Finite Group Databasebeta

← C2×#20292857 #28346022

Subgroups — C2×#20292857 #28346022

Order 13696.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 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×#13133807C211yesmaximal, characteristic, complemented
2C22×C8×D107C211yesmaximal, characteristic, complemented
3C2×#16812089C211yesmaximal, characteristic, complemented
4C2×#16812084C211yesmaximal, characteristic
5#20292857C218yesmaximal, direct product factor
6C2×#16812032C211yesmaximal, characteristic, complemented
7C2×#16812029C211yesmaximal, characteristic, complemented
8C2×#106×C107C211yesmaximal, characteristic, complemented
9C22×Dic214C2211yescharacteristic
10C22×D428—21no—
11C22×D428C2211yescharacteristic
12C22×C4×D107C2211yescharacteristic
13C2×#13133807—24no—
14C2×#13133807C2214yescomplemented
15C22×C107⋊C8C2211yescharacteristic
16C2×C8×D107—24no—
17C2×C8×D107C2214yescomplemented
18#16812089—22no—
19#16812089C2214yescomplemented
20#16812084—22no—
21#16812084C2214yes—
22C2×C4⋊Dic107—21no—
23C2×(C2×D107)⋊C4—21no—
24C2×(C2×D107)⋊C4—21no—
25#16812032C2214yescomplemented
26#16812029C2214yescomplemented
27C22×C8×C107C2211yescharacteristic
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C4⋊C4×C107C2211yescharacteristic
30#106×C107C2214yescomplemented
31C23×D107—21no—
32C23×D107D411yescharacteristic
33C2×Dic214—22no—
34C2×Dic214—22no—
35C2×Dic214—21no—
36C2×Dic214C2312yes—
37C2×D428—22no—
38C2×D428—22no—
39C2×D428—24no—
40C2×D428—21no—
41C2×D428—44no—
42C2×D428C2312yes—
43C22×Dic107—21no—
44C22×Dic107D411yescharacteristic
45C2×C4×D107—22no—
46C2×C4×D107—22no—
47C2×C4×D107D412yes—
48C2×C4×D107C2312yes—
49#13133807—28no—
50C2×C107⋊C8—22no—
51C2×C107⋊C8C2312yes—
52C8×D107—28no—
53C4⋊Dic107—24no—
54(C2×D107)⋊C4—24no—
55(C2×D107)⋊C4—24no—
56C22×C4×C107—21no—
57C22×C4×C107—21no—
58C22×C4×C107C2311yescharacteristic
59C2×C8×C107D414yescomplemented
60C2×C8×C107C2312yes—
61C4⋊C4×C107—21no—
62C4⋊C4×C107—21no—
63C4⋊C4×C107C2312yes—
64C4⋊C4×C107C2312yes—
65C23×C107C2×D411yescharacteristic
66C22×D107—22no—
67C22×D107—22no—
68C22×D107—22no—
69C22×D107—42no—
70C22×D107—42no—
71C22×D107—42no—
72C22×D107SD1614yescomplemented
73C22×D107C2×D412yes—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—42no—
77C2×C4×C107—42no—
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D412yes—
80C2×C4×C107#4112yes—
81C2×C4×C107C2411yescharacteristic
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—42no—
85C2×Dic107C2×D412yes—
86D428—22no—
87D428—22no—
88D428—48no—
89D428—42no—
90C4×D107—24no—
91C4×D107—22no—
92C4×D107—22no—
93Dic214—22no—
94Dic214—22no—
95Dic214—42no—
96C8×C107C2×D414yescomplemented
97C107⋊C8—22no—
98C22×C107#12112yes—
99C22×C107C2×SD1612yes—
100C22×C107#13512yes—
101C22×C107C22×D411yescharacteristic
102C2×D107—22no—
103C2×D107—22no—
104C2×D107—28no—
105C2×D107—42no—
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—84no—
109C2×D107C2×SD1614yescomplemented
110C4×C107—42no—
111C4×C107—42no—
112C4×C107#12112yes—
113C4×C107C22×D411yescharacteristic
114C4×C107C2×#4111yescharacteristic
115Dic107—22no—
116Dic107—42no—
117C2×C107#46514yescomplemented
118C2×C107C2×#12111yescharacteristic
119C2×C107C22×SD1611yescharacteristic
120C2×C107C2×#13511yescharacteristic
121D107—24no—
122D107—82no—
123C2×#465—1071nomaximal, Sylow
124C107C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16—1071no—
126C23×C8—1071no—
127C2×C22⋊Q8—1071no—
128C2×#121—1071no—
129#465—1078no—
130C2×#106D10711yescharacteristic, complemented
131C2×#103—1071no—
132C2×#102—1071no—
133C22×Q8—1071no—
134C22×D4—1071no—
135C22×D4—2141no—
136C23×C4—1071no—
137C2×SD16—1074no—
138C2×SD16—2144no—
139C22×C8—1071no—
140C22×C8—1074no—
141C22×C8—2144no—
142C22×C8C2×D10711yescharacteristic
143C22⋊Q8—1074no—
144C22⋊Q8—2142no—
145#121—1074no—
146#121—2142no—
147C2×C4⋊C4—2141no—
148C2×C4⋊C4C2×D10711yescharacteristic
149C2×C4⋊C4C2×D10711yescharacteristic
150C2×C22⋊C4—2141no—
151C2×C22⋊C4—2141no—
152#106C2×D10714yescomplemented
153#103—1074no—
154#102—1074no—
155C24—1071no—
156C24—2141no—
157C2×Q8—1072no—
158C2×Q8—2142no—
159C2×Q8—2142no—
160C2×Q8—2141no—
161C2×D4—1072no—
162C2×D4—4284no—
163C2×D4—2144no—
164C2×D4—2141no—
165C2×D4—2142no—
166C2×D4—2142no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—1071no—
170C22×C4—1072no—
171C22×C4—1072no—
172C22×C4—2141no—
173C22×C4—2142no—
174C22×C4—2142no—
175C22×C4C22×D10711yescharacteristic
176SD16—2148no—
177C2×C8—1072no—
178C2×C8—2142no—
179C2×C8—2148no—
180C2×C8D42814yescomplemented
181C2×C8C22×D10712yes—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—2144no—
185C4⋊C4C22×D10712yes—
186C4⋊C4C22×D10712yes—
187C22⋊C4—2144no—
188C22⋊C4—2144no—
189C23—1074no—
190C23—1072no—
191C23—4282no—
192C23—4282no—
193C23—4282no—
194C23—2142no—
195C23—2142no—
196C23—2142no—
197C23D4×D10711yescharacteristic
198Q8—4282no—
199Q8—2142no—
200Q8—2142no—
201D4—4288no—
202D4—4282no—
203D4—2142no—
204D4—2142no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—1072no—
210C2×C4—4282no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2144no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4D4×D10711yescharacteristic
217C2×C4C2×D42812yes—
218C2×C4#1313381912yes—
219C2×C4C23×D10711yescharacteristic
220C8—2142no—
221C8C2×D42814yescomplemented
222C22—8564no—
223C22—1074no—
224C22—4282no—
225C22—4282no—
226C22—4282no—
227C22—2142no—
228C22—2142no—
229C22—2148no—
230C22#1681208412yes—
231C22SD16×D10712yes—
232C22#1681213012yes—
233C22C2×D4×D10711yescharacteristic
234C4—42no—
235C4—42no—
236C4—4282no—
237C4—2142no—
238C4#1681208412yes—
239C4C22×D42811yescharacteristic
240C4C2×#1313381911yescharacteristic
241C2—8562no—
242C2—2144no—
243C2#2029285714yesdirect product factor
244C2C2×#1681208411yescharacteristic
245C2C2×SD16×D10711yescharacteristic
246C2C2×#1681213011yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#106×C107C2
socleC23×C107C2×D4
prime radicalC2×#16812084C2
hypercenterC2×#106D107
hypocenterC107C2×#465
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
1. C22×#13133807C2semidirect
2. C22×C8×D107C2semidirect
3. C2×#16812089C2semidirect
5. #20292857C2direct (N × H)
6. C2×#16812032C2semidirect
7. C2×#16812029C2semidirect
8. C2×#106×C107C2semidirect
14. C2×#13133807C22semidirect
17. C2×C8×D107C22semidirect
19. #16812089C22semidirect
25. #16812032C22semidirect
26. #16812029C22semidirect
30. #106×C107C22semidirect
59. C2×C8×C107D4semidirect
72. C22×D107SD16semidirect
96. C8×C107C2×D4semidirect
109. C2×D107C2×SD16semidirect
117. C2×C107#465semidirect
124. C107C2×#465semidirect
130. C2×#106D107semidirect
152. #106C2×D107semidirect
180. C2×C8D428semidirect
221. C8C2×D428semidirect
243. C2#20292857direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338072
C22×C8×D1072
C2×#168120892
C2×#168120842
#202928572
C2×#168120322
C2×#168120292
C2×#106×C1072
C2×#465107