Finite Group Databasebeta

← C2×#20292825 #28345891

Subgroups — C2×#20292825 #28345891

Order 13696.

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

normal subgroups only

This group has 584 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×#13133805C211yesmaximal, characteristic, complemented
2C2×#16812105C211yesmaximal, characteristic
3C2×Q8×Dic107C211yesmaximal, characteristic, complemented
4#20292825C218yesmaximal, direct product factor
5C2×Dic107⋊C8C211yesmaximal, characteristic, complemented
6C2×#16812031C211yesmaximal, characteristic, complemented
7C2×#16812030C211yesmaximal, characteristic, complemented
8C22×SD16×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107—21no—
11C22×C22⋊D107—21no—
12C2×#13133805—24no—
13C2×#13133805C2214yescomplemented
14C22×C107⋊C8C2211yescharacteristic
15#16812105—22no—
16#16812105C2214yes—
17Q8×Dic107—22no—
18Q8×Dic107C2214yescomplemented
19C2×C4⋊Dic107—21no—
20C2×C4⋊Dic107C2211yescharacteristic
21C2×C4×Dic107—21no—
22C2×C4×Dic107C2211yescharacteristic
23Dic107⋊C8C2214yescomplemented
24#16812031C2214yescomplemented
25#16812030C2214yescomplemented
26C22×Q8×C107C2211yescharacteristic
27C22×D4×C107C2211yescharacteristic
28C2×SD16×C107—24no—
29C2×SD16×C107C2214yescomplemented
30C22×C8×C107C2211yescharacteristic
31C23×D107—21no—
32C2×D428—22no—
33C2×D428—22no—
34C2×D428—21no—
35C2×D428C2312yes—
36C2×C22⋊D107—24no—
37C2×C22⋊D107—24no—
38C2×C22⋊D107—44no—
39C2×C22⋊D107—44no—
40C22×Dic107—21no—
41C22×Dic107D411yescharacteristic
42C22×Dic107D411yescharacteristic
43#13133805—28no—
44C2×C107⋊C8—22no—
45C2×C107⋊C8C2312yes—
46C4⋊Dic107—21no—
47C4⋊Dic107—24no—
48C4⋊Dic107C2312yes—
49C4×Dic107—24no—
50C4×Dic107D412yes—
51C4×Dic107C2312yes—
52C24×C107—21no—
53C2×Q8×C107—21no—
54C2×Q8×C107—22no—
55C2×Q8×C107D414yescomplemented
56C2×Q8×C107C2312yes—
57C2×D4×C107—22no—
58C2×D4×C107—21no—
59C2×D4×C107—22no—
60C2×D4×C107C2312yes—
61C22×C4×C107—21no—
62C22×C4×C107C2311yescharacteristic
63SD16×C107—28no—
64C2×C8×C107—22no—
65C2×C8×C107C2312yes—
66C23×C107—22no—
67C23×C107—42no—
68C23×C107—42no—
69C23×C107—42no—
70C23×C107C2×D411yescharacteristic
71C22×D107—22no—
72C22×D107—42no—
73C22×D107—42no—
74C22×D107—42no—
75Q8×C107—24no—
76Q8×C107—22no—
77Q8×C107C2×D414yescomplemented
78D4×C107—22no—
79D4×C107—22no—
80D4×C107—42no—
81C2×C4×C107—24no—
82C2×C4×C107—22no—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×D412yes—
85C2×C4×C107#4112yes—
86C2×C4×C107C2411yescharacteristic
87C22⋊D107—48no—
88C22⋊D107—48no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—42no—
92C2×Dic107SD1614yescomplemented
93C2×Dic107C2×D412yes—
94C2×Dic107C2×D412yes—
95D428—22no—
96D428—22no—
97D428—42no—
98C8×C107—22no—
99C107⋊C8—22no—
100C22×C107—42no—
101C22×C107—42no—
102C22×C107—42no—
103C22×C107—84no—
104C22×C107#12112yes—
105C22×C107C2×SD1612yes—
106C22×C107AGL(1,ℤ/8)12yes—
107C22×C107C22×D411yescharacteristic
108C2×D107—42no—
109C2×D107—42no—
110C2×D107—42no—
111C2×D107—84no—
112C4×C107—24no—
113C4×C107#12112yes—
114C4×C107C22×D411yescharacteristic
115C4×C107C2×#4111yescharacteristic
116Dic107—22no—
117Dic107—42no—
118Dic107C2×SD1614yescomplemented
119C2×C107—82no—
120C2×C107#46014yescomplemented
121C2×C107C2×#12111yescharacteristic
122C2×C107C22×SD1611yescharacteristic
123C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
124D107—82no—
125C2×#460—1071nomaximal, Sylow
126C107C2×#46011yescharacteristic, complemented, Sylow
127C22×SD16—1071no—
128C22×SD16D10711yescharacteristic, complemented
129C2×#127—1071no—
130C2×C4×Q8—1071no—
131#460—1078no—
132C2×C4⋊C8—1071no—
133C2×#102—1071no—
134C2×#102—1071no—
135C22×Q8C2×D10711yescharacteristic
136C22×D4—1071no—
137C22×D4—2141no—
138C22×D4—2141no—
139C22×D4C2×D10711yescharacteristic
140C2×SD16—24no—
141C2×SD16—1074no—
142C2×SD16—2144no—
143C2×SD16C2×D10714yescomplemented
144C22×C8—1071no—
145C22×C8C2×D10711yescharacteristic
146#127—1074no—
147#127—2142no—
148C4×Q8—1074no—
149C4×Q8—2142no—
150C2×C4⋊C4—1071no—
151C2×C4⋊C4—2141no—
152C2×C42—1071no—
153C2×C42—2141no—
154C4⋊C8—1074no—
155#102—1074no—
156#102—1074no—
157C24—21no—
158C24—2141no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8C22⋊D10714yescomplemented
162C2×Q8C22×D10712yes—
163C2×D4—21no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—1072no—
167C2×D4—4284no—
168C2×D4—4284no—
169C2×D4—2144no—
170C2×D4—2144no—
171C2×D4—2141no—
172C2×D4—2142no—
173C2×D4—2142no—
174C2×D4C22×D10712yes—
175C22×C4—21no—
176C22×C4—1071no—
177C22×C4—1071no—
178C22×C4—2141no—
179C22×C4C22×D10711yescharacteristic
180SD16—28no—
181SD16—2148no—
182C2×C8—22no—
183C2×C8—1072no—
184C2×C8—2142no—
185C2×C8C22×D10712yes—
186C4⋊C4—1072no—
187C4⋊C4—2144no—
188C4⋊C4—2141no—
189C42—1072no—
190C42—1072no—
191C42—2144no—
192C23—22no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—4282no—
197C23—4282no—
198C23—4282no—
199C23—2142no—
200C23D4×D10711yescharacteristic
201Q8—24no—
202Q8—22no—
203Q8C2×C22⋊D10714yescomplemented
204D4—22no—
205D4—22no—
206D4—42no—
207D4—4288no—
208D4—4288no—
209D4—4282no—
210D4—2142no—
211D4—2142no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—1074no—
215C2×C4—1072no—
216C2×C4—1072no—
217C2×C4—4282no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4D4×D10711yescharacteristic
221C2×C4C2×C22⋊D10712yes—
222C2×C4#1313381712yes—
223C2×C4C23×D10711yescharacteristic
224C8—22no—
225C8—2142no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22—8564no—
231C22—4282no—
232C22—4282no—
233C22—4282no—
234C22#1681208112yes—
235C22SD16×D10712yes—
236C22#1681213612yes—
237C22C2×D4×D10711yescharacteristic
238C4—24no—
239C4—1074no—
240C4—4282no—
241C4—2142no—
242C4#1681208112yes—
243C4C22×C22⋊D10711yescharacteristic
244C4C2×#1313381711yescharacteristic
245C2—82no—
246C2—8562no—
247C2#2029282514yesdirect product factor
248C2C2×#1681208111yescharacteristic
249C2C2×SD16×D10711yescharacteristic
250C2C2×#1681213611yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×SD16×C107C2
socleC23×C107C2×D4
prime radicalC2×#16812105C2
hypercenterC22×SD16D107
hypocenterC107C2×#460
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13133805C2semidirect
3. C2×Q8×Dic107C2semidirect
4. #20292825C2direct (N × H)
5. C2×Dic107⋊C8C2semidirect
6. C2×#16812031C2semidirect
7. C2×#16812030C2semidirect
8. C22×SD16×C107C2semidirect
13. C2×#13133805C22semidirect
18. Q8×Dic107C22semidirect
23. Dic107⋊C8C22semidirect
24. #16812031C22semidirect
25. #16812030C22semidirect
29. C2×SD16×C107C22semidirect
55. C2×Q8×C107D4semidirect
77. Q8×C107C2×D4semidirect
92. C2×Dic107SD16semidirect
118. Dic107C2×SD16semidirect
120. C2×C107#460semidirect
126. C107C2×#460semidirect
128. C22×SD16D107semidirect
143. C2×SD16C2×D107semidirect
161. C2×Q8C22⋊D107semidirect
203. Q8C2×C22⋊D107semidirect
247. C2#20292825direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338052
C2×#168121052
C2×Q8×Dic1072
#202928252
C2×Dic107⋊C82
C2×#168120312
C2×#168120302
C22×SD16×C1072
C2×#460107