Finite Group Databasebeta

← C2×#26042322 #37550486

Subgroups — C2×#26042322 #37550486

Order 23168.

472 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 162 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×#19153904C211yesmaximal, characteristic, complemented
2C2×#19153903C211yesmaximal, characteristic, complemented
3#26042322C218yesmaximal, direct product factor
4C2×#19153886C211yesmaximal, characteristic, complemented
5C2×#19153881C211yesmaximal, characteristic, complemented
6C2×#19153880C211yesmaximal, characteristic, complemented
7C2×C8×#10624382C211yesmaximal, characteristic, complemented
8C2×#19153837C211yesmaximal, characteristic
9#19153904—22no—
10#19153904C2214yescomplemented
11#19153903—22no—
12#19153903C2214yescomplemented
13C2×#15668459—21no—
14C2×#15668459C2211yescharacteristic
15C2×#15668459C2211yescharacteristic
16C2×#15668458—21no—
17C2×D181⋊C8—21no—
18C2×D181⋊C8C2211yescharacteristic
19C22×#13089685—21no—
20C2×C4×#10624382C2211yescharacteristic
21#19153886C2214yescomplemented
22#19153881C2214yescomplemented
23#19153880C2214yescomplemented
24C8×#10624382C2214yescomplemented
25C2×#15668450C2211yescharacteristic
26C2×#15668448C2211yescharacteristic
27#19153837C414yescomplemented
28#19153837C2214yes—
29C2×#15668439C411yescharacteristic, complemented
30C2×#15668438C411yescharacteristic, complemented
31C2×#15668434C411yescharacteristic, complemented
32C2×C8×D181C2211yescharacteristic
33C2×SD16×C181C411yescharacteristic, complemented
34C22×#10624382—21no—
35#15668459—24no—
36#15668459—21no—
37#15668459—21no—
38#15668459C2312yes—
39#15668459C2312yes—
40#15668458—24no—
41D181⋊C8—24no—
42D181⋊C8D412yes—
43D181⋊C8C2312yes—
44C2×#13089685—24no—
45C2×#13089685—21no—
46C2×#13089685—21no—
47C2×#13089685D411yescharacteristic
48C2×#13089685D411yescharacteristic
49C4×#10624382—21no—
50C4×#10624382C2312yes—
51#15668450—21no—
52#15668450—21no—
53#15668450C2×C412yes—
54#15668450C2312yes—
55#15668448—21no—
56#15668448—21no—
57#15668448C2×C412yes—
58#15668448C2312yes—
59C2×Dic362C2×C411yescharacteristic
60C2×D724—21no—
61C2×D724C2×C411yescharacteristic
62C2×C22⋊D181—21no—
63C22×Dic181—21no—
64C2×C4×D181—21no—
65C2×C4×D181C2311yescharacteristic
66#15668439C2×C414yescomplemented
67#15668438C2×C414yescomplemented
68#15668434C2×C414yescomplemented
69C2×#13089678C2×C411yescharacteristic
70C8×D181C2×C412yes—
71C8×D181C2312yes—
72C2×Q8×C181C2×C411yescharacteristic
73C2×D4×C181C2×C411yescharacteristic
74SD16×C181C2×C414yescomplemented
75C2×C8×C181C2×C411yescharacteristic
76C2×#10624382—24no—
77C2×#10624382—22no—
78#13089685—22no—
79#13089685—22no—
80#13089685C2×D412yes—
81#13089685C2×D412yes—
82C22×D181—21no—
83C22×D181#4111yescharacteristic
84Dic362—21no—
85Dic362C22×C412yes—
86D724—22no—
87D724—22no—
88D724—21no—
89D724C22×C412yes—
90C22⋊D181—22no—
91C22⋊D181—22no—
92C2×Dic181—21no—
93C2×Dic181—22no—
94C2×Dic181—22no—
95C2×Dic181C2×D411yescharacteristic
96C4×D181—22no—
97C4×D181—22no—
98C4×D181C22×C411yescharacteristic
99C4×D181C2×D411yescharacteristic
100C4×D181#4111yescharacteristic
101C4×D181C2411yescharacteristic
102#13089678C22×C412yes—
103C23×C181—21no—
104Q8×C181—21no—
105Q8×C181C22×C412yes—
106D4×C181—21no—
107D4×C181C22×C412yes—
108C2×C4×C181—21no—
109C2×C4×C181C22×C411yescharacteristic
110C8×C181C22×C412yes—
111C22×C181—22no—
112C22×C181—42no—
113C22×C181C4×D411yescharacteristic
114C2×D181—22no—
115C2×D181—22no—
116C2×D181—42no—
117C2×D181C4×D411yescharacteristic
118C2×D181C2×#4111yescharacteristic
119#10624382—24no—
120C4×C181—22no—
121C4×C181C4×D411yescharacteristic
122C4×C181C23×C411yescharacteristic
123Dic181—22no—
124Dic181C4×D411yescharacteristic
125Dic181C22×D411yescharacteristic
126C2×C181—42no—
127C2×C181#44312yescomplemented
128C2×C181C2×C4×D411yescharacteristic
129D181—21no—
130D181—21no—
131D181—42no—
132C181C2×#44311yescharacteristic, complemented, Sylow
133C2×#443—1811nomaximal, Sylow
134C2×#135—1811no—
135C2×#131—1811no—
136C2×#131—1811no—
137#443—1818no—
138C2×#108—1811no—
139C2×#104—1811no—
140C2×#104—1811no—
141C2×C4×C8—1811no—
142C2×#41—1811no—
143C2×#41—1811no—
144#135—1814no—
145#135—1814no—
146C2×Q16—1811no—
147C2×SD16—1811no—
148C2×SD16#1062438211yescharacteristic, complemented
149C2×D8—1811no—
150#131—3622no—
151#131—3622no—
152#131—1814no—
153#131—1814no—
154C2×M16—3621no—
155C2×M16—3621no—
156C2×M16—1811no—
157C2×M16—1811no—
158C22×C8—3621no—
159C22×C8—3621no—
160C22×C8—1811no—
161C22×C8—1811no—
162C2×C42—1811no—
163#108—1814no—
164#104—1814no—
165#104—1814no—
166C4×C8—1814no—
167#41—3621no—
168#41—3621no—
169#41—3621no—
170#41—3621no—
171#41—1812no—
172#41—1812no—
173#41—1812no—
174#41—1812no—
175C2×Q8—1811no—
176C2×Q8C2×#1062438211yescharacteristic
177C2×D4—3621no—
178C2×D4—3621no—
179C2×D4—1811no—
180C2×D4C2×#1062438211yescharacteristic
181C22×C4—3621no—
182C22×C4—3621no—
183C22×C4—3621no—
184C22×C4—1811no—
185Q16—1814no—
186SD16—1814no—
187SD16C2×#1062438214yescomplemented
188D8—1814no—
189M16—3621no—
190M16—3621no—
191M16—3624no—
192M16—3624no—
193M16—1812no—
194M16—1812no—
195C2×C8—3621no—
196C2×C8—3621no—
197C2×C8—3624no—
198C2×C8—3624no—
199C2×C8—1812no—
200C2×C8—1811no—
201C2×C8—1811no—
202C2×C8—1812no—
203C2×C8—1811no—
204C2×C8—1812no—
205C2×C8—1812no—
206C2×C8C2×#1062438211yescharacteristic
207C42—3621no—
208C42—1812no—
209C23—21no—
210C23—3621no—
211C23—1811no—
212Q8—21no—
213Q8—3621no—
214Q8—1812no—
215Q8C22×#1062438212yes—
216D4—21no—
217D4—3622no—
218D4—3622no—
219D4—3622no—
220D4—3622no—
221D4—3621no—
222D4—1812no—
223D4C22×#1062438212yes—
224C2×C4—21no—
225C2×C4—3622no—
226C2×C4—3622no—
227C2×C4—3624no—
228C2×C4—3622no—
229C2×C4—3621no—
230C2×C4—3622no—
231C2×C4—3622no—
232C2×C4—1811no—
233C2×C4—1811no—
234C2×C4—1811no—
235C2×C4—1811no—
236C2×C4—1811no—
237C2×C4C22×#1062438211yescharacteristic
238C8—3622no—
239C8—3622no—
240C8—1812no—
241C8—1812no—
242C8—1812no—
243C8C22×#1062438212yes—
244C22—22no—
245C22—42no—
246C22—3622no—
247C22—3622no—
248C22—1811no—
249C22—1811no—
250C22—7242no—
251C22D4×#1062438211yescharacteristic
252C4—22no—
253C4—3624no—
254C4—3622no—
255C4—1811no—
256C4—1811no—
257C4D4×#1062438211yescharacteristic
258C4C23×#1062438211yescharacteristic
259C2—42no—
260C2—3621no—
261C2—3621no—
262C2—7242no—
263C2#2604232212yesdirect product factor
264C2C2×D4×#1062438211yescharacteristic
subgroupG/H
centerC22D4×#10624382
commutatorC4×C181C23×C4
FrattiniC4C23×#10624382
FittingC2×SD16×C181C4
socleC22×C181C4×D4
prime radicalC2×#19153837C2
cosocleC4×D181C24
hypercenterC2×SD16#10624382
hypocenterC181C2×#443
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19153904C2semidirect
2. C2×#19153903C2semidirect
3. #26042322C2direct (N × H)
4. C2×#19153886C2semidirect
5. C2×#19153881C2semidirect
6. C2×#19153880C2semidirect
7. C2×C8×#10624382C2semidirect
10. #19153904C22semidirect
12. #19153903C22semidirect
21. #19153886C22semidirect
22. #19153881C22semidirect
23. #19153880C22semidirect
24. C8×#10624382C22semidirect
27. #19153837C4semidirect
29. C2×#15668439C4semidirect
30. C2×#15668438C4semidirect
31. C2×#15668434C4semidirect
33. C2×SD16×C181C4semidirect
66. #15668439C2×C4semidirect
67. #15668438C2×C4semidirect
68. #15668434C2×C4semidirect
74. SD16×C181C2×C4semidirect
127. C2×C181#443semidirect
132. C181C2×#443semidirect
148. C2×SD16#10624382semidirect
187. SD16C2×#10624382semidirect
263. C2#26042322direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191539042
C2×#191539032
#260423222
C2×#191538862
C2×#191538812
C2×#191538802
C2×C8×#106243822
C2×#191538372
C2×#443181