Finite Group Databasebeta

← C2×#20292807 #28345793

Subgroups — C2×#20292807 #28345793

Order 13696.

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

normal subgroups only

This group has 760 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
2C2×#16812142C212yesmaximal, complemented
3C2×#16812096C211yesmaximal, characteristic, complemented
4#20292807C218yesmaximal, direct product factor
5C2×#16812035C212yesmaximal, complemented
6C2×#100×C107C211yesmaximal, characteristic, complemented
7#16812154—24no—
8#16812154—22no—
9#16812154C2214yes—
10C2×#13133818C2212yes—
11C2×#13133817—22no—
12C22×Dic214C2211yescharacteristic, complemented
13C22×C22⋊D107—22no—
14C2×D4×D107—22no—
15#16812142—24no—
16#16812142C2218yescomplemented
17C2×Dic428—22no—
18C2×#13133807—22no—
19#16812096—22no—
20#16812096C2214yescomplemented
21C2×C22⋊Dic107—22no—
22C2×C4×Dic107C2211yescharacteristic, complemented
23#16812035C2218yescomplemented
24C22×D4×C107C2211yescharacteristic
25C2×M16×C107C2212yescomplemented
26#100×C107C2214yescomplemented
27C23×D107—22no—
28#13133818—22no—
29#13133818—22no—
30#13133818D414yes—
31#13133818C2314yes—
32#13133817—28no—
33#13133817—28no—
34C2×Dic214—22no—
35C2×Dic214—22no—
36C2×Dic214—21no—
37C2×Dic214D412yes—
38C2×Dic214C2312yescomplemented
39C2×D428D412yes—
40C2×C22⋊D107—24no—
41C2×C22⋊D107—28no—
42C2×C22⋊D107—22no—
43C2×C22⋊D107—28no—
44D4×D107—28no—
45D4×D107—28no—
46C22×Dic107—22no—
47C22×Dic107—21no—
48C22×Dic107—21no—
49C2×C4×D107—22no—
50Dic428—28no—
51#13133807—28no—
52C22⋊Dic107—28no—
53C4×Dic107—21no—
54C4×Dic107C2312yescomplemented
55C24×C107D412yes—
56C2×D4×C107—22no—
57C2×D4×C107—22no—
58C2×D4×C107D412yes—
59C2×D4×C107C2312yes—
60C22×C4×C107C2311yescharacteristic
61M16×C107—22no—
62M16×C107C2314yescomplemented
63C2×C8×C107—22no—
64C23×C107—22no—
65C23×C107—22no—
66C23×C107—44no—
67C23×C107C2×D411yescharacteristic
68C23×C107C2×D414yes—
69C22×D107—22no—
70C22×D107—28no—
71C22×D107—84no—
72D4×C107—24no—
73D4×C107—24no—
74C2×C4×C107C2×D412yes—
75C2×C4×C107C2×D412yes—
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2411yescharacteristic
78C22⋊D107—216no—
79C22⋊D107—24no—
80C22⋊D107—24no—
81C2×Dic107—28no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—42no—
86C2×Dic107—41no—
87C2×Dic107—41no—
88D428—22no—
89D428C2×D414yes—
90C4×D107—24no—
91C4×D107—24no—
92Dic214—22no—
93Dic214—24no—
94Dic214—42no—
95Dic214C2×D414yes—
96C8×C107—24no—
97C22×C107—22no—
98C22×C107—24no—
99C22×C107—44no—
100C22×C107—44no—
101C22×C107—82no—
102C22×C107#12011yescharacteristic
103C22×C107#12011yescharacteristic
104C22×C107C22×D411yescharacteristic
105C2×D107—24no—
106C2×D107—44no—
107C2×D107—88no—
108C4×C107#12012yes—
109C4×C107C22×D412yes—
110Dic107—24no—
111Dic107—42no—
112Dic107—42no—
113C2×C107—21no—
114C2×C107—21no—
115C2×C107—44no—
116C2×C107#45512yescomplemented
117C2×C107C2×#12011yescharacteristic
118D107—44no—
119C2×#455—1071nomaximal, Sylow
120C107C2×#45511yescharacteristic, complemented, Sylow
121C2×H(5,𝔽2)—1071no—
122C2×#137—1072no—
123C2×#124—1071no—
124#455—1078no—
125C2×#104—1072no—
126C2×#100D10711yescharacteristic, complemented
127H(5,𝔽2)—1074no—
128H(5,𝔽2)—2144no—
129H(5,𝔽2)—2142no—
130C2×#41—1072no—
131C2×#41—2142no—
132C22×Q8—1071no—
133C22×D4—2142no—
134C22×D4—2142no—
135C22×D4C2×D10711yescharacteristic
136#137—1078no—
137#137—2144no—
138C2×Q16—2142no—
139C2×SD16—2142no—
140C2×M16C2×D10712yescomplemented
141#124—1074no—
142#124—2142no—
143C2×C22⋊C4—2142no—
144C2×C42—1071no—
145#104—1078no—
146#100C2×D10714yescomplemented
147C24—2142no—
148C24D42812yes—
149#41—1074no—
150#41—1074no—
151#41—2148no—
152#41—2148no—
153#41—2142no—
154#41—2142no—
155C2×Q8—1072no—
156C2×Q8—1072no—
157C2×Q8—2142no—
158C2×Q8—2142no—
159C2×Q8—2141no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—1072no—
163C2×D4—2148no—
164C2×D4—2148no—
165C2×D4—2144no—
166C2×D4—2148no—
167C2×D4—2142no—
168C2×D4—2148no—
169C2×D4D42812yes—
170C2×D4C22×D10712yes—
171C22×C4—2141no—
172C22×C4—2141no—
173C22×C4—2142no—
174C22×C4—2142no—
175C22×C4C22×D10711yescharacteristic
176Q16—2148no—
177SD16—2148no—
178M16—22no—
179M16C22×D10714yescomplemented
180C2×C8—22no—
181C22⋊C4—2148no—
182C42—1072no—
183C42—2141no—
184C23—22no—
185C23—22no—
186C23—44no—
187C23—8564no—
188C23—2148no—
189C23—2142no—
190C23C2×D42811yescharacteristic
191C23C2×D42814yes—
192Q8—1074no—
193Q8—4282no—
194Q8—2144no—
195Q8—2142no—
196D4—24no—
197D4—24no—
198D4—1074no—
199D4—21416no—
200D4—2144no—
201D4—2144no—
202D4—2142no—
203C2×C4—4282no—
204C2×C4—4281no—
205C2×C4—4281no—
206C2×C4—2142no—
207C2×C4—2142no—
208C2×C4—2148no—
209C2×C4—2142no—
210C2×C4—2144no—
211C2×C4—2144no—
212C2×C4D4×D10712yes—
213C2×C4D4×D10712yes—
214C2×C4C2×D42811yescharacteristic
215C2×C4C23×D10711yescharacteristic
216C8—24no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—8568no—
223C22—4284no—
224C22—2144no—
225C22#1681207911yescharacteristic
226C22#1681207911yescharacteristic
227C22C22×D42811yescharacteristic
228C4—4282no—
229C4—4282no—
230C4—2144no—
231C4#1681207912yes—
232C4C2×D4×D10712yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—4284no—
237C2#2029280712yesdirect product factor
238C2C2×#1681207911yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#100×C107C2
socleC22×C107#120
prime radicalC2×#16812154C2
hypercenterC2×#100D107
hypocenterC107C2×#455
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
2. C2×#16812142C2semidirect
3. C2×#16812096C2semidirect
4. #20292807C2direct (N × H)
5. C2×#16812035C2semidirect
6. C2×#100×C107C2semidirect
12. C22×Dic214C22semidirect
16. #16812142C22semidirect
20. #16812096C22semidirect
22. C2×C4×Dic107C22semidirect
23. #16812035C22semidirect
25. C2×M16×C107C22semidirect
26. #100×C107C22semidirect
38. C2×Dic214C23semidirect
54. C4×Dic107C23semidirect
62. M16×C107C23semidirect
116. C2×C107#455semidirect
120. C107C2×#455semidirect
126. C2×#100D107semidirect
140. C2×M16C2×D107semidirect
146. #100C2×D107semidirect
179. M16C22×D107semidirect
237. C2#20292807direct (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×#168121422
C2×#168120962
#202928072
C2×#168120352
C2×#100×C1072
C2×#455107