Finite Group Databasebeta

← #34114277

Subgroups — #34114277

Order 20096.

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#23758022C212yesmaximal, complemented
2#23758021C212yesmaximal, complemented
3#23758019C214yesmaximal, complemented
4#23758015C211yesmaximal, characteristic, complemented
5#23758014C214yesmaximal, complemented
6#23758013C212yesmaximal, complemented
7#23757824C212yesmaximal, complemented
8#23757822C211yesmaximal, characteristic, complemented
9#23757808C212yesmaximal, complemented
10#23757806C214yesmaximal, complemented
11#23757786C214yesmaximal, complemented
12#23757781C212yesmaximal, complemented
13#563×C157C211yesmaximal, characteristic, complemented
14C2×#14578955C2212yes—
15C2×#14578955C2211yescharacteristic, complemented
16C2×#14578953C2211yescharacteristic
17C22×C4×D157C2212yescomplemented
18#19097769C2211yescharacteristic, complemented
19#19097768C2212yes—
20#19097767C2211yescharacteristic
21#19097766C2214yescomplemented
22#19097765C2214yescomplemented
23#19097761C2214yescomplemented
24#19097760C2212yes—
25#19097760C2212yescomplemented
26#19097759C2212yes—
27#19097759C2211yescharacteristic, complemented
28#19097757C2214yes—
29#19097756C2214yes—
30#19097756C2214yescomplemented
31#19097756C2214yescomplemented
32#19097756C2214yescomplemented
33#19097754C2212yescomplemented
34#19097753C2214yescomplemented
35#19097751C2212yescomplemented
36#19097749C2214yescomplemented
37#19097749C2214yescomplemented
38#19097747C2214yescomplemented
39#19097746C2212yes—
40#19097746C2214yes—
41#19097746C2214yescomplemented
42#19097746C2214yescomplemented
43#19097745C2214yes—
44#19097744C2214yes—
45#19097744C2214yes—
46#19097744C2214yescomplemented
47#19097741C2214yes—
48#19097741C2214yescomplemented
49#19097741C2212yescomplemented
50#19097739C2212yes—
51#19097736C2212yescomplemented
52#19097733C2212yes—
53#19097733C2214yescomplemented
54#19097733C2214yescomplemented
55#19097731C2214yes—
56#19097731C2214yescomplemented
57#19097729C2214yescomplemented
58#19097727C2212yescomplemented
59C42×D157C2212yescomplemented
60C42×D157C2211yescharacteristic, complemented
61C4⋊Q8×C157C2212yescomplemented
62#126×C157C2214yescomplemented
63#125×C157C2212yescomplemented
64#124×C157C2211yescharacteristic
65#123×C157C2212yes—
66C22⋊Q8×C157C2214yescomplemented
67C23×D157C2312yes—
68#14578955—44no—
69#14578955—42no—
70#14578953—42no—
71C2×Dic314C2312yes—
72C2×Dic314C2311yescharacteristic
73C2×D628C2312yes—
74C2×D628C2314yes—
75C2×D628C2314yes—
76C2×D628C2312yes—
77C2×D628C2314yes—
78C2×D628C2312yes—
79C2×D628C2311yescharacteristic
80C2×C22⋊D157C2314yes—
81C2×C22⋊D157C2314yes—
82C2×C22⋊D157C2312yes—
83C22×Dic157C2312yes—
84C2×C4×D157—44no—
85C2×C4×D157C2312yes—
86C2×C4×D157C2311yescharacteristic
87C2×C4×D157C2314yes—
88C2×C4×D157C2314yes—
89C2×C4×D157C2312yes—
90C2×C4×D157C2314yes—
91C4⋊Dic157C2314yes—
92#14578933C2314yes—
93#14578933C2314yes—
94#14578933C2314yes—
95#14578933C2314yes—
96C22⋊Dic157C2312yes—
97#14578931C2314yes—
98#14578931C2314yes—
99#14578931C2314yes—
100#14578931C2314yes—
101#14578931C2314yes—
102#14578931C2314yes—
103#14578931C2314yes—
104#14578931C2314yes—
105#14578931C2314yes—
106C4×Dic157C2314yes—
107C4×Dic157C2312yes—
108C4×Dic157C2312yes—
109C4×Dic157C2311yescharacteristic
110C2×Q8×C157C2312yes—
111C2×Q8×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C22×C4×C157C2312yes—
114C4⋊C4×C157C2312yes—
115C4⋊C4×C157C2314yes—
116C4⋊C4×C157C2314yes—
117C4⋊C4×C157C2314yes—
118C4⋊C4×C157C2312yes—
119C22⋊C4×C157C2314yes—
120C22⋊C4×C157C2314yes—
121C22⋊C4×C157C2312yes—
122C42×C157C2311yescharacteristic
123C42×C157C2312yes—
124C22×D157—42no—
125C22×D157—44no—
126C22×D157C2411yescharacteristic
127C22×D157C2412yes—
128C22×D157C2414yes—
129Dic314—42no—
130Dic314—41no—
131D628—44no—
132D628—44no—
133D628—42no—
134D628—44no—
135D628—41no—
136D628—42no—
137D628—42no—
138C22⋊D157—44no—
139C22⋊D157—44no—
140C22⋊D157—42no—
141C2×Dic157—24no—
142C2×Dic157C2412yes—
143C2×Dic157C2412yes—
144C2×Dic157C2414yes—
145C2×Dic157C2411yescharacteristic
146C4×D157—28no—
147C4×D157—22no—
148C4×D157—24no—
149C4×D157—44no—
150C4×D157—42no—
151C4×D157—44no—
152C23×C157C2412yes—
153Q8×C157—42no—
154Q8×C157—41no—
155D4×C157—41no—
156C2×C4×C157—42no—
157C2×C4×C157C2414yes—
158C2×C4×C157C2414yes—
159C2×C4×C157C2412yes—
160C2×C4×C157C2412yes—
161C2×C4×C157C2411yescharacteristic
162C22×C157—22no—
163C22×C157—24no—
164C22×C157C2511yescharacteristic
165C2×D157—24no—
166C2×D157—21no—
167C2×D157—22no—
168C2×D157—24no—
169C2×D157—22no—
170C2×D157—24no—
171C2×D157—24no—
172C2×D157—48no—
173C4×C157—24no—
174C4×C157—21no—
175C4×C157—22no—
176C4×C157—24no—
177C4×C157—22no—
178Dic157—22no—
179Dic157—22no—
180Dic157—24no—
181Dic157H(5,𝔽2)12yes—
182C2×C157—42no—
183C2×C157C2×H(5,𝔽2)11yescharacteristic
184C2×C157#58512yes—
185D157—41no—
186D157—42no—
187D157—44no—
188C157#352911yescharacteristic, complemented, Sylow
189#3529—1571nomaximal, Sylow
190#563—1572no—
191#563D15711yescharacteristic, complemented
192#562—1572no—
193#562—1574no—
194H(3,𝔽4)—1571no—
195#560—1574no—
196#560—1572no—
197#532—1572no—
198#532—1571no—
199#529—1574no—
200#529—1572no—
201#525—1572no—
202#525—1574no—
203C2×#41—1572no—
204C2×#41—1571no—
205C2×#41—1571no—
206C23×C4—1572no—
207C4⋊Q8—1571no—
208C4⋊Q8C2×D15712yescomplemented
209#127—1571no—
210#127—1572no—
211#126—1574no—
212#126—1574no—
213#126C2×D15714yescomplemented
214#125—1574no—
215#125C2×D15712yescomplemented
216#124—1572no—
217#124—1572no—
218#124—1574no—
219#124—1572no—
220#124—1571no—
221#124C2×D15711yescharacteristic
222#123—1572no—
223#123—1574no—
224#123—1574no—
225#123—1574no—
226#123—1574no—
227#123—1574no—
228#123C2×D15712yes—
229C22⋊Q8—1574no—
230C22⋊Q8—1574no—
231C22⋊Q8—1574no—
232C22⋊Q8—1572no—
233C22⋊Q8C2×D15714yescomplemented
234#121—1574no—
235#121—1574no—
236#121—1572no—
237#121—1574no—
238#121—1574no—
239#121—1574no—
240#121—1574no—
241#121—1574no—
242#120—1572no—
243#120—1574no—
244#120—1572no—
245#120—1574no—
246C4×Q8—1572no—
247C4×D4—1572no—
248C4×D4—1574no—
249C4×D4—1574no—
250C4×D4—1574no—
251C4×D4—1574no—
252#117—1574no—
253#117—1572no—
254C2×C42—1572no—
255C2×C42—1571no—
256C24—1572no—
257#41—6282no—
258#41—6284no—
259#41—6282no—
260C2×Q8—1572no—
261C2×Q8—1571no—
262C2×Q8C22×D15712yes—
263C2×Q8C22×D15711yescharacteristic
264C2×D4—1574no—
265C2×D4—1574no—
266C2×D4—1572no—
267C2×D4—1572no—
268C2×D4—1574no—
269C2×D4—1574no—
270C2×D4—1572no—
271C2×D4—1574no—
272C2×D4—1571no—
273C2×D4—1572no—
274C2×D4C22×D15711yescharacteristic
275C22×C4—6284no—
276C22×C4—1572no—
277C22×C4—1571no—
278C22×C4—1572no—
279C22×C4—1574no—
280C22×C4—1574no—
281C22×C4—1572no—
282C22×C4—1574no—
283C22×C4C22×D15712yes—
284C4⋊C4—1574no—
285C4⋊C4—1574no—
286C4⋊C4—1574no—
287C4⋊C4—1574no—
288C4⋊C4—1574no—
289C4⋊C4C22×D15714yes—
290C4⋊C4C22×D15714yes—
291C4⋊C4C22×D15714yes—
292C4⋊C4C22×D15712yes—
293C4⋊C4C22×D15712yes—
294C22⋊C4—1572no—
295C22⋊C4—1574no—
296C22⋊C4—1574no—
297C22⋊C4—1574no—
298C22⋊C4—1574no—
299C22⋊C4—1574no—
300C22⋊C4—1574no—
301C22⋊C4—1574no—
302C22⋊C4—1574no—
303C22⋊C4—1574no—
304C22⋊C4C22×D15712yes—
305C22⋊C4C22×D15714yes—
306C22⋊C4C22×D15714yes—
307C42—1574no—
308C42—1572no—
309C42—1572no—
310C42—1571no—
311C42C22×D15711yescharacteristic
312C42C22×D15712yes—
313C23—6282no—
314C23—6284no—
315C23—1574no—
316C23—1572no—
317C23—1571no—
318C23C23×D15712yes—
319Q8—42no—
320Q8—41no—
321Q8—6281no—
322Q8—6282no—
323D4—41no—
324D4—6284no—
325D4—6281no—
326D4—6282no—
327D4—6282no—
328D4—6284no—
329D4—6284no—
330D4—6282no—
331D4—6282no—
332D4—6284no—
333D4—6284no—
334C2×C4—42no—
335C2×C4—3148no—
336C2×C4—3142no—
337C2×C4—3144no—
338C2×C4—3144no—
339C2×C4—6284no—
340C2×C4—6284no—
341C2×C4—6282no—
342C2×C4—1574no—
343C2×C4—1572no—
344C2×C4—1572no—
345C2×C4—1571no—
346C2×C4C23×D15712yes—
347C2×C4C23×D15711yescharacteristic
348C2×C4C23×D15714yes—
349C2×C4C23×D15712yes—
350C2×C4C23×D15714yes—
351C22—22no—
352C22—24no—
353C22—3144no—
354C22—3142no—
355C22—3141no—
356C22—3142no—
357C22—3144no—
358C22—3144no—
359C22—3144no—
360C22—6288no—
361C22C24×D15711yescharacteristic
362C4—24no—
363C4—21no—
364C4—22no—
365C4—24no—
366C4—22no—
367C4—3142no—
368C4—3142no—
369C4—3144no—
370C4—1572no—
371C2—42no—
372C2—6284no—
373C2—6282no—
374C2—6281no—
375C2H(5,𝔽2)×D15711yescharacteristic
376C2#2375811012yes—
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#563×C157C2
hypercenter#563D157
hypocenterC157#3529
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #23758022C2semidirect
2. #23758021C2semidirect
3. #23758019C2semidirect
4. #23758015C2semidirect
5. #23758014C2semidirect
6. #23758013C2semidirect
7. #23757824C2semidirect
8. #23757822C2semidirect
9. #23757808C2semidirect
10. #23757806C2semidirect
11. #23757786C2semidirect
12. #23757781C2semidirect
13. #563×C157C2semidirect
15. C2×#14578955C22semidirect
17. C22×C4×D157C22semidirect
18. #19097769C22semidirect
21. #19097766C22semidirect
22. #19097765C22semidirect
23. #19097761C22semidirect
25. #19097760C22semidirect
27. #19097759C22semidirect
30. #19097756C22semidirect
31. #19097756C22semidirect
32. #19097756C22semidirect
33. #19097754C22semidirect
34. #19097753C22semidirect
35. #19097751C22semidirect
36. #19097749C22semidirect
37. #19097749C22semidirect
38. #19097747C22semidirect
41. #19097746C22semidirect
42. #19097746C22semidirect
46. #19097744C22semidirect
48. #19097741C22semidirect
49. #19097741C22semidirect
51. #19097736C22semidirect
53. #19097733C22semidirect
54. #19097733C22semidirect
56. #19097731C22semidirect
57. #19097729C22semidirect
58. #19097727C22semidirect
59. C42×D157C22semidirect
60. C42×D157C22semidirect
61. C4⋊Q8×C157C22semidirect
62. #126×C157C22semidirect
63. #125×C157C22semidirect
66. C22⋊Q8×C157C22semidirect
188. C157#3529semidirect
191. #563D157semidirect
208. C4⋊Q8C2×D157semidirect
213. #126C2×D157semidirect
215. #125C2×D157semidirect
233. C22⋊Q8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580222
#237580212
#237580192
#237580152
#237580142
#237580132
#237578242
#237578222
#237578082
#237578062
#237577862
#237577812
#563×C1572
#3529157