Finite Group Databasebeta

← C2×#20292797 #28345787

Subgroups — C2×#20292797 #28345787

Order 13696.

844 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16812154C211yesmaximal, characteristic, complemented
2C2×#16812137C212yesmaximal, complemented
3C2×#16812105C211yesmaximal, characteristic, complemented
4#20292797C218yesmaximal, direct product factor
5C2×#16812035C212yesmaximal, complemented
6C2×#100×C107C211yesmaximal, characteristic, complemented
7#16812154—24no—
8#16812154—22no—
9#16812154C2214yescomplemented
10C2×#13133818C2212yes—
11C2×#13133817—22no—
12C22×D428C2211yescharacteristic, complemented
13C22×C22⋊D107—22no—
14C22×C22⋊D107—22no—
15C2×D4×D107—22no—
16#16812137—24no—
17#16812137C2218yescomplemented
18C2×#13133807—22no—
19C2×D856—22no—
20#16812105—22no—
21#16812105C2214yescomplemented
22C2×C4×Dic107C2211yescharacteristic, complemented
23#16812035C2218yescomplemented
24C22×D4×C107C2211yescharacteristic
25C2×M16×C107C2212yescomplemented
26#100×C107C2214yescomplemented
27C23×D107—22no—
28C23×D107—21no—
29#13133818—22no—
30#13133818—22no—
31#13133818D414yescomplemented
32#13133818C2314yes—
33#13133817—28no—
34#13133817—28no—
35C2×Dic214D412yescomplemented
36C2×D428—22no—
37C2×D428—22no—
38C2×D428—21no—
39C2×D428D412yescomplemented
40C2×D428C2312yescomplemented
41C2×C22⋊D107—24no—
42C2×C22⋊D107—28no—
43C2×C22⋊D107—22no—
44C2×C22⋊D107—28no—
45C2×C22⋊D107—28no—
46C2×C22⋊D107—48no—
47D4×D107—28no—
48D4×D107—28no—
49C22×Dic107—22no—
50C22×Dic107—21no—
51C2×C4×D107—22no—
52#13133807—28no—
53D856—28no—
54C4×Dic107—21no—
55C4×Dic107C2312yescomplemented
56C24×C107D412yes—
57C2×D4×C107—22no—
58C2×D4×C107—22no—
59C2×D4×C107D412yes—
60C2×D4×C107C2312yes—
61C22×C4×C107C2311yescharacteristic
62M16×C107—22no—
63M16×C107C2314yescomplemented
64C2×C8×C107—22no—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107—44no—
68C23×C107C2×D411yescharacteristic
69C23×C107C2×D414yes—
70C22×D107—22no—
71C22×D107—28no—
72C22×D107—22no—
73C22×D107—44no—
74C22×D107—41no—
75C22×D107—41no—
76C22×D107—84no—
77D4×C107—24no—
78D4×C107—24no—
79C2×C4×C107C2×D412yes—
80C2×C4×C107C2×D412yes—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2411yescharacteristic
83C22⋊D107—216no—
84C22⋊D107—24no—
85C22⋊D107—24no—
86C22⋊D107—416no—
87C2×Dic107—28no—
88C2×Dic107—24no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91D428—22no—
92D428—24no—
93D428—42no—
94D428C2×D414yescomplemented
95C4×D107—24no—
96C4×D107—24no—
97Dic214—22no—
98Dic214C2×D414yescomplemented
99C8×C107—24no—
100C22×C107—22no—
101C22×C107—24no—
102C22×C107—44no—
103C22×C107—44no—
104C22×C107—82no—
105C22×C107#12011yescharacteristic
106C22×C107#12011yescharacteristic
107C22×C107C22×D411yescharacteristic
108C2×D107—24no—
109C2×D107—44no—
110C2×D107—44no—
111C2×D107—42no—
112C2×D107—88no—
113C2×D107—82no—
114C2×D107—82no—
115C4×C107#12012yes—
116C4×C107C22×D412yes—
117Dic107—24no—
118Dic107—24no—
119C2×C107—21no—
120C2×C107—21no—
121C2×C107—44no—
122C2×C107#45312yescomplemented
123C2×C107C2×#12011yescharacteristic
124D107—44no—
125D107—82no—
126C2×#453—1071nomaximal, Sylow
127C107C2×#45311yescharacteristic, complemented, Sylow
128C2×H(5,𝔽2)—1071no—
129C2×AGL(1,ℤ/8)—1072no—
130C2×#127—1071no—
131#453—1078no—
132C2×#104—1072no—
133C2×#100D10711yescharacteristic, complemented
134H(5,𝔽2)—1074no—
135H(5,𝔽2)—2144no—
136H(5,𝔽2)—2142no—
137C2×#41—1072no—
138C2×#41—2142no—
139C22×D4—1071no—
140C22×D4—2142no—
141C22×D4—2142no—
142C22×D4—2142no—
143C22×D4C2×D10711yescharacteristic
144AGL(1,ℤ/8)—1078no—
145AGL(1,ℤ/8)—2144no—
146C2×SD16—2142no—
147C2×D8—2142no—
148C2×M16C2×D10712yescomplemented
149#127—1074no—
150#127—2142no—
151C2×C42—1071no—
152#104—1078no—
153#100C2×D10714yescomplemented
154C24—2142no—
155C24—2141no—
156C24D42812yes—
157#41—1074no—
158#41—1074no—
159#41—2148no—
160#41—2148no—
161#41—2142no—
162#41—2142no—
163C2×Q8—1072no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—1072no—
167C2×D4—1072no—
168C2×D4—4288no—
169C2×D4—2148no—
170C2×D4—2148no—
171C2×D4—2148no—
172C2×D4—2144no—
173C2×D4—2148no—
174C2×D4—2142no—
175C2×D4—2148no—
176C2×D4—2142no—
177C2×D4—2142no—
178C2×D4—2141no—
179C2×D4D42812yes—
180C2×D4C22×D10712yes—
181C22×C4—2142no—
182C22×C4—2141no—
183C22×C4—2142no—
184C22×C4C22×D10711yescharacteristic
185SD16—2148no—
186D8—2148no—
187M16—22no—
188M16C22×D10714yescomplemented
189C2×C8—22no—
190C42—1072no—
191C42—2141no—
192C23—22no—
193C23—22no—
194C23—44no—
195C23—8564no—
196C23—4284no—
197C23—4281no—
198C23—4281no—
199C23—2148no—
200C23—2142no—
201C23—2142no—
202C23C2×D42811yescharacteristic
203C23C2×D42814yes—
204Q8—1074no—
205Q8—2142no—
206D4—24no—
207D4—24no—
208D4—1074no—
209D4—42816no—
210D4—4282no—
211D4—21416no—
212D4—2144no—
213D4—2144no—
214D4—2142no—
215D4—2144no—
216C2×C4—2144no—
217C2×C4—2148no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4—2144no—
221C2×C4—2144no—
222C2×C4D4×D10712yes—
223C2×C4D4×D10712yes—
224C2×C4C2×D42811yescharacteristic
225C2×C4C23×D10711yescharacteristic
226C8—24no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—82no—
232C22—8562no—
233C22—8568no—
234C22—8562no—
235C22—4284no—
236C22—4282no—
237C22—4284no—
238C22—2144no—
239C22#1681207911yescharacteristic
240C22#1681207911yescharacteristic
241C22C22×D42811yescharacteristic
242C4—2144no—
243C4—2144no—
244C4#1681207912yes—
245C4C2×D4×D10712yes—
246C2—21no—
247C2—21no—
248C2—44no—
249C2—8562no—
250C2—4284no—
251C2#2029279712yesdirect product factor
252C2C2×#1681207911yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#100×C107C2
socleC22×C107#120
hypercenterC2×#100D107
hypocenterC107C2×#453
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#16812154C2semidirect
2. C2×#16812137C2semidirect
3. C2×#16812105C2semidirect
4. #20292797C2direct (N × H)
5. C2×#16812035C2semidirect
6. C2×#100×C107C2semidirect
9. #16812154C22semidirect
12. C22×D428C22semidirect
17. #16812137C22semidirect
21. #16812105C22semidirect
22. C2×C4×Dic107C22semidirect
23. #16812035C22semidirect
25. C2×M16×C107C22semidirect
26. #100×C107C22semidirect
31. #13133818D4semidirect
35. C2×Dic214D4semidirect
39. C2×D428D4semidirect
40. C2×D428C23semidirect
55. C4×Dic107C23semidirect
63. M16×C107C23semidirect
94. D428C2×D4semidirect
98. Dic214C2×D4semidirect
122. C2×C107#453semidirect
127. C107C2×#453semidirect
133. C2×#100D107semidirect
148. C2×M16C2×D107semidirect
153. #100C2×D107semidirect
188. M16C22×D107semidirect
251. C2#20292797direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168121542
C2×#168121372
C2×#168121052
#202927972
C2×#168120352
C2×#100×C1072
C2×#453107