Finite Group Databasebeta

← #37547862

Subgroups — #37547862

Order 23168.

1058 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042028C212yesmaximal, complemented
2#26042027C212yesmaximal, complemented
3#26042025C214yesmaximal, complemented
4#26042021C211yesmaximal, characteristic, complemented
5#26042020C214yesmaximal, complemented
6#26042019C212yesmaximal, complemented
7#26041830C212yesmaximal, complemented
8#26041828C211yesmaximal, characteristic, complemented
9#26041814C212yesmaximal, complemented
10#26041812C214yesmaximal, complemented
11#26041792C214yesmaximal, complemented
12#26041787C212yesmaximal, complemented
13#563×C181C211yesmaximal, characteristic, complemented
14C2×#15668450C2212yes—
15C2×#15668450C2211yescharacteristic, complemented
16C2×#15668448C2211yescharacteristic
17C22×C4×D181C2212yescomplemented
18#19153814C2211yescharacteristic, complemented
19#19153813C2212yes—
20#19153812C2211yescharacteristic
21#19153811C2214yescomplemented
22#19153810C2214yescomplemented
23#19153806C2214yescomplemented
24#19153805C2212yes—
25#19153805C2212yescomplemented
26#19153804C2212yes—
27#19153804C2211yescharacteristic, complemented
28#19153802C2214yes—
29#19153801C2214yes—
30#19153801C2214yescomplemented
31#19153801C2214yescomplemented
32#19153801C2214yescomplemented
33#19153799C2212yescomplemented
34#19153798C2214yescomplemented
35#19153796C2212yescomplemented
36#19153794C2214yescomplemented
37#19153794C2214yescomplemented
38#19153792C2214yescomplemented
39#19153791C2212yes—
40#19153791C2214yes—
41#19153791C2214yescomplemented
42#19153791C2214yescomplemented
43#19153790C2214yes—
44#19153789C2214yes—
45#19153789C2214yes—
46#19153789C2214yescomplemented
47#19153786C2214yes—
48#19153786C2214yescomplemented
49#19153786C2212yescomplemented
50#19153784C2212yes—
51#19153781C2212yescomplemented
52#19153778C2212yes—
53#19153778C2214yescomplemented
54#19153778C2214yescomplemented
55#19153776C2214yes—
56#19153776C2214yescomplemented
57#19153774C2214yescomplemented
58#19153772C2212yescomplemented
59C42×D181C2212yescomplemented
60C42×D181C2211yescharacteristic, complemented
61C4⋊Q8×C181C2212yescomplemented
62#126×C181C2214yescomplemented
63#125×C181C2212yescomplemented
64#124×C181C2211yescharacteristic
65#123×C181C2212yes—
66C22⋊Q8×C181C2214yescomplemented
67C23×D181C2312yes—
68#15668450—44no—
69#15668450—42no—
70#15668448—42no—
71C2×Dic362C2312yes—
72C2×Dic362C2311yescharacteristic
73C2×D724C2312yes—
74C2×D724C2314yes—
75C2×D724C2314yes—
76C2×D724C2312yes—
77C2×D724C2314yes—
78C2×D724C2312yes—
79C2×D724C2311yescharacteristic
80C2×C22⋊D181C2314yes—
81C2×C22⋊D181C2314yes—
82C2×C22⋊D181C2312yes—
83C22×Dic181C2312yes—
84C2×C4×D181—44no—
85C2×C4×D181C2312yes—
86C2×C4×D181C2311yescharacteristic
87C2×C4×D181C2314yes—
88C2×C4×D181C2314yes—
89C2×C4×D181C2312yes—
90C2×C4×D181C2314yes—
91C4⋊Dic181C2314yes—
92#15668428C2314yes—
93#15668428C2314yes—
94#15668428C2314yes—
95#15668428C2314yes—
96C22⋊Dic181C2312yes—
97#15668426C2314yes—
98#15668426C2314yes—
99#15668426C2314yes—
100#15668426C2314yes—
101#15668426C2314yes—
102#15668426C2314yes—
103#15668426C2314yes—
104#15668426C2314yes—
105#15668426C2314yes—
106C4×Dic181C2314yes—
107C4×Dic181C2312yes—
108C4×Dic181C2312yes—
109C4×Dic181C2311yescharacteristic
110C2×Q8×C181C2312yes—
111C2×Q8×C181C2311yescharacteristic
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181C2312yes—
114C4⋊C4×C181C2312yes—
115C4⋊C4×C181C2314yes—
116C4⋊C4×C181C2314yes—
117C4⋊C4×C181C2314yes—
118C4⋊C4×C181C2312yes—
119C22⋊C4×C181C2314yes—
120C22⋊C4×C181C2314yes—
121C22⋊C4×C181C2312yes—
122C42×C181C2311yescharacteristic
123C42×C181C2312yes—
124C22×D181—42no—
125C22×D181—44no—
126C22×D181C2414yes—
127C22×D181C2411yescharacteristic
128C22×D181C2412yes—
129Dic362—42no—
130Dic362—41no—
131D724—44no—
132D724—44no—
133D724—42no—
134D724—44no—
135D724—41no—
136D724—42no—
137D724—42no—
138C22⋊D181—44no—
139C22⋊D181—44no—
140C22⋊D181—42no—
141C2×Dic181—24no—
142C2×Dic181C2414yes—
143C2×Dic181C2412yes—
144C2×Dic181C2412yes—
145C2×Dic181C2411yescharacteristic
146C4×D181—28no—
147C4×D181—22no—
148C4×D181—24no—
149C4×D181—44no—
150C4×D181—42no—
151C4×D181—44no—
152C23×C181C2412yes—
153Q8×C181—42no—
154Q8×C181—41no—
155D4×C181—41no—
156C2×C4×C181—42no—
157C2×C4×C181C2412yes—
158C2×C4×C181C2414yes—
159C2×C4×C181C2412yes—
160C2×C4×C181C2411yescharacteristic
161C2×C4×C181C2414yes—
162C22×C181—22no—
163C22×C181—24no—
164C22×C181C2511yescharacteristic
165C2×D181—24no—
166C2×D181—21no—
167C2×D181—22no—
168C2×D181—24no—
169C2×D181—22no—
170C2×D181—24no—
171C2×D181—24no—
172C2×D181—48no—
173C4×C181—24no—
174C4×C181—21no—
175C4×C181—22no—
176C4×C181—24no—
177C4×C181—22no—
178Dic181—22no—
179Dic181—22no—
180Dic181—24no—
181Dic181H(5,𝔽2)12yes—
182C2×C181—42no—
183C2×C181C2×H(5,𝔽2)11yescharacteristic
184C2×C181#58512yes—
185D181—41no—
186D181—42no—
187D181—44no—
188C181#352911yescharacteristic, complemented, Sylow
189#3529—1811nomaximal, Sylow
190#563—1812no—
191#563D18111yescharacteristic, complemented
192#562—1812no—
193#562—1814no—
194H(3,𝔽4)—1811no—
195#560—1814no—
196#560—1812no—
197#532—1812no—
198#532—1811no—
199#529—1814no—
200#529—1812no—
201#525—1812no—
202#525—1814no—
203C2×#41—1812no—
204C2×#41—1811no—
205C2×#41—1811no—
206C23×C4—1812no—
207C4⋊Q8—1811no—
208C4⋊Q8C2×D18112yescomplemented
209#127—1811no—
210#127—1812no—
211#126—1814no—
212#126—1814no—
213#126C2×D18114yescomplemented
214#125—1814no—
215#125C2×D18112yescomplemented
216#124—1812no—
217#124—1812no—
218#124—1814no—
219#124—1812no—
220#124—1811no—
221#124C2×D18111yescharacteristic
222#123—1812no—
223#123—1814no—
224#123—1814no—
225#123—1814no—
226#123—1814no—
227#123—1814no—
228#123C2×D18112yes—
229C22⋊Q8—1814no—
230C22⋊Q8—1814no—
231C22⋊Q8—1814no—
232C22⋊Q8—1812no—
233C22⋊Q8C2×D18114yescomplemented
234#121—1814no—
235#121—1814no—
236#121—1812no—
237#121—1814no—
238#121—1814no—
239#121—1814no—
240#121—1814no—
241#121—1814no—
242#120—1812no—
243#120—1814no—
244#120—1812no—
245#120—1814no—
246C4×Q8—1812no—
247C4×D4—1812no—
248C4×D4—1814no—
249C4×D4—1814no—
250C4×D4—1814no—
251C4×D4—1814no—
252#117—1814no—
253#117—1812no—
254C2×C42—1812no—
255C2×C42—1811no—
256C24—1812no—
257#41—7242no—
258#41—7244no—
259#41—7242no—
260C2×Q8—1812no—
261C2×Q8—1811no—
262C2×Q8C22×D18112yes—
263C2×Q8C22×D18111yescharacteristic
264C2×D4—1814no—
265C2×D4—1814no—
266C2×D4—1812no—
267C2×D4—1812no—
268C2×D4—1814no—
269C2×D4—1814no—
270C2×D4—1812no—
271C2×D4—1814no—
272C2×D4—1811no—
273C2×D4—1812no—
274C2×D4C22×D18111yescharacteristic
275C22×C4—1812no—
276C22×C4—1811no—
277C22×C4—1812no—
278C22×C4—1814no—
279C22×C4—1814no—
280C22×C4—1812no—
281C22×C4—1814no—
282C22×C4—7244no—
283C22×C4C22×D18112yes—
284C4⋊C4—1814no—
285C4⋊C4—1814no—
286C4⋊C4—1814no—
287C4⋊C4—1814no—
288C4⋊C4—1814no—
289C4⋊C4C22×D18114yes—
290C4⋊C4C22×D18114yes—
291C4⋊C4C22×D18114yes—
292C4⋊C4C22×D18112yes—
293C4⋊C4C22×D18112yes—
294C22⋊C4—1812no—
295C22⋊C4—1814no—
296C22⋊C4—1814no—
297C22⋊C4—1814no—
298C22⋊C4—1814no—
299C22⋊C4—1814no—
300C22⋊C4—1814no—
301C22⋊C4—1814no—
302C22⋊C4—1814no—
303C22⋊C4—1814no—
304C22⋊C4C22×D18112yes—
305C22⋊C4C22×D18114yes—
306C22⋊C4C22×D18114yes—
307C42—1814no—
308C42—1812no—
309C42—1812no—
310C42—1811no—
311C42C22×D18111yescharacteristic
312C42C22×D18112yes—
313C23—1814no—
314C23—1812no—
315C23—1811no—
316C23—7242no—
317C23—7244no—
318C23C23×D18112yes—
319Q8—42no—
320Q8—41no—
321Q8—7241no—
322Q8—7242no—
323D4—41no—
324D4—7244no—
325D4—7241no—
326D4—7242no—
327D4—7242no—
328D4—7244no—
329D4—7244no—
330D4—7242no—
331D4—7242no—
332D4—7244no—
333D4—7244no—
334C2×C4—42no—
335C2×C4—3628no—
336C2×C4—3622no—
337C2×C4—3624no—
338C2×C4—3624no—
339C2×C4—1814no—
340C2×C4—1812no—
341C2×C4—1812no—
342C2×C4—1811no—
343C2×C4—7244no—
344C2×C4—7244no—
345C2×C4—7242no—
346C2×C4C23×D18112yes—
347C2×C4C23×D18111yescharacteristic
348C2×C4C23×D18114yes—
349C2×C4C23×D18112yes—
350C2×C4C23×D18114yes—
351C22—24no—
352C22—22no—
353C22—3622no—
354C22—3624no—
355C22—3624no—
356C22—3624no—
357C22—3624no—
358C22—3622no—
359C22—3621no—
360C22—7248no—
361C22C24×D18111yescharacteristic
362C4—24no—
363C4—22no—
364C4—24no—
365C4—21no—
366C4—22no—
367C4—3622no—
368C4—3624no—
369C4—3622no—
370C4—1812no—
371C2—42no—
372C2—7244no—
373C2—7241no—
374C2—7242no—
375C2H(5,𝔽2)×D18111yescharacteristic
376C2#2604211612yes—
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#563×C181C2
hypercenter#563D181
hypocenterC181#3529
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #26042028C2semidirect
2. #26042027C2semidirect
3. #26042025C2semidirect
4. #26042021C2semidirect
5. #26042020C2semidirect
6. #26042019C2semidirect
7. #26041830C2semidirect
8. #26041828C2semidirect
9. #26041814C2semidirect
10. #26041812C2semidirect
11. #26041792C2semidirect
12. #26041787C2semidirect
13. #563×C181C2semidirect
15. C2×#15668450C22semidirect
17. C22×C4×D181C22semidirect
18. #19153814C22semidirect
21. #19153811C22semidirect
22. #19153810C22semidirect
23. #19153806C22semidirect
25. #19153805C22semidirect
27. #19153804C22semidirect
30. #19153801C22semidirect
31. #19153801C22semidirect
32. #19153801C22semidirect
33. #19153799C22semidirect
34. #19153798C22semidirect
35. #19153796C22semidirect
36. #19153794C22semidirect
37. #19153794C22semidirect
38. #19153792C22semidirect
41. #19153791C22semidirect
42. #19153791C22semidirect
46. #19153789C22semidirect
48. #19153786C22semidirect
49. #19153786C22semidirect
51. #19153781C22semidirect
53. #19153778C22semidirect
54. #19153778C22semidirect
56. #19153776C22semidirect
57. #19153774C22semidirect
58. #19153772C22semidirect
59. C42×D181C22semidirect
60. C42×D181C22semidirect
61. C4⋊Q8×C181C22semidirect
62. #126×C181C22semidirect
63. #125×C181C22semidirect
66. C22⋊Q8×C181C22semidirect
188. C181#3529semidirect
191. #563D181semidirect
208. C4⋊Q8C2×D181semidirect
213. #126C2×D181semidirect
215. #125C2×D181semidirect
233. C22⋊Q8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420282
#260420272
#260420252
#260420212
#260420202
#260420192
#260418302
#260418282
#260418142
#260418122
#260417922
#260417872
#563×C1812
#3529181