Finite Group Databasebeta

← #27185273

Subgroups — #27185273

Order 12928.

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#20265926C212yesmaximal, complemented
2#20265925C212yesmaximal, complemented
3#20265923C214yesmaximal, complemented
4#20265919C211yesmaximal, characteristic, complemented
5#20265918C214yesmaximal, complemented
6#20265917C212yesmaximal, complemented
7#20265728C212yesmaximal, complemented
8#20265726C211yesmaximal, characteristic, complemented
9#20265712C212yesmaximal, complemented
10#20265710C214yesmaximal, complemented
11#20265690C214yesmaximal, complemented
12#20265685C212yesmaximal, complemented
13#563×C101C211yesmaximal, characteristic, complemented
14C2×#13108075C2212yes—
15C2×#13108075C2211yescharacteristic, complemented
16C2×#13108073C2211yescharacteristic
17C22×C4×D101C2212yescomplemented
18#15692253C2211yescharacteristic, complemented
19#15692252C2212yes—
20#15692251C2211yescharacteristic
21#15692250C2214yescomplemented
22#15692249C2214yescomplemented
23#15692245C2214yescomplemented
24#15692244C2212yes—
25#15692244C2212yescomplemented
26#15692243C2212yes—
27#15692243C2211yescharacteristic, complemented
28#15692241C2214yes—
29#15692240C2214yes—
30#15692240C2214yescomplemented
31#15692240C2214yescomplemented
32#15692240C2214yescomplemented
33#15692238C2212yescomplemented
34#15692237C2214yescomplemented
35#15692235C2212yescomplemented
36#15692233C2214yescomplemented
37#15692233C2214yescomplemented
38#15692231C2214yescomplemented
39#15692230C2212yes—
40#15692230C2214yes—
41#15692230C2214yescomplemented
42#15692230C2214yescomplemented
43#15692229C2214yes—
44#15692228C2214yes—
45#15692228C2214yes—
46#15692228C2214yescomplemented
47#15692225C2214yes—
48#15692225C2214yescomplemented
49#15692225C2212yescomplemented
50#15692223C2212yes—
51#15692220C2212yescomplemented
52#15692217C2212yes—
53#15692217C2214yescomplemented
54#15692217C2214yescomplemented
55#15692215C2214yes—
56#15692215C2214yescomplemented
57#15692213C2214yescomplemented
58#15692211C2212yescomplemented
59C42×D101C2212yescomplemented
60C42×D101C2211yescharacteristic, complemented
61C4⋊Q8×C101C2212yescomplemented
62#126×C101C2214yescomplemented
63#125×C101C2212yescomplemented
64#124×C101C2211yescharacteristic
65#123×C101C2212yes—
66C22⋊Q8×C101C2214yescomplemented
67C23×D101C2312yes—
68#13108075—44no—
69#13108075—42no—
70#13108073—42no—
71C2×Dic202C2311yescharacteristic
72C2×Dic202C2312yes—
73C2×D404C2312yes—
74C2×D404C2314yes—
75C2×D404C2312yes—
76C2×D404C2311yescharacteristic
77C2×D404C2312yes—
78C2×D404C2314yes—
79C2×D404C2314yes—
80C2×C22⋊D101C2312yes—
81C2×C22⋊D101C2314yes—
82C2×C22⋊D101C2314yes—
83C22×Dic101C2312yes—
84C2×C4×D101—44no—
85C2×C4×D101C2314yes—
86C2×C4×D101C2311yescharacteristic
87C2×C4×D101C2312yes—
88C2×C4×D101C2314yes—
89C2×C4×D101C2314yes—
90C2×C4×D101C2312yes—
91C4⋊Dic101C2314yes—
92#13108053C2314yes—
93#13108053C2314yes—
94#13108053C2314yes—
95#13108053C2314yes—
96C22⋊Dic101C2312yes—
97#13108051C2314yes—
98#13108051C2314yes—
99#13108051C2314yes—
100#13108051C2314yes—
101#13108051C2314yes—
102#13108051C2314yes—
103#13108051C2314yes—
104#13108051C2314yes—
105#13108051C2314yes—
106C4×Dic101C2314yes—
107C4×Dic101C2312yes—
108C4×Dic101C2312yes—
109C4×Dic101C2311yescharacteristic
110C2×Q8×C101C2312yes—
111C2×Q8×C101C2311yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C22×C4×C101C2312yes—
114C4⋊C4×C101C2312yes—
115C4⋊C4×C101C2314yes—
116C4⋊C4×C101C2314yes—
117C4⋊C4×C101C2314yes—
118C4⋊C4×C101C2312yes—
119C22⋊C4×C101C2314yes—
120C22⋊C4×C101C2314yes—
121C22⋊C4×C101C2312yes—
122C42×C101C2311yescharacteristic
123C42×C101C2312yes—
124C23×C101C2412yes—
125C22×D101—42no—
126C22×D101—44no—
127C22×D101C2414yes—
128C22×D101C2412yes—
129C22×D101C2411yescharacteristic
130Q8×C101—41no—
131Q8×C101—42no—
132D4×C101—41no—
133C2×C4×C101—42no—
134C2×C4×C101C2412yes—
135C2×C4×C101C2414yes—
136C2×C4×C101C2412yes—
137C2×C4×C101C2411yescharacteristic
138C2×C4×C101C2414yes—
139C22⋊D101—44no—
140C22⋊D101—44no—
141C22⋊D101—42no—
142C2×Dic101—24no—
143C2×Dic101C2414yes—
144C2×Dic101C2412yes—
145C2×Dic101C2412yes—
146C2×Dic101C2411yescharacteristic
147D404—42no—
148D404—44no—
149D404—44no—
150D404—41no—
151D404—44no—
152D404—42no—
153D404—42no—
154C4×D101—28no—
155C4×D101—22no—
156C4×D101—24no—
157C4×D101—44no—
158C4×D101—44no—
159C4×D101—42no—
160Dic202—42no—
161Dic202—41no—
162C22×C101—22no—
163C22×C101—24no—
164C22×C101C2511yescharacteristic
165C2×D101—24no—
166C2×D101—22no—
167C2×D101—21no—
168C2×D101—22no—
169C2×D101—24no—
170C2×D101—24no—
171C2×D101—24no—
172C2×D101—48no—
173C4×C101—21no—
174C4×C101—24no—
175C4×C101—22no—
176C4×C101—22no—
177C4×C101—24no—
178Dic101—22no—
179Dic101—24no—
180Dic101—22no—
181Dic101H(5,𝔽2)12yes—
182C2×C101—42no—
183C2×C101C2×H(5,𝔽2)11yescharacteristic
184C2×C101#58512yes—
185D101—42no—
186D101—41no—
187D101—44no—
188#3529—1011nomaximal, Sylow
189C101#352911yescharacteristic, complemented, Sylow
190#563—1012no—
191#563D10111yescharacteristic, complemented
192#562—1012no—
193#562—1014no—
194H(3,𝔽4)—1011no—
195#560—1014no—
196#560—1012no—
197#532—1012no—
198#532—1011no—
199#529—1014no—
200#529—1012no—
201#525—1012no—
202#525—1014no—
203C2×#41—1012no—
204C2×#41—1011no—
205C2×#41—1011no—
206C23×C4—1012no—
207C4⋊Q8—1011no—
208C4⋊Q8C2×D10112yescomplemented
209#127—1011no—
210#127—1012no—
211#126—1014no—
212#126—1014no—
213#126C2×D10114yescomplemented
214#125—1014no—
215#125C2×D10112yescomplemented
216#124—1012no—
217#124—1012no—
218#124—1014no—
219#124—1012no—
220#124—1011no—
221#124C2×D10111yescharacteristic
222#123—1012no—
223#123—1014no—
224#123—1014no—
225#123—1014no—
226#123—1014no—
227#123—1014no—
228#123C2×D10112yes—
229C22⋊Q8—1014no—
230C22⋊Q8—1014no—
231C22⋊Q8—1014no—
232C22⋊Q8—1012no—
233C22⋊Q8C2×D10114yescomplemented
234#121—1014no—
235#121—1014no—
236#121—1012no—
237#121—1014no—
238#121—1014no—
239#121—1014no—
240#121—1014no—
241#121—1014no—
242#120—1012no—
243#120—1014no—
244#120—1012no—
245#120—1014no—
246C4×Q8—1012no—
247C4×D4—1012no—
248C4×D4—1014no—
249C4×D4—1014no—
250C4×D4—1014no—
251C4×D4—1014no—
252#117—1014no—
253#117—1012no—
254C2×C42—1012no—
255C2×C42—1011no—
256C24—1012no—
257#41—4042no—
258#41—4044no—
259#41—4042no—
260C2×Q8—1012no—
261C2×Q8—1011no—
262C2×Q8C22×D10112yes—
263C2×Q8C22×D10111yescharacteristic
264C2×D4—1014no—
265C2×D4—1014no—
266C2×D4—1012no—
267C2×D4—1012no—
268C2×D4—1014no—
269C2×D4—1014no—
270C2×D4—1012no—
271C2×D4—1014no—
272C2×D4—1011no—
273C2×D4—1012no—
274C2×D4C22×D10111yescharacteristic
275C22×C4—1012no—
276C22×C4—1014no—
277C22×C4—1012no—
278C22×C4—1014no—
279C22×C4—1011no—
280C22×C4—1012no—
281C22×C4—1014no—
282C22×C4—4044no—
283C22×C4C22×D10112yes—
284C4⋊C4—1014no—
285C4⋊C4—1014no—
286C4⋊C4—1014no—
287C4⋊C4—1014no—
288C4⋊C4—1014no—
289C4⋊C4C22×D10114yes—
290C4⋊C4C22×D10114yes—
291C4⋊C4C22×D10114yes—
292C4⋊C4C22×D10112yes—
293C4⋊C4C22×D10112yes—
294C22⋊C4—1012no—
295C22⋊C4—1014no—
296C22⋊C4—1014no—
297C22⋊C4—1014no—
298C22⋊C4—1014no—
299C22⋊C4—1014no—
300C22⋊C4—1014no—
301C22⋊C4—1014no—
302C22⋊C4—1014no—
303C22⋊C4—1014no—
304C22⋊C4C22×D10112yes—
305C22⋊C4C22×D10114yes—
306C22⋊C4C22×D10114yes—
307C42—1014no—
308C42—1012no—
309C42—1012no—
310C42—1011no—
311C42C22×D10111yescharacteristic
312C42C22×D10112yes—
313C23—1014no—
314C23—1012no—
315C23—1011no—
316C23—4042no—
317C23—4044no—
318C23C23×D10112yes—
319Q8—42no—
320Q8—41no—
321Q8—4041no—
322Q8—4042no—
323D4—41no—
324D4—4044no—
325D4—4041no—
326D4—4042no—
327D4—4042no—
328D4—4044no—
329D4—4044no—
330D4—4042no—
331D4—4042no—
332D4—4044no—
333D4—4044no—
334C2×C4—42no—
335C2×C4—1011no—
336C2×C4—1014no—
337C2×C4—1012no—
338C2×C4—1012no—
339C2×C4—4044no—
340C2×C4—4044no—
341C2×C4—4042no—
342C2×C4—2028no—
343C2×C4—2022no—
344C2×C4—2024no—
345C2×C4—2024no—
346C2×C4C23×D10112yes—
347C2×C4C23×D10111yescharacteristic
348C2×C4C23×D10114yes—
349C2×C4C23×D10112yes—
350C2×C4C23×D10114yes—
351C22—22no—
352C22—24no—
353C22—4048no—
354C22—2024no—
355C22—2022no—
356C22—2021no—
357C22—2022no—
358C22—2024no—
359C22—2024no—
360C22—2024no—
361C22C24×D10111yescharacteristic
362C4—24no—
363C4—21no—
364C4—22no—
365C4—24no—
366C4—22no—
367C4—1012no—
368C4—2022no—
369C4—2022no—
370C4—2024no—
371C2—42no—
372C2—4044no—
373C2—4042no—
374C2—4041no—
375C2H(5,𝔽2)×D10111yescharacteristic
376C2#2026601412yes—
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#563×C101C2
hypercenter#563D101
hypocenterC101#3529
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20265926C2semidirect
2. #20265925C2semidirect
3. #20265923C2semidirect
4. #20265919C2semidirect
5. #20265918C2semidirect
6. #20265917C2semidirect
7. #20265728C2semidirect
8. #20265726C2semidirect
9. #20265712C2semidirect
10. #20265710C2semidirect
11. #20265690C2semidirect
12. #20265685C2semidirect
13. #563×C101C2semidirect
15. C2×#13108075C22semidirect
17. C22×C4×D101C22semidirect
18. #15692253C22semidirect
21. #15692250C22semidirect
22. #15692249C22semidirect
23. #15692245C22semidirect
25. #15692244C22semidirect
27. #15692243C22semidirect
30. #15692240C22semidirect
31. #15692240C22semidirect
32. #15692240C22semidirect
33. #15692238C22semidirect
34. #15692237C22semidirect
35. #15692235C22semidirect
36. #15692233C22semidirect
37. #15692233C22semidirect
38. #15692231C22semidirect
41. #15692230C22semidirect
42. #15692230C22semidirect
46. #15692228C22semidirect
48. #15692225C22semidirect
49. #15692225C22semidirect
51. #15692220C22semidirect
53. #15692217C22semidirect
54. #15692217C22semidirect
56. #15692215C22semidirect
57. #15692213C22semidirect
58. #15692211C22semidirect
59. C42×D101C22semidirect
60. C42×D101C22semidirect
61. C4⋊Q8×C101C22semidirect
62. #126×C101C22semidirect
63. #125×C101C22semidirect
66. C22⋊Q8×C101C22semidirect
189. C101#3529semidirect
191. #563D101semidirect
208. C4⋊Q8C2×D101semidirect
213. #126C2×D101semidirect
215. #125C2×D101semidirect
233. C22⋊Q8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659262
#202659252
#202659232
#202659192
#202659182
#202659172
#202657282
#202657262
#202657122
#202657102
#202656902
#202656852
#563×C1012
#3529101