Finite Group Databasebeta

← #21435110

Subgroups — #21435110

Order 7552.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#127×D59C211yesmaximal, characteristic, complemented
2#16837931C211yesmaximal, characteristic, complemented
3#16837792C211yesmaximal, characteristic, complemented
4#16837767C211yesmaximal, characteristic, complemented
5#16837766C211yesmaximal, characteristic, complemented
6#16837600C212yesmaximal, complemented
7#16837595C212yesmaximal, complemented
8#16837551C212yesmaximal, complemented
9#16837534C212yesmaximal, complemented
10#16837457C211yesmaximal, characteristic, complemented
11C59×#496C211yesmaximal, characteristic, complemented
12C2×D4×D59—21no—
13C2×D4×D59—21no—
14C2×D4×D59C2212yes—
15C2×#11745671C2212yescomplemented
16C2×#11745669C2212yescomplemented
17#14273724C2211yescharacteristic
18#14273723—21no—
19#14273723—21no—
20#14273723C2212yes—
21#14273720—21no—
22#14273719C2211yescharacteristic, complemented
23#14273709—21no—
24#14273703C2211yescharacteristic
25#14273698C2211yescharacteristic
26Q8×Dic59C2211yescharacteristic, complemented
27D4×Dic59C2211yescharacteristic, complemented
28(C22⋊D59)⋊C4—21no—
29#14273683—21no—
30C42×D59C2211yescharacteristic
31Dic59⋊C8C2212yescomplemented
32#14273649C2212yescomplemented
33#14273649C2212yescomplemented
34#14273648C2212yescomplemented
35#14273648C2212yescomplemented
36(C2×D59)⋊C8C2212yes—
37#14273638C2211yescharacteristic, complemented
38C2×SD16×C59C2212yescomplemented
39C2×D8×C59C2212yescomplemented
40#127×C59C2211yescharacteristic
41#124×C59C2211yescharacteristic, complemented
42#97×C59C2211yescharacteristic, complemented
43C23×D59—22no—
44C2×Q8×C59C2311yescharacteristic
45C2×D4×C59—21no—
46C2×D4×C59—21no—
47C2×D4×C59C2311yescharacteristic
48C2×D4×C59C2312yes—
49C2×C22⋊D59—21no—
50C2×C22⋊D59—22no—
51C2×C22⋊D59—22no—
52C2×C22⋊D59—22no—
53C22×Dic59—21no—
54D4×D59—42no—
55D4×D59—42no—
56D4×D59—44no—
57D4×D59—44no—
58C2×D236—21no—
59C2×D236—21no—
60C2×D236C2312yes—
61C2×C4×D59D411yescharacteristic
62C2×C4×D59D411yescharacteristic
63C2×C4×D59C2311yescharacteristic
64SD16×C59—42no—
65D8×C59—42no—
66C2×C8×C59C2312yes—
67C22⋊C4×C59—21no—
68C22⋊C4×C59—21no—
69C42×C59C2311yescharacteristic
70C22⋊Dic59—21no—
71C22⋊Dic59—21no—
72#11745671—42no—
73#11745669—42no—
74(C2×D59)⋊C4—21no—
75C4⋊Dic59—21no—
76C4⋊Dic59C2311yescharacteristic
77C4⋊Dic59C2311yescharacteristic
78Dic59⋊C4—21no—
79Dic59⋊C4—21no—
80Dic59⋊C4—21no—
81C4×Dic59—21no—
82C4×Dic59D411yescharacteristic
83C4×Dic59D411yescharacteristic
84C4×Dic59C2311yescharacteristic
85C2×C59⋊C8C2312yes—
86C23×C59—21no—
87C23×C59—22no—
88C22×D59—22no—
89C22×D59—44no—
90C22×D59—44no—
91C22×D59—44no—
92C22×D59C2×D411yescharacteristic
93Q8×C59—22no—
94D4×C59—22no—
95D4×C59—22no—
96D4×C59—22no—
97D4×C59—42no—
98D4×C59—42no—
99C2×C4×C59—21no—
100C2×C4×C59C2×D411yescharacteristic
101C2×C4×C59C2×D411yescharacteristic
102C2×C4×C59C2411yescharacteristic
103C22⋊D59—44no—
104C22⋊D59—44no—
105C22⋊D59—42no—
106C22⋊D59—44no—
107C2×Dic59—21no—
108C2×Dic59—21no—
109C2×Dic59—42no—
110C2×Dic59C2×D411yescharacteristic
111C2×Dic59C2×D411yescharacteristic
112C2×Dic59C2×D411yescharacteristic
113D236—22no—
114D236—22no—
115D236—42no—
116D236—42no—
117C4×D59—22no—
118C4×D59—22no—
119C4×D59—22no—
120C8×C59—22no—
121C59⋊C8—22no—
122C22×C59—42no—
123C22×C59—41no—
124C22×C59—41no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—41no—
128C22×C59C22×D411yescharacteristic
129C2×D59—21no—
130C2×D59—21no—
131C2×D59—42no—
132C2×D59—42no—
133C2×D59—42no—
134C2×D59—88no—
135C2×D59AGL(1,ℤ/8)12yes—
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—41no—
139C4×C59H(5,𝔽2)12yes—
140Dic59—21no—
141Dic59—21no—
142Dic59—41no—
143Dic59—41no—
144Dic59AGL(1,ℤ/8)12yes—
145#3217—591nomaximal, Sylow
146C2×C59—82no—
147C2×C59—81no—
148C2×C59#53511yescharacteristic
149C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
151D59—22no—
152D59—82no—
153C2×#127—591no—
154#529—591no—
155#496—591no—
156#496D5911yescharacteristic, complemented
157#490—591no—
158#490—591no—
159#460—592no—
160#459—592no—
161#450—592no—
162#447—592no—
163#432—591no—
164C59#321711yescharacteristic, complemented, Sylow
165C22×D4—592no—
166C22×D4—1181no—
167C22×D4—1181no—
168C2×SD16—592no—
169C2×SD16C2×D5912yescomplemented
170C2×D8—592no—
171C2×D8C2×D5912yescomplemented
172#127—592no—
173#127—591no—
174#127—1181no—
175#127—1181no—
176#127C2×D5911yescharacteristic
177#126—1181no—
178#125—591no—
179#124C2×D5911yescharacteristic, complemented
180#123—1181no—
181C22⋊Q8—591no—
182#121—591no—
183C4×Q8—591no—
184C4×D4—591no—
185C4×D4—1181no—
186#117—1181no—
187C2×C42—591no—
188C4⋊C8—592no—
189#102—592no—
190#102—592no—
191#102—592no—
192#102—592no—
193C22⋊C8—592no—
194#97—591no—
195#97C2×D5911yescharacteristic, complemented
196C24—1182no—
197C2×Q8C22×D5911yescharacteristic
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—592no—
201C2×D4—1182no—
202C2×D4—1182no—
203C2×D4—1182no—
204C2×D4—1181no—
205C2×D4—1181no—
206C2×D4—1181no—
207C2×D4—2362no—
208C2×D4—2362no—
209C2×D4—2364no—
210C2×D4—2364no—
211C2×D4C22×D5912yes—
212C2×D4C22×D5911yescharacteristic
213C22×C4—591no—
214C22×C4—591no—
215C22×C4—591no—
216C22×C4—1181no—
217SD16—42no—
218SD16—2362no—
219D8—42no—
220D8—2362no—
221C2×C8—592no—
222C2×C8C22×D5912yes—
223C4⋊C4—591no—
224C4⋊C4—591no—
225C4⋊C4—1181no—
226C4⋊C4—1181no—
227C4⋊C4—1181no—
228C4⋊C4—1181no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—1181no—
232C22⋊C4—1181no—
233C22⋊C4—1181no—
234C42—591no—
235C42—591no—
236C42—591no—
237C42—1181no—
238C42C22×D5911yescharacteristic
239C23—21no—
240C23—22no—
241C23—591no—
242C23—1182no—
243C23—2364no—
244C23—2364no—
245C23—2364no—
246Q8—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—1182no—
253D4—1182no—
254D4—2362no—
255D4—2362no—
256D4—2364no—
257D4—2364no—
258D4—2364no—
259D4—2362no—
260C2×C4—21no—
261C2×C4—591no—
262C2×C4—591no—
263C2×C4—591no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—1181no—
268C2×C4—1181no—
269C2×C4—2362no—
270C2×C4D4×D5911yescharacteristic
271C2×C4D4×D5911yescharacteristic
272C2×C4C23×D5911yescharacteristic
273C8—22no—
274C8—1182no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—592no—
282C22—1181no—
283C22—1181no—
284C22—4728no—
285C22—2362no—
286C22—2362no—
287C22—2362no—
288C22C2×D4×D5911yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—592no—
293C4—1181no—
294C4—1181no—
295C4—2361no—
296C4—2361no—
297C4#1427377212yes—
298C2—82no—
299C2—81no—
300C2—1182no—
301C2—4722no—
302C2#1683796511yescharacteristic
303C2AGL(1,ℤ/8)×D5911yescharacteristic
304C2AGL(1,ℤ/8)×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#496C2
socleC22×C59C22×D4
hypercenter#496D59
hypocenterC59#3217
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #127×D59C2semidirect
2. #16837931C2semidirect
3. #16837792C2semidirect
4. #16837767C2semidirect
5. #16837766C2semidirect
6. #16837600C2semidirect
7. #16837595C2semidirect
8. #16837551C2semidirect
9. #16837534C2semidirect
10. #16837457C2semidirect
11. C59×#496C2semidirect
15. C2×#11745671C22semidirect
16. C2×#11745669C22semidirect
22. #14273719C22semidirect
26. Q8×Dic59C22semidirect
27. D4×Dic59C22semidirect
31. Dic59⋊C8C22semidirect
32. #14273649C22semidirect
33. #14273649C22semidirect
34. #14273648C22semidirect
35. #14273648C22semidirect
37. #14273638C22semidirect
38. C2×SD16×C59C22semidirect
39. C2×D8×C59C22semidirect
41. #124×C59C22semidirect
42. #97×C59C22semidirect
156. #496D59semidirect
164. C59#3217semidirect
169. C2×SD16C2×D59semidirect
171. C2×D8C2×D59semidirect
179. #124C2×D59semidirect
195. #97C2×D59semidirect