Finite Group Databasebeta

← #37536702

Subgroups — #37536702

Order 23168.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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×#19153784C211yesmaximal, characteristic
2C2×#19153776C212yesmaximal, complemented
3C2×C22⋊C4×D181C212yesmaximal, complemented
4#26041167C212yesmaximal, complemented
5#26041159C211yesmaximal, characteristic, complemented
6#26041155C212yesmaximal, complemented
7#26041150C212yesmaximal, complemented
8C4×#15668426C212yesmaximal, complemented
9#390×C181C211yesmaximal, characteristic, complemented
10C24×D181C2211yescharacteristic
11C22×C22⋊D181C2212yes—
12C23×Dic181C2211yescharacteristic, complemented
13C22×C4×D181C2212yes—
14C22×C4×D181C2212yes—
15#19153784C418yescomplemented
16#19153776—28no—
17C2×#15668428C2212yescomplemented
18C2×C22⋊Dic181C2212yes—
19C2×C22⋊Dic181C2211yescharacteristic
20C2×C22⋊Dic181C2211yescharacteristic, complemented
21C2×#15668426C2212yes—
22C2×#15668426C2212yes—
23C2×#15668426C2212yes—
24C22⋊C4×D181—28no—
25C2×C4×Dic181C2212yescomplemented
26#19153723C2212yescomplemented
27#19153723C2212yescomplemented
28#19153723C2212yescomplemented
29C22×D4×C181C2211yescharacteristic
30C2×C4⋊C4×C181C2211yescharacteristic, complemented
31C2×C22⋊C4×C181C2212yes—
32C2×C22⋊C4×C181C2212yescomplemented
33C2×C42×C181C2211yescharacteristic, complemented
34C23×D181—24no—
35C23×D181—24no—
36C23×D181—41no—
37C23×D181—41no—
38C23×D181C2×C414yescomplemented
39C23×D181C2312yes—
40C2×C22⋊D181—24no—
41C2×C22⋊D181—24no—
42C2×C22⋊D181C2×C418yescomplemented
43C22×Dic181—24no—
44C22×Dic181—41no—
45C22×Dic181C2312yes—
46C22×Dic181C2312yes—
47C22×Dic181C2312yes—
48C2×C4×D181—24no—
49C2×C4×D181—24no—
50C2×C4×D181—24no—
51C2×C4×D181—24no—
52C2×C4×D181—42no—
53C2×C4×D181—42no—
54#15668428—24no—
55C22⋊Dic181—24no—
56C22⋊Dic181—41no—
57C22⋊Dic181C2×C414yescomplemented
58#15668426—24no—
59#15668426C2×C418yescomplemented
60#15668426D418yescomplemented
61C4×Dic181—24no—
62C24×C181C2311yescharacteristic
63C24×C181C2311yescharacteristic
64C2×D4×C181—41no—
65C2×D4×C181—41no—
66C2×D4×C181C2×C414yescomplemented
67C22×C4×C181C2312yes—
68C22×C4×C181C2311yescharacteristic
69C22×C4×C181C2312yes—
70C4⋊C4×C181—41no—
71C22⋊C4×C181—24no—
72C22⋊C4×C181—24no—
73C42×C181—41no—
74C22×D181—22no—
75C22×D181—22no—
76C22×D181—22no—
77C22×D181—22no—
78C22×D181—216no—
79C22×D181—44no—
80C22×D181—44no—
81C22×D181—48no—
82C22×D181—44no—
83C22×D181C22×C414yes—
84C22×D181C2×D414yes—
85C22×D181#4114yes—
86C22⋊D181—216no—
87C2×Dic181—22no—
88C2×Dic181—22no—
89C2×Dic181—22no—
90C2×Dic181—22no—
91C2×Dic181—22no—
92C2×Dic181—22no—
93C2×Dic181—22no—
94C2×Dic181—44no—
95C2×Dic181C22×C414yes—
96C2×Dic181C2×D414yes—
97C4×D181—48no—
98C4×D181—48no—
99C23×C181—22no—
100C23×C181—22no—
101C23×C181—22no—
102C23×C181—22no—
103C23×C181—21no—
104C23×C181—21no—
105C23×C181—21no—
106C23×C181—21no—
107C23×C181C22×C412yes—
108C23×C181C22×C412yes—
109C23×C181C2411yescharacteristic
110D4×C181—44no—
111C2×C4×C181—22no—
112C2×C4×C181—21no—
113C2×C4×C181—22no—
114C2×C4×C181—21no—
115C2×C4×C181—22no—
116C2×C4×C181—22no—
117C2×C4×C181—22no—
118C2×C4×C181C22×C412yes—
119C2×C4×C181C2×D414yes—
120C22×C181—24no—
121C22×C181—24no—
122C22×C181—22no—
123C22×C181—22no—
124C22×C181—41no—
125C22×C181—41no—
126C22×C181—42no—
127C22×C181—42no—
128C22×C181—41no—
129C22×C181—41no—
130C22×C181C23×C411yescharacteristic
131C22×C181C22×D412yes—
132C22×C181C2×#4112yes—
133C22×C181H(5,𝔽2)11yescharacteristic
134C22×C181H(5,𝔽2)11yescharacteristic
135C2×D181—24no—
136C2×D181—24no—
137C2×D181—24no—
138C2×D181—24no—
139C2×D181—24no—
140C2×D181—24no—
141C2×D181—416no—
142C2×D181C4×D418yescomplemented
143C4×C181—22no—
144C4×C181—24no—
145C4×C181—42no—
146Dic181—24no—
147Dic181—24no—
148Dic181—42no—
149C2×C181—42no—
150C2×C181—42no—
151C2×C181C2×C4×D412yes—
152C2×C181#51811yescharacteristic
153C2×C181D4211yescharacteristic
154C2×C181#54612yes—
155C2×C181#55111yescharacteristic
156D181—28no—
157C181#235811yescharacteristic, complemented, Sylow
158#2358—1811nomaximal, Sylow
159C2×#120—1811no—
160C2×C4×D4—1812no—
161C22×C22⋊C4—1812no—
162#390D18111yescharacteristic, complemented
163#388—1812no—
164#386—1811no—
165#386—1812no—
166#385—1812no—
167C4×C22⋊C4—1812no—
168C25—1811no—
169C22×D4—1812no—
170C22×D4C2×D18111yescharacteristic
171C23×C4—1811no—
172C23×C4—1812no—
173C23×C4—1812no—
174#120—1818no—
175C4×D4—3628no—
176C2×C4⋊C4—1812no—
177C2×C4⋊C4C2×D18111yescharacteristic, complemented
178C2×C22⋊C4—3628no—
179C2×C22⋊C4—1812no—
180C2×C22⋊C4—1812no—
181C2×C22⋊C4—1811no—
182C2×C22⋊C4—1812no—
183C2×C22⋊C4—1812no—
184C2×C22⋊C4—1811no—
185C2×C22⋊C4C2×D18112yes—
186C2×C22⋊C4C2×D18112yescomplemented
187C2×C42—1812no—
188C2×C42C2×D18111yescharacteristic, complemented
189#95—1812no—
190#95—1812no—
191#95—1812no—
192C24—3624no—
193C24—3624no—
194C24—1812no—
195C24—1814no—
196C24—7241no—
197C24—7241no—
198C24C22×D18111yescharacteristic
199C24C22×D18111yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—3624no—
203C2×D4—3624no—
204C2×D4—1818no—
205C2×D4C4×D18114yescomplemented
206C22×C4—3624no—
207C22×C4—3624no—
208C22×C4—3624no—
209C22×C4—3624no—
210C22×C4—3624no—
211C22×C4—1812no—
212C22×C4—1812no—
213C22×C4—1812no—
214C22×C4—7242no—
215C22×C4—7242no—
216C22×C4—7241no—
217C22×C4C22×D18112yes—
218C22×C4C22×D18111yescharacteristic
219C22×C4C22×D18112yes—
220C4⋊C4—41no—
221C4⋊C4—3624no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—3624no—
225C22⋊C4—3624no—
226C22⋊C4—1814no—
227C22⋊C4—1818no—
228C22⋊C4—1818no—
229C22⋊C4—7241no—
230C42—41no—
231C42—3624no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—36216no—
241C23—3622no—
242C23—3622no—
243C23—3622no—
244C23—3622no—
245C23—1814no—
246C23—1814no—
247C23—1814no—
248C23—7244no—
249C23—7244no—
250C23—7248no—
251C23—7244no—
252C23C2×C4×D18112yes—
253C23C2×C4×D18112yes—
254C23C23×D18111yescharacteristic
255D4—44no—
256D4—36216no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—3622no—
265C2×C4—3622no—
266C2×C4—3622no—
267C2×C4—3622no—
268C2×C4—3622no—
269C2×C4—3622no—
270C2×C4—3622no—
271C2×C4—1814no—
272C2×C4—1814no—
273C2×C4—7248no—
274C2×C4—7248no—
275C2×C4—7244no—
276C2×C4C2×C4×D18112yes—
277C2×C4D4×D18114yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—3624no—
289C22—3624no—
290C22—3624no—
291C22—3624no—
292C22—3624no—
293C22—3624no—
294C22—1818no—
295C22—72416no—
296C22C22×C4×D18111yescharacteristic
297C22C2×D4×D18112yes—
298C22#41×D18112yes—
299C22#1915386111yescharacteristic
300C22#1915386111yescharacteristic
301C4—24no—
302C4—22no—
303C4—42no—
304C4—3624no—
305C4—3624no—
306C4—7242no—
307C2—42no—
308C2—42no—
309C2—3628no—
310C2C4×D4×D18112yes—
311C2#2604175211yescharacteristic
312C2#2604191411yescharacteristic
313C2#2604191812yes—
314C2#2604195911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D181
commutatorC22×C181C23×C4
Fitting#390×C181C2
socle, cosocleC23×C181C24
prime radicalC2×#19153784C2
hypercenter#390D181
hypocenterC181#2358
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#19153776C2semidirect
3. C2×C22⋊C4×D181C2semidirect
4. #26041167C2semidirect
5. #26041159C2semidirect
6. #26041155C2semidirect
7. #26041150C2semidirect
8. C4×#15668426C2semidirect
9. #390×C181C2semidirect
12. C23×Dic181C22semidirect
15. #19153784C4semidirect
17. C2×#15668428C22semidirect
20. C2×C22⋊Dic181C22semidirect
25. C2×C4×Dic181C22semidirect
26. #19153723C22semidirect
27. #19153723C22semidirect
28. #19153723C22semidirect
30. C2×C4⋊C4×C181C22semidirect
32. C2×C22⋊C4×C181C22semidirect
33. C2×C42×C181C22semidirect
38. C23×D181C2×C4semidirect
42. C2×C22⋊D181C2×C4semidirect
57. C22⋊Dic181C2×C4semidirect
59. #15668426C2×C4semidirect
60. #15668426D4semidirect
66. C2×D4×C181C2×C4semidirect
142. C2×D181C4×D4semidirect
157. C181#2358semidirect
162. #390D181semidirect
177. C2×C4⋊C4C2×D181semidirect
186. C2×C22⋊C4C2×D181semidirect
188. C2×C42C2×D181semidirect
205. C2×D4C4×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191537842
C2×#191537762
C2×C22⋊C4×D1812
#260411672
#260411592
#260411552
#260411502
C4×#156684262
#390×C1812
#2358181