Finite Group Databasebeta

← #21435715

Subgroups — #21435715

Order 7552.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#14273751C212yesmaximal, complemented
2#16838072C211yesmaximal, characteristic, complemented
3#16838037C211yesmaximal, characteristic, complemented
4#16837652C212yesmaximal, complemented
5#16837599C211yesmaximal, characteristic, complemented
6#16837594C211yesmaximal, characteristic, complemented
7#16837544C212yesmaximal, complemented
8#16837534C212yesmaximal, complemented
9C4×#11745660C211yesmaximal, characteristic, complemented
10#16837481C211yesmaximal, characteristic, complemented
11C59×#493C211yesmaximal, characteristic, complemented
12C2×#11745687C2212yescomplemented
13C2×D4×D59—21no—
14C2×D4×D59C2212yes—
15#14273751—28no—
16C2×#11745660C2211yescharacteristic
17C2×#11745670C2212yescomplemented
18C2×#11745669C2212yes—
19C2×#11745659C2212yescomplemented
20#14273723—21no—
21#14273699—22no—
22#14273698—21no—
23#14273698C2212yes—
24#14273698C2212yescomplemented
25#14273695—22no—
26C4×Dic118C2211yescharacteristic, complemented
27C4×D236C2211yescharacteristic
28C4×D236C2211yescharacteristic
29D4×Dic59—21no—
30#14273659C2211yescharacteristic, complemented
31C4⋊(C59⋊C8)C2211yescharacteristic, complemented
32#14273652C2211yescharacteristic, complemented
33#14273649C2211yescharacteristic
34#14273648C2212yescomplemented
35#14273648C2212yescomplemented
36(C2×D59)⋊C8C2212yescomplemented
37C2×D8×C59C2212yescomplemented
38C2×D8×C59C2212yescomplemented
39#127×C59C2211yescharacteristic
40#127×C59C2211yescharacteristic, complemented
41C4×C8×C59C2211yescharacteristic, complemented
42C23×D59—22no—
43C2×D4×C59—21no—
44C2×D4×C59—21no—
45C2×D4×C59—21no—
46C2×D4×C59—21no—
47C2×D4×C59C2312yes—
48C2×D4×C59C2312yes—
49C2×C22⋊D59—22no—
50C2×C22⋊D59—22no—
51C2×C22⋊D59—22no—
52C2×C22⋊D59—22no—
53C22×Dic59—22no—
54#11745687—24no—
55#11745687—42no—
56D4×D59—24no—
57D4×D59—42no—
58D4×D59—44no—
59C2×D236D411yescharacteristic
60C2×D236C2311yescharacteristic
61C2×C4×D59—21no—
62C2×C4×D59C2312yes—
63C2×Dic118C2311yescharacteristic
64D8×C59—24no—
65D8×C59—42no—
66C2×C8×C59C2311yescharacteristic
67C2×C8×C59C2311yescharacteristic
68C42×C59C2311yescharacteristic
69C22⋊Dic59—22no—
70C22⋊Dic59—22no—
71#11745670—24no—
72#11745669—24no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4D412yes—
75C4⋊Dic59D411yescharacteristic
76C4⋊Dic59C2311yescharacteristic
77C4⋊Dic59C2311yescharacteristic
78Dic59⋊C4—21no—
79C4×Dic59—21no—
80C2×C59⋊C8C2312yes—
81#11745660D414yescomplemented
82#11745659—24no—
83C23×C59—22no—
84C23×C59—22no—
85C22×D59—21no—
86C22×D59—44no—
87C22×D59—44no—
88C22×D59—82no—
89C22×D59C2×D412yes—
90D4×C59—22no—
91D4×C59—22no—
92D4×C59—22no—
93D4×C59—22no—
94D4×C59—42no—
95D4×C59—42no—
96D4×C59—81no—
97D4×C59—81no—
98C2×C4×C59C2×D411yescharacteristic
99C2×C4×C59C2×D411yescharacteristic
100C2×C4×C59C2411yescharacteristic
101C22⋊D59—44no—
102C22⋊D59—44no—
103C22⋊D59—44no—
104C22⋊D59—82no—
105C2×Dic59—21no—
106C2×Dic59—21no—
107C2×Dic59—21no—
108C2×Dic59—44no—
109C2×Dic59C2×D412yes—
110D236—22no—
111D236—21no—
112D236C2×D412yes—
113C4×D59—22no—
114C4×D59—22no—
115C4×D59—42no—
116Dic118—21no—
117Dic118C2×D412yescomplemented
118C8×C59—21no—
119C8×C59C2×D412yescomplemented
120C59⋊C8—22no—
121C22×C59—42no—
122C22×C59—42no—
123C22×C59—42no—
124C22×C59—42no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59C22×D411yescharacteristic
128C2×D59—22no—
129C2×D59—22no—
130C2×D59—22no—
131C2×D59—22no—
132C2×D59—41no—
133C2×D59—41no—
134C2×D59—88no—
135C4×C59—21no—
136C4×C59AGL(1,ℤ/8)12yescomplemented
137C4×C59C22×D411yescharacteristic
138C4×C59H(5,𝔽2)11yescharacteristic
139Dic59—22no—
140Dic59—22no—
141Dic59—41no—
142Dic59—41no—
143#3274—591nomaximal, Sylow
144C2×C59—82no—
145C2×C59—82no—
146C2×C59D4211yescharacteristic
147C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C59#57611yescharacteristic
149D59—42no—
150D59—42no—
151C2×AGL(1,ℤ/8)—592no—
152#550—591no—
153D42—591no—
154#493D5911yescharacteristic, complemented
155#469—592no—
156#460—591no—
157#459—591no—
158#449—592no—
159#447—592no—
160C4×SD16—591no—
161#436—591no—
162C59#327411yescharacteristic, complemented, Sylow
163C2×#41—592no—
164C22×D4—592no—
165C22×D4—1181no—
166AGL(1,ℤ/8)—1188no—
167C2×SD16—592no—
168C2×SD16—591no—
169C2×D8—592no—
170C2×D8C2×D5912yescomplemented
171C2×D8C2×D5912yescomplemented
172C2×M16—592no—
173#127—1181no—
174#127C2×D5911yescharacteristic
175#127C2×D5911yescharacteristic, complemented
176#121—592no—
177#121—592no—
178#121—1182no—
179#121—1181no—
180#120—1182no—
181C4×Q8—591no—
182C4×D4—591no—
183C4×D4—591no—
184C4×D4—1181no—
185#106—591no—
186C4⋊C8—591no—
187#103—591no—
188#102—592no—
189#102—592no—
190#102—591no—
191C22⋊C8—592no—
192C4×C8C2×D5911yescharacteristic, complemented
193C24—1182no—
194#41—1184no—
195#41—2362no—
196C2×Q8—591no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—591no—
202C2×D4—591no—
203C2×D4—1182no—
204C2×D4—1184no—
205C2×D4—1182no—
206C2×D4—1182no—
207C2×D4—1182no—
208C2×D4—2362no—
209C2×D4—2364no—
210C2×D4C22×D5912yes—
211C2×D4C22×D5912yes—
212C22×C4—592no—
213C22×C4—1182no—
214C22×C4—1181no—
215SD16—594no—
216SD16—1184no—
217D8—24no—
218D8—42no—
219D8—1184no—
220M16—1184no—
221C2×C8—592no—
222C2×C8C22×D5911yescharacteristic
223C2×C8C22×D5911yescharacteristic
224C4⋊C4—591no—
225C4⋊C4—591no—
226C4⋊C4—591no—
227C4⋊C4—1181no—
228C22⋊C4—592no—
229C22⋊C4—1182no—
230C22⋊C4—1182no—
231C22⋊C4—1181no—
232C42—1181no—
233C42C22×D5911yescharacteristic
234C23—22no—
235C23—22no—
236C23—592no—
237C23—1181no—
238C23—4722no—
239C23—2364no—
240C23—2364no—
241Q8—592no—
242Q8—1181no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—592no—
252D4—1182no—
253D4—1181no—
254D4—4722no—
255D4—2364no—
256D4—2364no—
257D4—2364no—
258C2×C4—592no—
259C2×C4—1181no—
260C2×C4—1182no—
261C2×C4—1182no—
262C2×C4—1181no—
263C2×C4—1181no—
264C2×C4—2364no—
265C2×C4—2362no—
266C2×C4D4×D5911yescharacteristic
267C2×C4D4×D5911yescharacteristic
268C2×C4C23×D5911yescharacteristic
269C8—21no—
270C8—1182no—
271C8D4×D5912yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—1182no—
279C22—1182no—
280C22—1182no—
281C22—1182no—
282C22—4728no—
283C22—2361no—
284C22—2361no—
285C22C2×D4×D5911yescharacteristic
286C4—21no—
287C4—1182no—
288C4—1182no—
289C4—2361no—
290C4—2361no—
291C4#1427375112yescomplemented
292C4C2×D4×D5911yescharacteristic
293C4#1427377211yescharacteristic
294C2—82no—
295C2—82no—
296C2—2362no—
297C2—2362no—
298C2#1683803711yescharacteristic
299C2C2×#1427375111yescharacteristic
300C2#1683820311yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#493C2
socleC22×C59C22×D4
hypercenter#493D59
hypocenterC59#3274
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#14273751C2semidirect
2. #16838072C2semidirect
3. #16838037C2semidirect
4. #16837652C2semidirect
5. #16837599C2semidirect
6. #16837594C2semidirect
7. #16837544C2semidirect
8. #16837534C2semidirect
9. C4×#11745660C2semidirect
10. #16837481C2semidirect
11. C59×#493C2semidirect
12. C2×#11745687C22semidirect
17. C2×#11745670C22semidirect
19. C2×#11745659C22semidirect
24. #14273698C22semidirect
26. C4×Dic118C22semidirect
30. #14273659C22semidirect
31. C4⋊(C59⋊C8)C22semidirect
32. #14273652C22semidirect
34. #14273648C22semidirect
35. #14273648C22semidirect
36. (C2×D59)⋊C8C22semidirect
37. C2×D8×C59C22semidirect
38. C2×D8×C59C22semidirect
40. #127×C59C22semidirect
41. C4×C8×C59C22semidirect
81. #11745660D4semidirect
117. Dic118C2×D4semidirect
119. C8×C59C2×D4semidirect
136. C4×C59AGL(1,ℤ/8)semidirect
154. #493D59semidirect
162. C59#3274semidirect
170. C2×D8C2×D59semidirect
171. C2×D8C2×D59semidirect
175. #127C2×D59semidirect
192. C4×C8C2×D59semidirect
271. C8D4×D59semidirect
291. C4#14273751semidirect