Finite Group Databasebeta

← #37547048

Subgroups — #37547048

Order 23168.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D181C212yesmaximal, complemented
2#26041856C214yesmaximal, complemented
3#26041854C212yesmaximal, complemented
4#26041849C212yesmaximal, complemented
5#26041847C212yesmaximal, complemented
6#26041842C212yesmaximal, complemented
7#26041837C214yesmaximal, complemented
8#26041829C211yesmaximal, characteristic, complemented
9#26041825C212yesmaximal, complemented
10#26041788C218yesmaximal, complemented
11C2×#19153773C211yesmaximal, characteristic, complemented
12#532×C181C211yesmaximal, characteristic, complemented
13C2×#15668450C2212yes—
14C2×#15668449C2214yes—
15C2×#15668449C2212yescomplemented
16C2×#15668448C2212yes—
17C2×#15668448C2214yescomplemented
18#41×D181—216no—
19C2×Q8×D181C2212yescomplemented
20C2×D4×D181C2214yes—
21C2×D4×D181C2212yescomplemented
22C22×C4×D181C2211yescharacteristic
23C22×C4×D181C2214yescomplemented
24#19153816C2211yescharacteristic, complemented
25#19153814C2212yescomplemented
26#19153813C2211yescharacteristic, complemented
27#19153812C2212yescomplemented
28#19153805C2212yescomplemented
29#19153804C2212yescomplemented
30#19153803C2212yescomplemented
31#19153801C2218yescomplemented
32#19153798C2218yescomplemented
33#19153794C2218yescomplemented
34#19153793C2218yescomplemented
35#19153790C2214yescomplemented
36#19153789C2218yescomplemented
37#19153788C2218yescomplemented
38#19153788C2218yescomplemented
39#19153786C2218yescomplemented
40#19153784C2218yescomplemented
41C4×D724C2214yescomplemented
42C4×C22⋊D181C2218yescomplemented
43#19153773C2214yescomplemented
44#19153773C2212yescomplemented
45#19153773C2212yescomplemented
46C2×#15668428C2212yescomplemented
47C2×#15668426C2212yes—
48C2×C4×Dic181C2211yescharacteristic, complemented
49C2×#41×C181C2212yescomplemented
50C4⋊Q8×C181C2211yescharacteristic, complemented
51#127×C181C2211yescharacteristic, complemented
52#124×C181C2212yescomplemented
53#121×C181C2214yescomplemented
54C4×D4×C181C2214yescomplemented
55C2×C42×C181C2211yescharacteristic, complemented
56C23×D181C2314yes—
57C23×D181C2311yescharacteristic
58#15668450—28no—
59#15668449—216no—
60#15668449—28no—
61#15668448—216no—
62#15668448—28no—
63C2×Dic362C2314yes—
64C2×Dic362C2312yes—
65Q8×D181—28no—
66C2×D724C2314yes—
67C2×D724C2312yes—
68C2×D724C2314yes—
69C2×C22⋊D181C2314yes—
70C2×C22⋊D181C2314yes—
71C2×C22⋊D181C2314yes—
72C2×C22⋊D181C2318yes—
73D4×D181—216no—
74D4×D181—28no—
75C22×Dic181C2314yes—
76C22×Dic181C2312yes—
77C22×Dic181C2311yescharacteristic
78C2×C4×D181—216no—
79C2×C4×D181D418yescomplemented
80C2×C4×D181C2314yes—
81C2×C4×D181C2314yes—
82C2×C4×D181C2314yes—
83C2×C4×D181C2312yes—
84C2×C4×D181C2312yes—
85C4⋊Dic181C2314yes—
86#15668428C2314yes—
87#15668428C2314yes—
88#15668428C2318yes—
89C22⋊Dic181C2318yes—
90#15668426C2314yes—
91#15668426C2314yes—
92#15668426C2318yes—
93#15668426C2318yes—
94C4×Dic181C2312yes—
95C4×Dic181C2312yes—
96#41×C181—28no—
97C2×Q8×C181C2312yes—
98C2×D4×C181C2314yes—
99C2×D4×C181C2312yes—
100C2×D4×C181C2314yes—
101C22×C4×C181C2314yes—
102C22×C4×C181C2312yes—
103C22×C4×C181C2311yescharacteristic
104C4⋊C4×C181C2314yes—
105C22⋊C4×C181C2318yes—
106C42×C181C2312yes—
107C42×C181C2311yescharacteristic
108C42×C181C2311yescharacteristic
109C22×D181—28no—
110C22×D181—42no—
111C22×D181—48no—
112C22×D181C2×D414yes—
113C22×D181C2414yes—
114C22×D181C2412yes—
115Dic362—24no—
116Dic362—28no—
117D724—28no—
118D724—24no—
119D724—44no—
120C22⋊D181—28no—
121C22⋊D181—28no—
122C22⋊D181—28no—
123C22⋊D181—48no—
124C2×Dic181—28no—
125C2×Dic181—42no—
126C2×Dic181C2×D414yes—
127C2×Dic181C2414yes—
128C2×Dic181C2414yes—
129C2×Dic181C2412yes—
130C4×D181—28no—
131C4×D181—28no—
132C4×D181—28no—
133C4×D181C2×D418yes—
134C4×D181C2×D418yes—
135C23×C181C2414yes—
136C23×C181C2411yescharacteristic
137Q8×C181—24no—
138D4×C181—28no—
139D4×C181—24no—
140D4×C181—44no—
141C2×C4×C181—28no—
142C2×C4×C181—42no—
143C2×C4×C181C2×D414yes—
144C2×C4×C181C2412yes—
145C2×C4×C181C2412yes—
146C2×C4×C181C2414yes—
147C2×C4×C181C2411yescharacteristic
148C2×C4×C181C2411yescharacteristic
149C22×C181—22no—
150C22×C181—24no—
151C22×C181—28no—
152C22×C181C22×D412yes—
153C22×C181C2511yescharacteristic
154C2×D181—24no—
155C2×D181—24no—
156C2×D181—28no—
157C2×D181—44no—
158C2×D181—416no—
159C2×D181C22×D414yes—
160C4×C181—24no—
161C4×C181—22no—
162C4×C181—22no—
163C4×C181C22×D412yes—
164C4×C181C22×D412yes—
165Dic181—24no—
166Dic181—24no—
167Dic181C22×D414yes—
168C2×C181—22no—
169C2×C181—44no—
170C2×C181C23×D411yescharacteristic
171C2×C181#58512yes—
172D181—24no—
173D181—44no—
174C181#344911yescharacteristic, complemented, Sylow
175#3449—1811nomaximal, Sylow
176C22×#41—1812no—
177#537—1814no—
178#536—1812no—
179#536—1812no—
180#535—1812no—
181#535—1812no—
182#534—1814no—
183#532—1812no—
184#532—1811no—
185#532D18111yescharacteristic, complemented
186#525—1818no—
187C2×#117—1811no—
188C2×#41—36216no—
189C2×#41—1812no—
190C2×#41—1812no—
191C2×#41—1814no—
192C2×#41—1814no—
193C2×#41—1812no—
194C2×#41C2×D18112yescomplemented
195C22×Q8—1812no—
196C22×D4—1814no—
197C22×D4—1812no—
198C23×C4—1814no—
199C23×C4—1811no—
200C4⋊Q8—1812no—
201C4⋊Q8—1811no—
202C4⋊Q8C2×D18111yescharacteristic, complemented
203#127—1812no—
204#127—1811no—
205#127C2×D18111yescharacteristic, complemented
206#124—1812no—
207#124—1812no—
208#124—1812no—
209#124C2×D18112yescomplemented
210#123—1818no—
211#123—1818no—
212C22⋊Q8—1818no—
213C22⋊Q8—1818no—
214#121—1818no—
215#121—1818no—
216#121—1814no—
217#121—1818no—
218#121C2×D18114yescomplemented
219#120—1818no—
220#120—1818no—
221C4×D4—1818no—
222C4×D4—1814no—
223C4×D4C2×D18114yescomplemented
224#117—1814no—
225#117—1812no—
226#117—1812no—
227C2×C4⋊C4—1812no—
228C2×C22⋊C4—1812no—
229C2×C42—1811no—
230C2×C42C2×D18111yescharacteristic, complemented
231C24—1814no—
232C24—1811no—
233#41—28no—
234#41—36216no—
235#41—36216no—
236#41—3628no—
237#41—3628no—
238#41—3628no—
239C2×Q8—3628no—
240C2×Q8—1814no—
241C2×Q8—1812no—
242C2×Q8C22×D18112yes—
243C2×D4—36216no—
244C2×D4—3628no—
245C2×D4—1814no—
246C2×D4—1814no—
247C2×D4—1814no—
248C2×D4—1818no—
249C2×D4—1814no—
250C2×D4—1812no—
251C2×D4—1814no—
252C2×D4C22×D18114yes—
253C2×D4C22×D18112yes—
254C2×D4C22×D18114yes—
255C22×C4—36216no—
256C22×C4—1814no—
257C22×C4—1814no—
258C22×C4—1814no—
259C22×C4—1814no—
260C22×C4—1811no—
261C22×C4—1818no—
262C22×C4—1812no—
263C22×C4—1812no—
264C22×C4—1812no—
265C22×C4C22×D18114yes—
266C22×C4C22×D18112yes—
267C22×C4C22×D18111yescharacteristic
268C4⋊C4—1814no—
269C4⋊C4—1814no—
270C4⋊C4—1818no—
271C4⋊C4—1814no—
272C4⋊C4C22×D18114yes—
273C22⋊C4—1818no—
274C22⋊C4—1814no—
275C22⋊C4—1814no—
276C22⋊C4—1818no—
277C22⋊C4—1818no—
278C22⋊C4C22×D18118yes—
279C42—1812no—
280C42—1812no—
281C42C22×D18111yescharacteristic
282C42C22×D18111yescharacteristic
283C42C22×D18112yes—
284C23—3628no—
285C23—1814no—
286C23—1812no—
287C23—1814no—
288C23—7242no—
289C23—7248no—
290C23C23×D18114yes—
291C23C23×D18111yescharacteristic
292Q8—24no—
293Q8—3628no—
294Q8—3624no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—3628no—
299D4—3628no—
300D4—3628no—
301D4—3628no—
302D4—3624no—
303D4—7244no—
304D4—7248no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—3628no—
308C2×C4—3628no—
309C2×C4—3628no—
310C2×C4—3628no—
311C2×C4—1814no—
312C2×C4—1814no—
313C2×C4—1814no—
314C2×C4—1812no—
315C2×C4—1818no—
316C2×C4—1818no—
317C2×C4—7242no—
318C2×C4D4×D18114yes—
319C2×C4C23×D18112yes—
320C2×C4C23×D18112yes—
321C2×C4C23×D18114yes—
322C2×C4C23×D18111yescharacteristic
323C2×C4C23×D18111yescharacteristic
324C22—22no—
325C22—28no—
326C22—24no—
327C22—3624no—
328C22—3628no—
329C22—3624no—
330C22—1814no—
331C22—7244no—
332C22—72416no—
333C22C2×D4×D18112yes—
334C22C24×D18111yescharacteristic
335C4—22no—
336C4—22no—
337C4—24no—
338C4—3624no—
339C4—3624no—
340C4—1814no—
341C4C2×D4×D18112yes—
342C4C2×D4×D18112yes—
343C2—22no—
344C2—44no—
345C2—3624no—
346C2—7244no—
347C2C22×D4×D18111yescharacteristic
348C2#2604211512yes—
subgroupG/H
centerC2×C4C23×D181
commutator, socle, cosocleC22×C181C25
FrattiniC22C24×D181
Fitting#532×C181C2
hypercenter#532D181
hypocenterC181#3449
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D181C2semidirect
2. #26041856C2semidirect
3. #26041854C2semidirect
4. #26041849C2semidirect
5. #26041847C2semidirect
6. #26041842C2semidirect
7. #26041837C2semidirect
8. #26041829C2semidirect
9. #26041825C2semidirect
10. #26041788C2semidirect
11. C2×#19153773C2semidirect
12. #532×C181C2semidirect
15. C2×#15668449C22semidirect
17. C2×#15668448C22semidirect
19. C2×Q8×D181C22semidirect
21. C2×D4×D181C22semidirect
23. C22×C4×D181C22semidirect
24. #19153816C22semidirect
25. #19153814C22semidirect
26. #19153813C22semidirect
27. #19153812C22semidirect
28. #19153805C22semidirect
29. #19153804C22semidirect
30. #19153803C22semidirect
31. #19153801C22semidirect
32. #19153798C22semidirect
33. #19153794C22semidirect
34. #19153793C22semidirect
35. #19153790C22semidirect
36. #19153789C22semidirect
37. #19153788C22semidirect
38. #19153788C22semidirect
39. #19153786C22semidirect
40. #19153784C22semidirect
41. C4×D724C22semidirect
42. C4×C22⋊D181C22semidirect
43. #19153773C22semidirect
44. #19153773C22semidirect
45. #19153773C22semidirect
46. C2×#15668428C22semidirect
48. C2×C4×Dic181C22semidirect
49. C2×#41×C181C22semidirect
50. C4⋊Q8×C181C22semidirect
51. #127×C181C22semidirect
52. #124×C181C22semidirect
53. #121×C181C22semidirect
54. C4×D4×C181C22semidirect
55. C2×C42×C181C22semidirect
79. C2×C4×D181D4semidirect
174. C181#3449semidirect
185. #532D181semidirect
194. C2×#41C2×D181semidirect
202. C4⋊Q8C2×D181semidirect
205. #127C2×D181semidirect
209. #124C2×D181semidirect
218. #121C2×D181semidirect
223. C4×D4C2×D181semidirect
230. C2×C42C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1812
#260418562
#260418542
#260418492
#260418472
#260418422
#260418372
#260418292
#260418252
#260417882
C2×#191537732
#532×C1812
#3449181