)
5. C2×#16812040C2semidirect
6. C2×#16812032C2semidirect
7. C2×#16812030C2semidirect
11. C2×#13133806C22semidirect
15. C22×C107⋊C8C22semidirect
17. #16812088C22semidirect
21. #16812040C22semidirect
22. #16812032C22semidirect
23. #16812030C22semidirect
40. C2×C107⋊C8D4semidirect
71. C23×C107SD16semidirect
97. C107⋊C8C2×D4semidirect
107. C22×C107C2×SD16semidirect
119. C2×C107#465semidirect
124. C107C2×#465semidirect
195. C23#13133806semidirect
231. C22C2×#13133806semidirect
243. C2#20292854direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338062
C23×C107⋊C82
C2×#168120882
#202928542
C2×#168120402
C2×#168120322
C2×#168120302
C2×#121×C1072
C2×#465107