Finite Group Databasebeta

← #37547980

Subgroups — #37547980

Order 23168.

1050 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042017C214yesmaximal, complemented
2#26041981C214yesmaximal, complemented
3#26041979C212yesmaximal, complemented
4#26041967C212yesmaximal, complemented
5#26041949C212yesmaximal, complemented
6#26041933C212yesmaximal, complemented
7#26041891C211yesmaximal, characteristic, complemented
8#26041881C214yesmaximal, complemented
9#26041873C214yesmaximal, complemented
10#26041858C212yesmaximal, complemented
11#26041791C211yesmaximal, characteristic, complemented
12#26041789C212yesmaximal, complemented
13#540×C181C211yesmaximal, characteristic, complemented
14C2×#15668449C2211yescharacteristic
15C2×#15668448C2212yes—
16C2×#15668448C2212yescomplemented
17C22×C4×D181C2211yescharacteristic
18#19153816C2211yescharacteristic, complemented
19#19153812C2214yes—
20#19153811C2214yes—
21#19153809C2214yes—
22#19153809C2214yescomplemented
23#19153808C2211yescharacteristic, complemented
24#19153807C2214yescomplemented
25#19153806C2214yescomplemented
26#19153801C2214yes—
27#19153800C2214yes—
28#19153800C2212yes—
29#19153800C2214yescomplemented
30#19153799C2214yes—
31#19153798C2214yes—
32#19153796C2212yes—
33#19153796C2212yes—
34#19153796C2212yescomplemented
35#19153795C2212yes—
36#19153793C2214yescomplemented
37#19153791C2212yes—
38#19153790C2214yes—
39#19153789C2214yes—
40#19153789C2214yescomplemented
41#19153788C2214yes—
42#19153788C2214yes—
43#19153788C2214yescomplemented
44#19153787C2214yes—
45#19153787C2212yes—
46#19153787C2212yescomplemented
47#19153786C2212yes—
48#19153784C2214yes—
49C4×Dic362C2212yescomplemented
50C4×D724C2212yes—
51C4×D724C2212yes—
52D4×Dic181C2214yes—
53D4×Dic181C2214yescomplemented
54C4×C22⋊D181C2214yescomplemented
55#19153776C2214yes—
56#19153774C2212yes—
57#19153774C2214yescomplemented
58C2×C4⋊Dic181C2214yescomplemented
59C4⋊C4×D181C2212yescomplemented
60#127×C181C2211yescharacteristic, complemented
61#125×C181C2211yescharacteristic, complemented
62#123×C181C2214yescomplemented
63#121×C181C2212yes—
64#121×C181C2214yes—
65C4×D4×C181C2212yescomplemented
66#117×C181C2211yescharacteristic
67C23×D181C2311yescharacteristic
68#15668449—42no—
69#15668448—44no—
70#15668448—44no—
71C2×Dic362C2312yes—
72C2×Dic362C2311yescharacteristic
73C2×D724C2312yes—
74C2×D724C2312yes—
75C2×D724C2311yescharacteristic
76C2×C22⋊D181C2314yes—
77C2×C22⋊D181C2314yes—
78C2×C22⋊D181C2314yes—
79C2×C22⋊D181C2312yes—
80C2×C22⋊D181C2314yes—
81C22×Dic181C2314yes—
82C22×Dic181C2314yes—
83C22×Dic181C2311yescharacteristic
84C2×C4×D181—42no—
85C2×C4×D181C2312yes—
86C2×C4×D181C2312yes—
87C2×C4×D181C2314yes—
88C2×C4×D181C2312yes—
89C4⋊Dic181C2312yes—
90C4⋊Dic181C2312yes—
91C4⋊Dic181C2314yes—
92C4⋊Dic181C2314yes—
93C4⋊Dic181C2314yes—
94C4⋊Dic181C2312yes—
95#15668428C2314yes—
96#15668428C2314yes—
97#15668428C2314yes—
98#15668428C2314yes—
99C22⋊Dic181C2314yes—
100C22⋊Dic181C2314yes—
101C22⋊Dic181C2314yes—
102#15668426C2314yes—
103#15668426C2312yes—
104#15668426C2314yes—
105#15668426C2314yes—
106#15668426C2312yes—
107#15668426C2312yes—
108#15668426C2312yes—
109C4×Dic181C2314yes—
110C4×Dic181C2314yes—
111C2×D4×C181C2312yes—
112C2×D4×C181C2314yes—
113C2×D4×C181C2314yes—
114C22×C4×C181C2311yescharacteristic
115C22×C4×C181C2314yes—
116C4⋊C4×C181C2312yes—
117C4⋊C4×C181C2312yes—
118C4⋊C4×C181C2314yes—
119C22⋊C4×C181C2312yes—
120C22⋊C4×C181C2314yes—
121C22⋊C4×C181C2314yes—
122C42×C181C2311yescharacteristic
123C42×C181C2311yescharacteristic
124C22×D181—41no—
125C22×D181—42no—
126C22×D181C2412yes—
127C22×D181C2412yes—
128Dic362—24no—
129Dic362—41no—
130D724—42no—
131D724—42no—
132D724—41no—
133C22⋊D181—44no—
134C22⋊D181—44no—
135C22⋊D181—44no—
136C22⋊D181—44no—
137C22⋊D181—42no—
138C2×Dic181—41no—
139C2×Dic181—44no—
140C2×Dic181—44no—
141C2×Dic181C2412yes—
142C2×Dic181C2414yes—
143C2×Dic181C2414yes—
144C2×Dic181C2412yes—
145C4×D181—24no—
146C4×D181—24no—
147C4×D181—42no—
148C4×D181—44no—
149C23×C181C2411yescharacteristic
150C23×C181C2414yes—
151D4×C181—24no—
152D4×C181—44no—
153D4×C181—44no—
154C2×C4×C181—22no—
155C2×C4×C181—44no—
156C2×C4×C181C2412yes—
157C2×C4×C181C2412yes—
158C2×C4×C181C2411yescharacteristic
159C2×C4×C181C2414yes—
160C2×C4×C181C2411yescharacteristic
161C22×C181—21no—
162C22×C181—24no—
163C22×C181—24no—
164C22×C181—24no—
165C22×C181—22no—
166C22×C181C2511yescharacteristic
167C2×D181—22no—
168C2×D181—22no—
169C2×D181—22no—
170C2×D181—22no—
171C2×D181—22no—
172C2×D181—22no—
173C2×D181—44no—
174C4×C181—21no—
175C4×C181—22no—
176C4×C181—24no—
177C4×C181—22no—
178C4×C181H(5,𝔽2)12yes—
179Dic181—22no—
180Dic181—22no—
181Dic181—24no—
182Dic181—24no—
183C2×C181—41no—
184C2×C181—44no—
185C2×C181C2×H(5,𝔽2)11yescharacteristic
186C2×C181#58512yes—
187D181—42no—
188D181—42no—
189C181#353911yescharacteristic, complemented, Sylow
190#3539—1811nomaximal, Sylow
191#560—1814no—
192#553—1814no—
193#553—1812no—
194#552—1812no—
195#550—1812no—
196#547—1812no—
197#541—1811no—
198#540—1814no—
199#540D18111yescharacteristic, complemented
200#539—1814no—
201#537—1812no—
202#525—1812no—
203#525—1811no—
204C2×#41—1812no—
205C2×#41—1812no—
206C2×#41—1811no—
207C23×C4—1811no—
208C4⋊Q8—1811no—
209#127—1814no—
210#127C2×D18111yescharacteristic, complemented
211#126—1814no—
212#126—1814no—
213#126—1814no—
214#125—1814no—
215#125—1814no—
216#125—1811no—
217#125C2×D18111yescharacteristic, complemented
218#123—1814no—
219#123—1814no—
220#123—1814no—
221#123—1812no—
222#123—1814no—
223#123—1814no—
224#123C2×D18114yescomplemented
225C22⋊Q8—1812no—
226C22⋊Q8—1812no—
227C22⋊Q8—1812no—
228C22⋊Q8—1812no—
229C22⋊Q8—1814no—
230#121—1812no—
231#121—1812no—
232#121—1814no—
233#121—1814no—
234#121—1814no—
235#121—1814no—
236#121—1814no—
237#121—1812no—
238#121—1814no—
239#121—1814no—
240#121C2×D18112yes—
241#121C2×D18114yes—
242#120—1814no—
243#120—1812no—
244C4×Q8—1812no—
245C4×D4—1814no—
246C4×D4—1814no—
247C4×D4—1814no—
248C4×D4—1814no—
249C4×D4—1812no—
250C4×D4—1812no—
251C4×D4C2×D18112yescomplemented
252#117—1812no—
253#117—1814no—
254#117C2×D18111yescharacteristic
255C2×C4⋊C4—1814no—
256C2×C4⋊C4—1812no—
257C24—1811no—
258#41—7242no—
259#41—7244no—
260#41—7244no—
261C2×Q8—1812no—
262C2×Q8—1811no—
263C2×D4—1814no—
264C2×D4—1814no—
265C2×D4—1812no—
266C2×D4—1814no—
267C2×D4—1814no—
268C2×D4—1812no—
269C2×D4—1812no—
270C2×D4—1811no—
271C2×D4C22×D18112yes—
272C2×D4C22×D18114yes—
273C2×D4C22×D18114yes—
274C22×C4—1812no—
275C22×C4—1812no—
276C22×C4—1814no—
277C22×C4—1814no—
278C22×C4—1811no—
279C22×C4—1814no—
280C22×C4—1812no—
281C22×C4—7242no—
282C22×C4C22×D18111yescharacteristic
283C22×C4C22×D18114yes—
284C4⋊C4—1812no—
285C4⋊C4—1814no—
286C4⋊C4—1812no—
287C4⋊C4—1814no—
288C4⋊C4—1814no—
289C4⋊C4—1814no—
290C4⋊C4—1812no—
291C4⋊C4—1814no—
292C4⋊C4—1814no—
293C4⋊C4—1814no—
294C4⋊C4C22×D18112yes—
295C4⋊C4C22×D18114yes—
296C4⋊C4C22×D18112yes—
297C22⋊C4—1814no—
298C22⋊C4—1814no—
299C22⋊C4—1814no—
300C22⋊C4—1812no—
301C22⋊C4—1814no—
302C22⋊C4—1814no—
303C22⋊C4—1812no—
304C22⋊C4—1812no—
305C22⋊C4—1814no—
306C22⋊C4—1812no—
307C22⋊C4C22×D18112yes—
308C22⋊C4C22×D18114yes—
309C22⋊C4C22×D18114yes—
310C42—1814no—
311C42—1814no—
312C42C22×D18111yescharacteristic
313C42C22×D18111yescharacteristic
314C23—1812no—
315C23—1812no—
316C23—7241no—
317C23—7242no—
318C23C23×D18111yescharacteristic
319C23C23×D18114yes—
320Q8—3624no—
321Q8—7241no—
322D4—24no—
323D4—44no—
324D4—44no—
325D4—7241no—
326D4—7242no—
327D4—7242no—
328D4—7242no—
329D4—7244no—
330D4—7244no—
331D4—7244no—
332D4—7244no—
333C2×C4—22no—
334C2×C4—44no—
335C2×C4—3624no—
336C2×C4—3624no—
337C2×C4—1814no—
338C2×C4—1812no—
339C2×C4—1814no—
340C2×C4—1812no—
341C2×C4—7242no—
342C2×C4—7244no—
343C2×C4—7241no—
344C2×C4—7244no—
345C2×C4—7244no—
346C2×C4C23×D18112yes—
347C2×C4C23×D18114yes—
348C2×C4C23×D18112yes—
349C2×C4C23×D18111yescharacteristic
350C2×C4C23×D18111yescharacteristic
351C22—24no—
352C22—24no—
353C22—22no—
354C22—21no—
355C22—24no—
356C22—3622no—
357C22—3622no—
358C22—3622no—
359C22—3622no—
360C22—3622no—
361C22—3622no—
362C22—7244no—
363C22C24×D18111yescharacteristic
364C4—22no—
365C4—24no—
366C4—22no—
367C4—21no—
368C4—3624no—
369C4—3622no—
370C4—3622no—
371C4—3624no—
372C4#1915386212yes—
373C2—41no—
374C2—44no—
375C2—7242no—
376C2—7242no—
377C2C2×#1915386211yescharacteristic
378C2#2604211412yes—
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#540×C181C2
hypercenter#540D181
hypocenterC181#3539
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #26042017C2semidirect
2. #26041981C2semidirect
3. #26041979C2semidirect
4. #26041967C2semidirect
5. #26041949C2semidirect
6. #26041933C2semidirect
7. #26041891C2semidirect
8. #26041881C2semidirect
9. #26041873C2semidirect
10. #26041858C2semidirect
11. #26041791C2semidirect
12. #26041789C2semidirect
13. #540×C181C2semidirect
16. C2×#15668448C22semidirect
18. #19153816C22semidirect
22. #19153809C22semidirect
23. #19153808C22semidirect
24. #19153807C22semidirect
25. #19153806C22semidirect
29. #19153800C22semidirect
34. #19153796C22semidirect
36. #19153793C22semidirect
40. #19153789C22semidirect
43. #19153788C22semidirect
46. #19153787C22semidirect
49. C4×Dic362C22semidirect
53. D4×Dic181C22semidirect
54. C4×C22⋊D181C22semidirect
57. #19153774C22semidirect
58. C2×C4⋊Dic181C22semidirect
59. C4⋊C4×D181C22semidirect
60. #127×C181C22semidirect
61. #125×C181C22semidirect
62. #123×C181C22semidirect
65. C4×D4×C181C22semidirect
189. C181#3539semidirect
199. #540D181semidirect
210. #127C2×D181semidirect
217. #125C2×D181semidirect
224. #123C2×D181semidirect
251. C4×D4C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420172
#260419812
#260419792
#260419672
#260419492
#260419332
#260418912
#260418812
#260418732
#260418582
#260417912
#260417892
#540×C1812
#3539181