Finite Group Databasebeta

← C2×#20292875 #28346051

Subgroups — C2×#20292875 #28346051

Order 13696.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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
2C22×#13133802C211yesmaximal, characteristic, complemented
3C2×#16812083C211yesmaximal, characteristic
4#20292875C218yesmaximal, direct product factor
5C2×#16812042C211yesmaximal, characteristic, complemented
6C2×#16812033C211yesmaximal, characteristic, complemented
7C2×#16812029C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107—21no—
11C2×#13133805—24no—
12C2×#13133805C2214yescomplemented
13C2×#13133802—24no—
14C2×#13133802C2214yescomplemented
15C22×C107⋊C8C2211yescharacteristic
16C22×C107⋊C8C2211yescharacteristic
17#16812083—22no—
18#16812083C2214yes—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×(C2×D107)⋊C4—21no—
21#16812042C2214yescomplemented
22#16812033C2214yescomplemented
23#16812029C2214yescomplemented
24C22×Q8×C107C2211yescharacteristic
25C23×C4×C107C2211yescharacteristic
26C22⋊Q8×C107—22no—
27C22⋊Q8×C107C2214yescomplemented
28C2×C4⋊C4×C107—21no—
29C2×C4⋊C4×C107C2211yescharacteristic
30C2×C22⋊C4×C107—21no—
31C23×D107—21no—
32C2×D428—22no—
33C2×D428—22no—
34C2×D428—21no—
35C2×D428C2312yes—
36C2×C22⋊D107—24no—
37C2×C22⋊D107—44no—
38C22×Dic107—21no—
39#13133805—28no—
40#13133802—28no—
41C2×C107⋊C8—22no—
42C2×C107⋊C8D414yescomplemented
43C2×C107⋊C8C2312yes—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107—21no—
46C4⋊Dic107C2312yes—
47(C2×D107)⋊C4—24no—
48C24×C107D411yescharacteristic
49C2×Q8×C107—21no—
50C2×Q8×C107—22no—
51C2×Q8×C107—22no—
52C2×Q8×C107C2312yes—
53C22×C4×C107—22no—
54C22×C4×C107—22no—
55C22×C4×C107—21no—
56C22×C4×C107—21no—
57C22×C4×C107D411yescharacteristic
58C22×C4×C107D412yes—
59C22×C4×C107C2312yes—
60C22×C4×C107C2311yescharacteristic
61C4⋊C4×C107—24no—
62C4⋊C4×C107—21no—
63C4⋊C4×C107C2312yes—
64C22⋊C4×C107—24no—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107—22no—
68C23×C107C2×D412yes—
69C23×C107C2×D411yescharacteristic
70C22×D107—22no—
71C22×D107—42no—
72C22×D107—42no—
73C22×D107—42no—
74Q8×C107—22no—
75Q8×C107—22no—
76Q8×C107—42no—
77C2×C4×C107—22no—
78C2×C4×C107—24no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—42no—
84C2×C4×C107—42no—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D412yes—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107#4112yes—
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—48no—
91C2×Dic107—22no—
92C2×Dic107—42no—
93D428—22no—
94D428—22no—
95D428—42no—
96C107⋊C8—22no—
97C107⋊C8C2×D414yescomplemented
98C22×C107—22no—
99C22×C107—22no—
100C22×C107—22no—
101C22×C107—44no—
102C22×C107#12112yes—
103C22×C107AGL(1,ℤ/8)12yes—
104C22×C107#13712yes—
105C22×C107C22×D411yescharacteristic
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—42no—
109C2×D107—84no—
110C4×C107—22no—
111C4×C107—42no—
112C4×C107—42no—
113C4×C107#12112yes—
114C4×C107C22×D411yescharacteristic
115C4×C107C2×#4111yescharacteristic
116Dic107—42no—
117C2×C107—42no—
118C2×C107#46814yescomplemented
119C2×C107C2×#12111yescharacteristic
120C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C107C2×#13711yescharacteristic
122D107—82no—
123C2×#468—1071nomaximal, Sylow
124C107C2×#46811yescharacteristic, complemented, Sylow
125C22×SD16—1071no—
126C22×M16—1071no—
127C2×C22⋊Q8D10711yescharacteristic, complemented
128C2×#121—1071no—
129#468—1078no—
130C2×#107—1071no—
131C2×#103—1071no—
132C2×#102—1071no—
133C22×Q8C2×D10711yescharacteristic
134C22×D4—1071no—
135C22×D4—2141no—
136C23×C4C2×D10711yescharacteristic
137C2×SD16—1074no—
138C2×SD16—2144no—
139C2×M16—1074no—
140C2×M16—2144no—
141C22×C8—1071no—
142C22×C8—1071no—
143C22⋊Q8—22no—
144C22⋊Q8C2×D10714yescomplemented
145#121—1074no—
146#121—2142no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—1071no—
149C2×C4⋊C4C2×D10711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2141no—
152#107—1074no—
153#103—1074no—
154#102—1074no—
155C24—2141no—
156C24C22⋊D10711yescharacteristic
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D10712yes—
161C2×D4—1072no—
162C2×D4—4284no—
163C2×D4—2144no—
164C2×D4—2142no—
165C2×D4—2141no—
166C2×D4—2142no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—2141no—
172C22×C4C22⋊D10712yes—
173C22×C4C22⋊D10711yescharacteristic
174C22×C4C22×D10712yes—
175C22×C4C22×D10711yescharacteristic
176SD16—2148no—
177M16—2148no—
178C2×C8—1074no—
179C2×C8—1072no—
180C2×C8—1072no—
181C2×C8—2142no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—1072no—
185C4⋊C4—2141no—
186C4⋊C4C22×D10712yes—
187C22⋊C4—24no—
188C22⋊C4—2144no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—4282no—
193C23—4282no—
194C23—4282no—
195C23—2142no—
196C23C2×C22⋊D10712yes—
197C23C2×C22⋊D10711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—4288no—
202D4—4282no—
203D4—2142no—
204D4—2142no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—4282no—
214C2×C4—2142no—
215C2×C4D4×D10712yes—
216C2×C4C2×C22⋊D10712yes—
217C2×C4C2×C22⋊D10711yescharacteristic
218C2×C4#1313381712yes—
219C2×C4C23×D10711yescharacteristic
220C8—1074no—
221C8—2142no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—44no—
226C22—8564no—
227C22—4282no—
228C22—4282no—
229C22—4282no—
230C22#1681208112yes—
231C22#1681213412yes—
232C22#1681214012yes—
233C22C22×C22⋊D10711yescharacteristic
234C4—22no—
235C4—42no—
236C4—42no—
237C4—4282no—
238C4#1681208112yes—
239C4C2×D4×D10711yescharacteristic
240C4C2×#1313381711yescharacteristic
241C2—42no—
242C2—8562no—
243C2#2029287514yesdirect product factor
244C2C2×#1681208111yescharacteristic
245C2C2×#1681213411yescharacteristic
246C2C2×#1681214011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C22⋊Q8×C107C2
socleC23×C107C2×D4
prime radicalC2×#16812083C2
hypercenterC2×C22⋊Q8D107
hypocenterC107C2×#468
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13133805C2semidirect
2. C22×#13133802C2semidirect
4. #20292875C2direct (N × H)
5. C2×#16812042C2semidirect
6. C2×#16812033C2semidirect
7. C2×#16812029C2semidirect
8. C2×C22⋊Q8×C107C2semidirect
12. C2×#13133805C22semidirect
14. C2×#13133802C22semidirect
21. #16812042C22semidirect
22. #16812033C22semidirect
23. #16812029C22semidirect
27. C22⋊Q8×C107C22semidirect
42. C2×C107⋊C8D4semidirect
97. C107⋊C8C2×D4semidirect
118. C2×C107#468semidirect
124. C107C2×#468semidirect
127. C2×C22⋊Q8D107semidirect
144. C22⋊Q8C2×D107semidirect
243. C2#20292875direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338052
C22×#131338022
C2×#168120832
#202928752
C2×#168120422
C2×#168120332
C2×#168120292
C2×C22⋊Q8×C1072
C2×#468107