Finite Group Databasebeta

← #470×D181 #37543319

Subgroups — #470×D181 #37543319

Order 23168.

492 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 135 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×Q16×D181C211yesmaximal, characteristic, complemented
2C2×M16×D181C211yesmaximal, characteristic
3C22⋊Q8×D181C212yesmaximal
4#26041538C211yesmaximal, characteristic, complemented
5#26041537C211yesmaximal, characteristic, complemented
6#26041536C212yesmaximal, complemented
7#26041535C212yesmaximal, complemented
8#26041534C211yesmaximal, characteristic, complemented
9#106×D181C211yesmaximal, characteristic, complemented
10#103×D181C212yesmaximal, complemented
11#470×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181C2212yes—
13C22×C4×D181C2211yescharacteristic
14C2×Dic724C2211yescharacteristic, complemented
15C2×#15668439C2212yescomplemented
16Q16×D181—24no—
17C2×#15668433C2211yescharacteristic
18C2×#15668432C2211yescharacteristic
19C2×#15668432C2211yescharacteristic
20M16×D181—24no—
21C2×C8×D181C2211yescharacteristic
22C2×C8×D181C2211yescharacteristic
23#19153796C2212yes—
24#19153795C2212yes—
25#19153794—22no—
26#19153793—22no—
27#19153792C2212yes—
28C4⋊C4×D181—22no—
29C4⋊C4×D181C2212yes—
30C22⋊C4×D181—22no—
31#19153748C2211yescharacteristic, complemented
32#19153747C2212yescomplemented
33#19153741C2212yescomplemented
34#19153740C2212yescomplemented
35#19153739C2212yescomplemented
36C2×Q16×C181C2211yescharacteristic, complemented
37C2×M16×C181C2211yescharacteristic
38C22⋊Q8×C181C2212yes—
39#106×C181C2211yescharacteristic, complemented
40#103×C181C2212yescomplemented
41C23×D181D411yescharacteristic
42C2×Dic362—22no—
43C2×Dic362C2312yes—
44Q8×D181—24no—
45Q8×D181—24no—
46C22×Dic181D411yescharacteristic
47C2×C4×D181—22no—
48C2×C4×D181—22no—
49C2×C4×D181—22no—
50C2×C4×D181—22no—
51C2×C4×D181D411yescharacteristic
52C2×C4×D181D411yescharacteristic
53C2×C4×D181C2311yescharacteristic
54C2×C4×D181C2311yescharacteristic
55Dic724—22no—
56#15668439—24no—
57#15668433—22no—
58#15668432—22no—
59#15668432—22no—
60C2×#13089678C2311yescharacteristic
61C2×#13089678C2311yescharacteristic
62C8×D181—22no—
63C8×D181D414yes—
64C4⋊Dic181—22no—
65C4⋊Dic181C2312yes—
66#15668428—22no—
67#15668428—22no—
68#15668428—22no—
69C22⋊Dic181—22no—
70#15668426—22no—
71#15668426—22no—
72C2×Q8×C181C2312yes—
73C22×C4×C181C2311yescharacteristic
74Q16×C181—22no—
75M16×C181—22no—
76C2×C8×C181C2311yescharacteristic
77C2×C8×C181C2311yescharacteristic
78C4⋊C4×C181—22no—
79C4⋊C4×C181C2312yes—
80C22⋊C4×C181—22no—
81C22×D181—22no—
82C22×D181—24no—
83C22×D181C2×D411yescharacteristic
84C22×D181C2×D411yescharacteristic
85Dic362—22no—
86Dic362—22no—
87Dic362—44no—
88C2×Dic181—22no—
89C2×Dic181—22no—
90C2×Dic181—22no—
91C2×Dic181C2×D411yescharacteristic
92C2×Dic181C2×D411yescharacteristic
93C4×D181—22no—
94C4×D181—21no—
95C4×D181—21no—
96C4×D181—22no—
97C4×D181—44no—
98C4×D181—44no—
99C4×D181C2×D412yes—
100C4×D181#4112yes—
101#13089678—21no—
102#13089678C2×D412yes—
103C23×C181C2×D411yescharacteristic
104Q8×C181—22no—
105Q8×C181—22no—
106C2×C4×C181—21no—
107C2×C4×C181—21no—
108C2×C4×C181—22no—
109C2×C4×C181—22no—
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2411yescharacteristic
112C8×C181—21no—
113C8×C181C2×D412yes—
114C22×C181—21no—
115C22×C181—22no—
116C22×C181C22×D411yescharacteristic
117C2×D181—21no—
118C2×D181—22no—
119C2×D181—44no—
120C2×D181#12112yes—
121C2×D181#13714yes—
122C4×C181—21no—
123C4×C181—42no—
124C4×C181—42no—
125C4×C181C22×D411yescharacteristic
126C4×C181C2×#4111yescharacteristic
127Dic181—21no—
128Dic181—42no—
129Dic181—42no—
130Dic181#12112yes—
131C2×C181—41no—
132C2×C181C2×#12111yescharacteristic
133C2×C181C2×#13712yes—
134D181—41no—
135D181#47014yesdirect product factor
136C181C2×#47011yescharacteristic, complemented, Sylow
137C2×#470—1811nomaximal, Sylow
138C22×Q16—1811no—
139C22×M16—1811no—
140C2×C22⋊Q8—1812no—
141#470—1811no—
142#470—1811no—
143#470—1812no—
144#470—1811no—
145#470—1812no—
146#470D18111yescharacteristic, direct product factor
147C2×#106—1811no—
148C2×#103—1812no—
149C22×Q8—1812no—
150C23×C4—1811no—
151C2×Q16—3624no—
152C2×Q16—1812no—
153C2×Q16—1811no—
154C2×Q16C2×D18111yescharacteristic, complemented
155C2×M16—3624no—
156C2×M16—1811no—
157C2×M16—1811no—
158C2×M16—1811no—
159C2×M16C2×D18111yescharacteristic
160C22×C8—1811no—
161C22×C8—1811no—
162C22⋊Q8—3622no—
163C22⋊Q8—3622no—
164C22⋊Q8—1812no—
165C22⋊Q8—1812no—
166C22⋊Q8—1812no—
167C22⋊Q8C2×D18112yes—
168C2×C4⋊C4—3622no—
169C2×C4⋊C4—1812no—
170C2×C22⋊C4—3622no—
171#106—1811no—
172#106—1812no—
173#106C2×D18111yescharacteristic, complemented
174#103—1812no—
175#103—1812no—
176#103—1812no—
177#103C2×D18112yescomplemented
178C24—1811no—
179C2×Q8—3622no—
180C2×Q8—3624no—
181C2×Q8—3624no—
182C2×Q8—1812no—
183C2×Q8C22×D18112yes—
184C22×C4—3622no—
185C22×C4—3622no—
186C22×C4—3622no—
187C22×C4—3622no—
188C22×C4—1811no—
189C22×C4—1811no—
190C22×C4—1811no—
191C22×C4—1811no—
192C22×C4—1811no—
193C22×C4C22×D18111yescharacteristic
194Q16—22no—
195Q16—3624no—
196Q16—3622no—
197M16—22no—
198M16—3622no—
199M16—3622no—
200M16—3622no—
201C2×C8—3622no—
202C2×C8—1811no—
203C2×C8—1811no—
204C2×C8—1814no—
205C2×C8C22×D18111yescharacteristic
206C2×C8C22×D18111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—3622no—
209C4⋊C4—3622no—
210C4⋊C4—3622no—
211C4⋊C4—3622no—
212C4⋊C4—1812no—
213C4⋊C4C22×D18112yes—
214C22⋊C4—22no—
215C22⋊C4—3622no—
216C22⋊C4—3622no—
217C22⋊C4—3622no—
218C23—3622no—
219C23—3624no—
220C23—1811no—
221C23—1811no—
222C23D4×D18111yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—3622no—
226Q8—3622no—
227Q8—7244no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—3622no—
233C2×C4—3622no—
234C2×C4—3622no—
235C2×C4—3621no—
236C2×C4—3621no—
237C2×C4—3622no—
238C2×C4—3622no—
239C2×C4—1811no—
240C2×C4—1811no—
241C2×C4—1812no—
242C2×C4—1812no—
243C2×C4—7244no—
244C2×C4—7244no—
245C2×C4D4×D18111yescharacteristic
246C2×C4C23×D18111yescharacteristic
247C8—21no—
248C8—3621no—
249C8—1812no—
250C8D4×D18112yes—
251C22—21no—
252C22—22no—
253C22—3621no—
254C22—3622no—
255C22—1812no—
256C22—1814no—
257C22—7244no—
258C22C2×D4×D18111yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—3621no—
263C4—1812no—
264C4—7242no—
265C4—7242no—
266C4C2×D4×D18111yescharacteristic
267C4#41×D18111yescharacteristic
268C2—41no—
269C2—1814no—
270C2—7241no—
271C2#121×D18111yescharacteristic
272C2#137×D18112yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#470×C181C2
socleC22×C181C22×D4
prime radicalC23×D181D4
hypercenter#470D181
hypocenterC181C2×#470
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D181C2semidirect
4. #26041538C2semidirect
5. #26041537C2semidirect
6. #26041536C2semidirect
7. #26041535C2semidirect
8. #26041534C2semidirect
9. #106×D181C2semidirect
10. #103×D181C2semidirect
11. #470×C181C2semidirect
14. C2×Dic724C22semidirect
15. C2×#15668439C22semidirect
31. #19153748C22semidirect
32. #19153747C22semidirect
33. #19153741C22semidirect
34. #19153740C22semidirect
35. #19153739C22semidirect
36. C2×Q16×C181C22semidirect
39. #106×C181C22semidirect
40. #103×C181C22semidirect
135. D181#470direct (N × H)
136. C181C2×#470semidirect
146. #470D181direct (N × H)
154. C2×Q16C2×D181semidirect
173. #106C2×D181semidirect
177. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1812
C2×M16×D1812
C22⋊Q8×D1812
#260415382
#260415372
#260415362
#260415352
#260415342
#106×D1812
#103×D1812
#470×C1812
C2×#470181