Finite Group Databasebeta

← #37543856

Subgroups — #37543856

Order 23168.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#26041810C211yesmaximal, characteristic, complemented
2C2×#19153778C211yesmaximal, characteristic, complemented
3#26041609C211yesmaximal, characteristic, complemented
4#26041582C212yesmaximal, complemented
5#26041493C212yesmaximal, complemented
6#26041432C211yesmaximal, characteristic, complemented
7#26041412C211yesmaximal, characteristic
8#26041388C212yesmaximal, complemented
9#26041378C212yesmaximal, complemented
10#26041333C211yesmaximal, characteristic, complemented
11#480×C181C211yesmaximal, characteristic, complemented
12C22×D724C2211yescharacteristic
13C22×C4×D181—21no—
14C2×#15668438C2212yes—
15C2×#15668434C2212yes—
16#19153809—21no—
17#19153807C2211yescharacteristic, complemented
18#19153803C2211yescharacteristic, complemented
19#19153797—21no—
20#19153795C2211yescharacteristic
21#19153781C2211yescharacteristic, complemented
22D4×Dic181—21no—
23D4×Dic181C2211yescharacteristic, complemented
24#19153778—21no—
25#19153778—21no—
26#19153778C2212yes—
27#19153778C2212yes—
28#19153772—21no—
29C2×#15668426—21no—
30C2×C4×Dic181C2211yescharacteristic
31#19153747C2212yescomplemented
32#19153745C2212yescomplemented
33#19153739C2212yescomplemented
34#19153738C2212yes—
35#19153736C2212yes—
36C22⋊#13089678C2211yescharacteristic
37#19153726C2212yescomplemented
38#121×C181C2211yescharacteristic
39C2×C4⋊C4×C181C2211yescharacteristic
40#107×C181C2212yescomplemented
41#102×C181C2212yescomplemented
42C22⋊C8×C181C2211yescharacteristic
43C23×D181—21no—
44C2×Dic362C2311yescharacteristic
45C2×D724—24no—
46C2×D724—21no—
47C2×D724C2312yes—
48C22×Dic181—21no—
49C22×Dic181D411yescharacteristic
50C22×Dic181D411yescharacteristic
51C2×C4×D181—22no—
52C2×C4×D181—22no—
53C2×C4×D181—44no—
54#15668438—24no—
55#15668434—24no—
56C2×#13089678C2312yes—
57C4⋊Dic181—21no—
58C4⋊Dic181C2311yescharacteristic
59#15668428—21no—
60#15668428—21no—
61#15668428—21no—
62C22⋊Dic181—21no—
63C22⋊Dic181—21no—
64C22⋊Dic181—21no—
65C22⋊Dic181—21no—
66#15668426—22no—
67#15668426—22no—
68C4×Dic181—21no—
69C4×Dic181D411yescharacteristic
70C4×Dic181D411yescharacteristic
71C4×Dic181C2311yescharacteristic
72C4×Dic181C2311yescharacteristic
73C2×D4×C181—21no—
74C2×D4×C181C2311yescharacteristic
75C22×C4×C181—21no—
76C22×C4×C181C2311yescharacteristic
77C2×C8×C181C2312yes—
78C4⋊C4×C181—21no—
79C4⋊C4×C181C2312yes—
80C4⋊C4×C181C2311yescharacteristic
81C22⋊C4×C181—21no—
82C22×D181—24no—
83C22×D181—22no—
84C22×D181—42no—
85C22×D181—42no—
86Dic362—22no—
87D724—24no—
88D724—22no—
89D724#4114yes—
90C2×Dic181—22no—
91C2×Dic181—21no—
92C2×Dic181—21no—
93C2×Dic181—41no—
94C2×Dic181—42no—
95C2×Dic181C2×D411yescharacteristic
96C2×Dic181C2×D411yescharacteristic
97C2×Dic181C2×D411yescharacteristic
98C2×Dic181C2×D411yescharacteristic
99C4×D181—44no—
100C4×D181—44no—
101#13089678—22no—
102C23×C181—21no—
103C23×C181C2×D411yescharacteristic
104D4×C181—22no—
105D4×C181—42no—
106C2×C4×C181—22no—
107C2×C4×C181—21no—
108C2×C4×C181—22no—
109C2×C4×C181—42no—
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2411yescharacteristic
112C8×C181—22no—
113C22×C181—21no—
114C22×C181—21no—
115C22×C181—41no—
116C22×C181—41no—
117C22×C181—41no—
118C22×C181AGL(1,ℤ/8)12yes—
119C22×C181C22×D411yescharacteristic
120C2×D181—24no—
121C2×D181—48no—
122C2×D181—42no—
123C2×D181—42no—
124C4×C181—21no—
125C4×C181—42no—
126C4×C181—41no—
127C4×C181C2×#4112yes—
128Dic181—21no—
129Dic181—21no—
130Dic181—21no—
131Dic181—21no—
132Dic181—41no—
133Dic181—41no—
134C2×C181—22no—
135C2×C181—81no—
136C2×C181#52511yescharacteristic
137C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C181#57511yescharacteristic
139D181—44no—
140C181#311411yescharacteristic, complemented, Sylow
141#3114—1811nomaximal, Sylow
142#529—1811no—
143C2×C4×D4—1811no—
144#483—1811no—
145#480D18111yescharacteristic, complemented
146#478—1812no—
147#463—1812no—
148#450—1811no—
149#447—1811no—
150#442—1812no—
151#440—1812no—
152#432—1811no—
153C22×D4—1811no—
154C23×C4—3621no—
155C2×SD16—1812no—
156C2×D8—1812no—
157#126—3621no—
158#125—1811no—
159#124—1811no—
160#123—3621no—
161C22⋊Q8—1811no—
162#121C2×D18111yescharacteristic
163C4×Q8—1811no—
164C4×D4—3621no—
165C4×D4—3621no—
166C4×D4—3621no—
167C4×D4—1811no—
168C4×D4—1812no—
169C4×D4—1812no—
170#117—3621no—
171C2×C4⋊C4C2×D18111yescharacteristic
172C2×C22⋊C4—3621no—
173C2×C42—1811no—
174#107C2×D18112yescomplemented
175#106—1812no—
176C4⋊C8—1812no—
177#103—1812no—
178#102—1812no—
179#102—1812no—
180#102C2×D18112yescomplemented
181C22⋊C8—1811no—
182C22⋊C8C2×D18111yescharacteristic
183#97—1812no—
184C24—3621no—
185C2×Q8—1811no—
186C2×D4—21no—
187C2×D4—3624no—
188C2×D4—3621no—
189C2×D4—1812no—
190C2×D4C22×D18111yescharacteristic
191C22×C4—21no—
192C22×C4—3622no—
193C22×C4—3622no—
194C22×C4—3621no—
195C22×C4—1811no—
196C22×C4—1811no—
197C22×C4—7244no—
198C22×C4C22×D18111yescharacteristic
199SD16—3624no—
200D8—3624no—
201C2×C8—1812no—
202C2×C8C22×D18112yes—
203C4⋊C4—21no—
204C4⋊C4—3621no—
205C4⋊C4—3621no—
206C4⋊C4—3621no—
207C4⋊C4—3621no—
208C4⋊C4—1811no—
209C4⋊C4C22×D18111yescharacteristic
210C4⋊C4C22×D18112yes—
211C22⋊C4—21no—
212C22⋊C4—3622no—
213C22⋊C4—3622no—
214C22⋊C4—3621no—
215C22⋊C4—3621no—
216C22⋊C4—3621no—
217C22⋊C4—3621no—
218C42—3621no—
219C42—1811no—
220C42—1811no—
221C42—1811no—
222C42—1811no—
223C23—21no—
224C23—3624no—
225C23—3622no—
226C23—7242no—
227C23—7242no—
228C23D4×D18111yescharacteristic
229Q8—3622no—
230D4—22no—
231D4—42no—
232D4—3624no—
233D4—3622no—
234D4—1814no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3622no—
240C2×C4—3621no—
241C2×C4—3621no—
242C2×C4—1811no—
243C2×C4—1811no—
244C2×C4—1811no—
245C2×C4—1811no—
246C2×C4—7244no—
247C2×C4—7244no—
248C2×C4—7241no—
249C2×C4—7242no—
250C2×C4D4×D18111yescharacteristic
251C2×C4C23×D18111yescharacteristic
252C8—22no—
253C8—3622no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—3624no—
260C22—7248no—
261C22—7242no—
262C22—7242no—
263C22#1915384012yes—
264C22C2×D4×D18111yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—3621no—
269C4—3621no—
270C4—3621no—
271C4—3621no—
272C4—7241no—
273C4—7241no—
274C4#41×D18112yes—
275C2—22no—
276C2—81no—
277C2—7244no—
278C2#2604178711yescharacteristic
279C2C2×#1915384011yescharacteristic
280C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#480×C181C2
socleC22×C181C22×D4
prime radical#26041412C2
hypercenter#480D181
hypocenterC181#3114
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #26041810C2semidirect
2. C2×#19153778C2semidirect
3. #26041609C2semidirect
4. #26041582C2semidirect
5. #26041493C2semidirect
6. #26041432C2semidirect
8. #26041388C2semidirect
9. #26041378C2semidirect
10. #26041333C2semidirect
11. #480×C181C2semidirect
17. #19153807C22semidirect
18. #19153803C22semidirect
21. #19153781C22semidirect
23. D4×Dic181C22semidirect
31. #19153747C22semidirect
32. #19153745C22semidirect
33. #19153739C22semidirect
37. #19153726C22semidirect
40. #107×C181C22semidirect
41. #102×C181C22semidirect
140. C181#3114semidirect
145. #480D181semidirect
174. #107C2×D181semidirect
180. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418102
C2×#191537782
#260416092
#260415822
#260414932
#260414322
#260414122
#260413882
#260413782
#260413332
#480×C1812
#3114181