Finite Group Databasebeta

← #37543299

Subgroups — #37543299

Order 23168.

520 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19153836C211yesmaximal, characteristic, complemented
2C23×#13089678C211yesmaximal, characteristic, complemented
3#26041790C211yesmaximal, characteristic, complemented
4#26041518C212yesmaximal, complemented
5#26041513C212yesmaximal, complemented
6#26041510C212yesmaximal, complemented
7#26041506C212yesmaximal, complemented
8#26041315C211yesmaximal, characteristic, complemented
9#26041275C211yesmaximal, characteristic, complemented
10#26041273C211yesmaximal, characteristic, complemented
11#525×C181C211yesmaximal, characteristic, complemented
12C2×#15668449C2211yescharacteristic
13#19153836—24no—
14C2×#15668439C2211yescharacteristic, complemented
15C2×#15668437C2211yescharacteristic, complemented
16C2×#15668436C2211yescharacteristic, complemented
17C2×#15668434C2211yescharacteristic, complemented
18C22×#13089678—24no—
19C22×#13089678C2212yes—
20C22×#13089678C2212yes—
21C22×#13089678C2211yescharacteristic
22C22×#13089678C2211yescharacteristic
23#19153799—21no—
24#19153795C2212yes—
25#19153790C2212yes—
26#19153785—21no—
27C4×C22⋊D181—22no—
28#19153775C2211yescharacteristic
29#19153749C2212yescomplemented
30#19153747C2212yescomplemented
31#19153740C2212yescomplemented
32#19153739C2212yescomplemented
33#19153737C2212yescomplemented
34#19153736C2212yescomplemented
35C2×#41×C181C2211yescharacteristic
36C23×C4×C181C2211yescharacteristic
37#123×C181—21no—
38C22⋊Q8×C181C2212yes—
39#121×C181C2212yes—
40#120×C181—21no—
41C4×D4×C181—22no—
42#117×C181C2211yescharacteristic
43#15668449—22no—
44#15668449—22no—
45C2×Dic362C2311yescharacteristic
46C2×D724C2311yescharacteristic
47C2×C22⋊D181—22no—
48C2×C22⋊D181—21no—
49C2×C4×D181—21no—
50#15668439—22no—
51#15668437—22no—
52#15668436—22no—
53#15668434—22no—
54C2×#13089678—24no—
55C2×#13089678—24no—
56C2×#13089678—22no—
57C2×#13089678D414yescomplemented
58C2×#13089678C2312yes—
59C2×#13089678C2311yescharacteristic
60C2×#13089678C2311yescharacteristic
61C4⋊Dic181C2312yes—
62#15668428—22no—
63C22⋊Dic181—21no—
64C22⋊Dic181—22no—
65#15668426—22no—
66C4×Dic181—21no—
67C24×C181D411yescharacteristic
68#41×C181—22no—
69#41×C181—22no—
70C2×Q8×C181C2311yescharacteristic
71C2×D4×C181—22no—
72C2×D4×C181—21no—
73C2×D4×C181C2311yescharacteristic
74C22×C4×C181—22no—
75C22×C4×C181—22no—
76C22×C4×C181—21no—
77C22×C4×C181D412yes—
78C22×C4×C181D411yescharacteristic
79C22×C4×C181C2312yes—
80C22×C4×C181C2311yescharacteristic
81C4⋊C4×C181—22no—
82C4⋊C4×C181C2312yes—
83C22⋊C4×C181—21no—
84C22⋊C4×C181—22no—
85C22⋊C4×C181—22no—
86C42×C181—21no—
87C22×D181—21no—
88Dic362—21no—
89Dic362—21no—
90D724—21no—
91D724—21no—
92C22⋊D181—42no—
93C22⋊D181—44no—
94C2×Dic181—22no—
95C2×Dic181—21no—
96C4×D181—42no—
97#13089678—22no—
98#13089678C2×D412yes—
99#13089678C2×D412yes—
100C23×C181—22no—
101C23×C181—24no—
102C23×C181—21no—
103C23×C181C2×D411yescharacteristic
104C23×C181C2×D412yes—
105Q8×C181—21no—
106Q8×C181—21no—
107D4×C181—21no—
108D4×C181—21no—
109D4×C181—42no—
110D4×C181—44no—
111C2×C4×C181—22no—
112C2×C4×C181—24no—
113C2×C4×C181—22no—
114C2×C4×C181—22no—
115C2×C4×C181—22no—
116C2×C4×C181—21no—
117C2×C4×C181—42no—
118C2×C4×C181C2×D412yes—
119C2×C4×C181C2×D412yes—
120C2×C4×C181C2×D411yescharacteristic
121C2×C4×C181#4112yes—
122C2×C4×C181C2411yescharacteristic
123C22×C181—28no—
124C22×C181—22no—
125C22×C181—22no—
126C22×C181—22no—
127C22×C181—41no—
128C22×C181—42no—
129C22×C181#12112yes—
130C22×C181#13514yescomplemented
131C22×C181C22×D411yescharacteristic
132C2×D181—41no—
133C2×D181—42no—
134C4×C181—22no—
135C4×C181—41no—
136C4×C181—42no—
137C4×C181#12112yes—
138C4×C181C22×D411yescharacteristic
139C4×C181C2×#4111yescharacteristic
140Dic181—41no—
141Dic181—42no—
142C2×C181—22no—
143C2×C181—24no—
144C2×C181—81no—
145C2×C181C2×#12111yescharacteristic
146C2×C181C2×#13512yes—
147D181—81no—
148C181#305011yescharacteristic, complemented, Sylow
149#3050—1811nomaximal, Sylow
150C2×#135—1811no—
151C23×C8—1811no—
152#525—1811no—
153#525D18111yescharacteristic, complemented
154#467—1812no—
155#466—1812no—
156#465—1812no—
157#465—1812no—
158#427—1811no—
159#416—1811no—
160#416—1811no—
161C2×#41—1811no—
162C2×#41C2×D18111yescharacteristic
163C23×C4C2×D18111yescharacteristic
164#135—3624no—
165C2×Q16—1811no—
166C2×SD16—1811no—
167C2×SD16—1811no—
168C2×D8—1811no—
169C22×C8—3624no—
170C22×C8—1811no—
171C22×C8—1812no—
172C22×C8—1812no—
173C22×C8—1811no—
174#123—21no—
175#123—3621no—
176C22⋊Q8—1812no—
177C22⋊Q8C2×D18112yes—
178#121—1812no—
179#121C2×D18112yes—
180#120—21no—
181#120—3621no—
182C4×D4—22no—
183C4×D4—3622no—
184#117—1811no—
185#117C2×D18111yescharacteristic
186#107—1812no—
187#106—1812no—
188#103—1812no—
189#103—1812no—
190#102—1812no—
191#102—1812no—
192C24C22⋊D18111yescharacteristic
193#41—22no—
194#41—22no—
195#41—3622no—
196#41—3622no—
197C2×Q8—1811no—
198C2×Q8C22×D18111yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—3622no—
202C2×D4—3621no—
203C2×D4—1811no—
204C2×D4C22×D18111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—3621no—
209C22×C4C22⋊D18111yescharacteristic
210C22×C4C22⋊D18112yes—
211C22×C4C22×D18112yes—
212C22×C4C22×D18111yescharacteristic
213Q16—3622no—
214SD16—3622no—
215SD16—3622no—
216D8—3622no—
217C2×C8—3622no—
218C2×C8—3624no—
219C2×C8—3624no—
220C2×C8—1812no—
221C2×C8—1814no—
222C2×C8—1811no—
223C2×C8—1811no—
224C4⋊C4—22no—
225C4⋊C4—3622no—
226C4⋊C4—1812no—
227C4⋊C4C22×D18112yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—3621no—
232C22⋊C4—3622no—
233C22⋊C4—3622no—
234C42—21no—
235C42—3621no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—3621no—
240C23C2×C22⋊D18111yescharacteristic
241C23C2×C22⋊D18112yes—
242Q8—21no—
243Q8—21no—
244Q8—3621no—
245Q8—3621no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—3621no—
251D4—3621no—
252D4—7242no—
253D4—7244no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—3622no—
262C2×C4—3621no—
263C2×C4—7242no—
264C2×C4D4×D18112yes—
265C2×C4C2×C22⋊D18112yes—
266C2×C4C2×C22⋊D18111yescharacteristic
267C2×C4#1566844812yes—
268C2×C4C23×D18111yescharacteristic
269C8—3622no—
270C8—1812no—
271C8—1812no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—7241no—
279C22—7242no—
280C22#1915378812yes—
281C22#1915383614yescomplemented
282C22C22×C22⋊D18111yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—7241no—
287C4—7242no—
288C4#1915378812yes—
289C4C2×D4×D18111yescharacteristic
290C4C2×#1566844811yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—14481no—
295C2C2×#1915378811yescharacteristic
296C2C2×#1915383612yes—
subgroupG/H
centerC2×C4C2×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#525×C181C2
socleC22×C181C22×D4
hypercenter#525D181
hypocenterC181#3050
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#19153836C2semidirect
2. C23×#13089678C2semidirect
3. #26041790C2semidirect
4. #26041518C2semidirect
5. #26041513C2semidirect
6. #26041510C2semidirect
7. #26041506C2semidirect
8. #26041315C2semidirect
9. #26041275C2semidirect
10. #26041273C2semidirect
11. #525×C181C2semidirect
14. C2×#15668439C22semidirect
15. C2×#15668437C22semidirect
16. C2×#15668436C22semidirect
17. C2×#15668434C22semidirect
29. #19153749C22semidirect
30. #19153747C22semidirect
31. #19153740C22semidirect
32. #19153739C22semidirect
33. #19153737C22semidirect
34. #19153736C22semidirect
57. C2×#13089678D4semidirect
130. C22×C181#135semidirect
148. C181#3050semidirect
153. #525D181semidirect
281. C22#19153836semidirect

Maximal subgroups

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

subgroupindex
C2×#191538362
C23×#130896782
#260417902
#260415182
#260415132
#260415102
#260415062
#260413152
#260412752
#260412732
#525×C1812
#3050181