Finite Group Databasebeta

← C2×#20292870 #28346046

Subgroups — C2×#20292870 #28346046

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×#13133801C211yesmaximal, characteristic, complemented
2C2×#16812085C211yesmaximal, characteristic, complemented
3C2×#16812080C211yesmaximal, characteristic
4#20292870C218yesmaximal, direct product factor
5C2×#16812043C211yesmaximal, characteristic, complemented
6C2×#16812033C211yesmaximal, characteristic, complemented
7C2×#16812030C211yesmaximal, characteristic, complemented
8C22×SD16×C107C211yesmaximal, characteristic, complemented
9C22×C22⋊D107—21no—
10C22×C4×D107C2211yescharacteristic
11C2×#13133801—24no—
12C2×#13133801C2214yescomplemented
13C22×C107⋊C8C2211yescharacteristic
14#16812085—22no—
15#16812085C2214yescomplemented
16#16812080—22no—
17#16812080C2214yes—
18C2×C4⋊Dic107C2211yescharacteristic
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4—21no—
21C2×C22⋊Dic107—21no—
22C2×(C2×D107)⋊C4—21no—
23#16812043C2214yescomplemented
24#16812033C2214yescomplemented
25#16812030C2214yescomplemented
26C22×Q8×C107C2211yescharacteristic
27C22×D4×C107C2211yescharacteristic
28C2×SD16×C107—24no—
29C2×SD16×C107C2214yescomplemented
30C22×C8×C107C2211yescharacteristic
31C23×D107D411yescharacteristic
32C2×C22⋊D107—24no—
33C2×C22⋊D107—44no—
34C22×Dic107—21no—
35C22×Dic107—21no—
36C22×Dic107D411yescharacteristic
37C2×C4×D107—22no—
38C2×C4×D107—22no—
39C2×C4×D107D412yes—
40C2×C4×D107C2312yes—
41#13133801—28no—
42C2×C107⋊C8—22no—
43C2×C107⋊C8C2312yes—
44C4⋊Dic107—21no—
45C4⋊Dic107—21no—
46C4⋊Dic107C2312yes—
47C4⋊Dic107C2312yes—
48Dic107⋊C4—24no—
49C22⋊Dic107—24no—
50(C2×D107)⋊C4—24no—
51C24×C107—21no—
52C2×Q8×C107—21no—
53C2×Q8×C107—22no—
54C2×Q8×C107—22no—
55C2×Q8×C107C2312yes—
56C2×D4×C107—22no—
57C2×D4×C107—21no—
58C2×D4×C107—22no—
59C2×D4×C107C2312yes—
60C22×C4×C107—21no—
61C22×C4×C107C2311yescharacteristic
62SD16×C107—28no—
63C2×C8×C107D414yescomplemented
64C2×C8×C107C2312yes—
65C23×C107—22no—
66C23×C107—42no—
67C23×C107—42no—
68C23×C107—42no—
69C23×C107C2×D411yescharacteristic
70C22×D107—22no—
71C22×D107—22no—
72C22×D107—22no—
73C22×D107C2×D412yes—
74Q8×C107—22no—
75Q8×C107—22no—
76Q8×C107—42no—
77D4×C107—22no—
78D4×C107—22no—
79D4×C107—42no—
80C2×C4×C107—22no—
81C2×C4×C107—42no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D412yes—
84C2×C4×C107#4112yes—
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—48no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—42no—
91C2×Dic107—42no—
92C2×Dic107C2×D412yes—
93C4×D107—24no—
94C4×D107—22no—
95C4×D107—22no—
96C8×C107C2×D414yescomplemented
97C107⋊C8—22no—
98C22×C107—42no—
99C22×C107—42no—
100C22×C107—42no—
101C22×C107—84no—
102C22×C107#12112yes—
103C22×C107AGL(1,ℤ/8)12yes—
104C22×C107#13712yes—
105C22×C107C22×D411yescharacteristic
106C2×D107—22no—
107C2×D107—22no—
108C2×D107—22no—
109C2×D107—44no—
110C4×C107—42no—
111C4×C107#12112yes—
112C4×C107C22×D411yescharacteristic
113C4×C107C2×#4111yescharacteristic
114Dic107—22no—
115Dic107—42no—
116Dic107—42no—
117C2×C107—82no—
118C2×C107#46814yescomplemented
119C2×C107C2×#12111yescharacteristic
120C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C107C2×#13711yescharacteristic
122D107—42no—
123C2×#468—1071nomaximal, Sylow
124C107C2×#46811yescharacteristic, complemented, Sylow
125C22×SD16D10711yescharacteristic, complemented
126C22×M16—1071no—
127C2×C22⋊Q8—1071no—
128C2×#121—1071no—
129#468—1078no—
130C2×#107—1071no—
131C2×#103—1071no—
132C2×#102—1071no—
133C22×Q8C2×D10711yescharacteristic
134C22×D4—2141no—
135C22×D4C2×D10711yescharacteristic
136C23×C4—1071no—
137C2×SD16—24no—
138C2×SD16C2×D10714yescomplemented
139C2×M16—1074no—
140C2×M16—2144no—
141C22×C8—1071no—
142C22×C8C2×D10711yescharacteristic
143C22⋊Q8—1074no—
144C22⋊Q8—2142no—
145#121—1074no—
146#121—2142no—
147C2×C4⋊C4—1071no—
148C2×C4⋊C4—1071no—
149C2×C4⋊C4—2141no—
150C2×C22⋊C4—2141no—
151C2×C22⋊C4—2141no—
152#107—1074no—
153#103—1074no—
154#102—1074no—
155C24—21no—
156C24—1071no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D10712yes—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—4284no—
165C2×D4—2144no—
166C2×D4C22×D10712yes—
167C22×C4—21no—
168C22×C4—1071no—
169C22×C4—1072no—
170C22×C4—1072no—
171C22×C4—2141no—
172C22×C4—2141no—
173C22×C4—2142no—
174C22×C4—2142no—
175C22×C4C22×D10711yescharacteristic
176SD16—28no—
177M16—2148no—
178C2×C8—1072no—
179C2×C8—2142no—
180C2×C8C22⋊D10714yescomplemented
181C2×C8C22×D10712yes—
182C4⋊C4—1072no—
183C4⋊C4—1072no—
184C4⋊C4—2144no—
185C4⋊C4—2141no—
186C4⋊C4—2141no—
187C22⋊C4—2144no—
188C22⋊C4—2144no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—1072no—
194C23—2142no—
195C23—2142no—
196C23—2142no—
197C23D4×D10711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—4288no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—1072no—
208C2×C4—4282no—
209C2×C4—4282no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2144no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4D4×D10711yescharacteristic
217C2×C4C2×C22⋊D10712yes—
218C2×C4#1313381712yes—
219C2×C4C23×D10711yescharacteristic
220C8—2142no—
221C8C2×C22⋊D10714yescomplemented
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—4284no—
227C22—2142no—
228C22—2142no—
229C22—2142no—
230C22#1681208012yes—
231C22#1681213612yes—
232C22#1681213812yes—
233C22C2×D4×D10711yescharacteristic
234C4—42no—
235C4—4282no—
236C4—4282no—
237C4—2142no—
238C4#1681208012yes—
239C4C22×C22⋊D10711yescharacteristic
240C4C2×#1313381711yescharacteristic
241C2—82no—
242C2—4282no—
243C2#2029287014yesdirect product factor
244C2C2×#1681208011yescharacteristic
245C2C2×#1681213611yescharacteristic
246C2C2×#1681213811yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×SD16×C107C2
socleC23×C107C2×D4
prime radicalC2×#16812080C2
hypercenterC22×SD16D107
hypocenterC107C2×#468
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×#13133801C2semidirect
2. C2×#16812085C2semidirect
4. #20292870C2direct (N × H)
5. C2×#16812043C2semidirect
6. C2×#16812033C2semidirect
7. C2×#16812030C2semidirect
8. C22×SD16×C107C2semidirect
12. C2×#13133801C22semidirect
15. #16812085C22semidirect
23. #16812043C22semidirect
24. #16812033C22semidirect
25. #16812030C22semidirect
29. C2×SD16×C107C22semidirect
63. C2×C8×C107D4semidirect
96. C8×C107C2×D4semidirect
118. C2×C107#468semidirect
124. C107C2×#468semidirect
125. C22×SD16D107semidirect
138. C2×SD16C2×D107semidirect
180. C2×C8C22⋊D107semidirect
221. C8C2×C22⋊D107semidirect
243. C2#20292870direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131338012
C2×#168120852
C2×#168120802
#202928702
C2×#168120432
C2×#168120332
C2×#168120302
C22×SD16×C1072
C2×#468107