Finite Group Databasebeta

← C2×#20292804 #28345790

Subgroups — C2×#20292804 #28345790

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×#16812140C212yesmaximal, complemented
3#20292804C218yesmaximal, direct product factor
4C2×#16812037C212yesmaximal, complemented
5C2×#16812026C211yesmaximal, characteristic, complemented
6C2×#124×C107C211yesmaximal, characteristic, complemented
7#16812154—24no—
8#16812154—22no—
9#16812154C2214yes—
10C2×#13133818C2212yes—
11C2×#13133817—22no—
12C22×C22⋊D107—22no—
13C2×D4×D107—22no—
14#16812140—24no—
15#16812140C2218yescomplemented
16C2×#13133808—22no—
17C2×#13133805—22no—
18C2×#13133802C2212yescomplemented
19#16812037C2218yescomplemented
20#16812026C2214yescomplemented
21C22×Q8×C107C2211yescharacteristic, complemented
22C22×D4×C107C2211yescharacteristic
23#124×C107—22no—
24#124×C107C2214yescomplemented
25C2×C22⋊C4×C107—22no—
26C2×C42×C107C2211yescharacteristic, complemented
27C23×D107—22no—
28#13133818—22no—
29#13133818—22no—
30#13133818D414yes—
31#13133818C2314yes—
32#13133817—28no—
33#13133817—28no—
34C2×Dic214D412yes—
35C2×D428D412yes—
36C2×C22⋊D107—24no—
37C2×C22⋊D107—28no—
38C2×C22⋊D107—22no—
39C2×C22⋊D107—28no—
40D4×D107—28no—
41D4×D107—28no—
42C22×Dic107—22no—
43C2×C4×D107—22no—
44#13133808—28no—
45#13133805—28no—
46#13133802—22no—
47#13133802C2314yescomplemented
48C2×C107⋊C8—22no—
49C24×C107D412yes—
50C2×Q8×C107—21no—
51C2×Q8×C107—22no—
52C2×Q8×C107—22no—
53C2×Q8×C107C2312yescomplemented
54C2×D4×C107—22no—
55C2×D4×C107—22no—
56C2×D4×C107D412yes—
57C2×D4×C107C2312yes—
58C22×C4×C107—21no—
59C22×C4×C107—21no—
60C22×C4×C107C2311yescharacteristic
61C22⋊C4×C107—28no—
62C42×C107—21no—
63C42×C107C2312yescomplemented
64C23×C107—22no—
65C23×C107—22no—
66C23×C107—44no—
67C23×C107C2×D411yescharacteristic
68C23×C107C2×D414yes—
69C22×D107—22no—
70C22×D107—28no—
71C22×D107—84no—
72Q8×C107—24no—
73Q8×C107—42no—
74D4×C107—24no—
75D4×C107—24no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—42no—
79C2×C4×C107—41no—
80C2×C4×C107—41no—
81C2×C4×C107C2×D412yes—
82C2×C4×C107C2×D412yes—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—216no—
86C22⋊D107—24no—
87C22⋊D107—24no—
88C2×Dic107—28no—
89C2×Dic107—22no—
90D428—22no—
91D428C2×D414yes—
92C4×D107—24no—
93C4×D107—24no—
94Dic214—22no—
95Dic214C2×D414yes—
96C107⋊C8—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—42no—
109C4×C107—42no—
110C4×C107#12012yes—
111C4×C107C22×D412yes—
112Dic107—24no—
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×#124D10711yescharacteristic, complemented
124#455—1078no—
125C2×#104—1072no—
126C2×#100—1071no—
127H(5,𝔽2)—1074no—
128H(5,𝔽2)—2144no—
129H(5,𝔽2)—2142no—
130C2×#41—1072no—
131C2×#41—2142no—
132C22×Q8C2×D10711yescharacteristic, complemented
133C22×D4—2142no—
134C22×D4—2142no—
135C22×D4C2×D10711yescharacteristic
136#137—1078no—
137#137—2144no—
138C2×Q16—2142no—
139C2×SD16—2142no—
140C2×M16—1072no—
141#124—22no—
142#124C2×D10714yescomplemented
143C2×C22⋊C4—22no—
144C2×C42C2×D10711yescharacteristic, complemented
145#104—1078no—
146#100—1074no—
147C24—2142no—
148C24C22⋊D10712yes—
149#41—1074no—
150#41—1074no—
151#41—2148no—
152#41—2148no—
153#41—2142no—
154#41—2142no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—1072no—
159C2×Q8C22×D10712yescomplemented
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×D4C22⋊D10712yes—
170C2×D4C22×D10712yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—2142no—
174C22×C4—2142no—
175C22×C4C22×D10711yescharacteristic
176Q16—2148no—
177SD16—2148no—
178M16—1074no—
179M16—2142no—
180C2×C8—2142no—
181C22⋊C4—28no—
182C42—21no—
183C42C22×D10712yescomplemented
184C23—22no—
185C23—22no—
186C23—44no—
187C23—8564no—
188C23—2148no—
189C23—2142no—
190C23C2×C22⋊D10711yescharacteristic
191C23C2×C22⋊D10714yes—
192Q8—24no—
193Q8—42no—
194Q8—1074no—
195Q8—2142no—
196D4—24no—
197D4—24no—
198D4—1074no—
199D4—21416no—
200D4—2144no—
201D4—2144no—
202D4—2142no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—2148no—
209C2×C4—2142no—
210C2×C4—2144no—
211C2×C4—2144no—
212C2×C4D4×D10712yes—
213C2×C4D4×D10712yes—
214C2×C4C2×C22⋊D10711yescharacteristic
215C2×C4C23×D10711yescharacteristic
216C8—2144no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—8568no—
223C22—4284no—
224C22—2144no—
225C22#1681207711yescharacteristic
226C22#1681207711yescharacteristic
227C22C22×C22⋊D10711yescharacteristic
228C4—42no—
229C4—42no—
230C4—2144no—
231C4#1681207712yes—
232C4C2×D4×D10712yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—4284no—
237C2#2029280412yesdirect product factor
238C2C2×#1681207711yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#124×C107C2
socleC22×C107#120
prime radicalC2×#16812154C2
hypercenterC2×#124D107
hypocenterC107C2×#455
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#16812140C2semidirect
3. #20292804C2direct (N × H)
4. C2×#16812037C2semidirect
5. C2×#16812026C2semidirect
6. C2×#124×C107C2semidirect
15. #16812140C22semidirect
18. C2×#13133802C22semidirect
19. #16812037C22semidirect
20. #16812026C22semidirect
21. C22×Q8×C107C22semidirect
24. #124×C107C22semidirect
26. C2×C42×C107C22semidirect
47. #13133802C23semidirect
53. C2×Q8×C107C23semidirect
63. C42×C107C23semidirect
116. C2×C107#455semidirect
120. C107C2×#455semidirect
123. C2×#124D107semidirect
132. C22×Q8C2×D107semidirect
142. #124C2×D107semidirect
144. C2×C42C2×D107semidirect
159. C2×Q8C22×D107semidirect
183. C42C22×D107semidirect
237. C2#20292804direct (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×#168121402
#202928042
C2×#168120372
C2×#168120262
C2×#124×C1072
C2×#455107