Finite Group Databasebeta

← #21435711

Subgroups — #21435711

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×#14273754C212yesmaximal, complemented
2#16838074C211yesmaximal, characteristic, complemented
3#16838037C211yesmaximal, characteristic, complemented
4#16837778C211yesmaximal, characteristic, complemented
5#16837654C212yesmaximal, complemented
6#16837598C211yesmaximal, characteristic, complemented
7#16837546C212yesmaximal, complemented
8#16837537C212yesmaximal, complemented
9C4×#11745671C211yesmaximal, characteristic, complemented
10#16837479C211yesmaximal, characteristic, complemented
11C59×#460C211yesmaximal, characteristic, complemented
12C2×#11745689C2212yescomplemented
13C2×D4×D59—21no—
14C2×D4×D59C2212yes—
15#14273754—28no—
16C2×#11745671C2211yescharacteristic
17C2×D472C2212yes—
18C2×#11745669C2212yescomplemented
19C2×#11745669C2212yescomplemented
20C2×#11745659C2212yescomplemented
21#14273724—21no—
22#14273724C2211yescharacteristic, complemented
23#14273702—22no—
24#14273702C2212yescomplemented
25#14273698—21no—
26#14273698C2212yes—
27#14273695—22no—
28C4×D236—21no—
29C4×D236C2211yescharacteristic
30C4×D236C2211yescharacteristic
31#14273658C2211yescharacteristic, complemented
32#14273651C2211yescharacteristic, complemented
33#14273649C2212yescomplemented
34#14273647C2211yescharacteristic
35(C2×D59)⋊C8C2212yescomplemented
36C4×C59⋊C8C2211yescharacteristic, complemented
37C2×SD16×C59C2212yescomplemented
38#127×C59C2211yescharacteristic
39C4×Q8×C59C2211yescharacteristic, complemented
40C4⋊C8×C59C2211yescharacteristic, complemented
41#102×C59C2212yescomplemented
42C23×D59—22no—
43C2×Q8×C59C2311yescharacteristic
44C2×D4×C59—21no—
45C2×D4×C59—21no—
46C2×D4×C59C2312yes—
47C2×C22⋊D59—22no—
48C2×C22⋊D59—22no—
49#11745689—24no—
50#11745689—42no—
51D4×D59—24no—
52D4×D59—42no—
53D4×D59—44no—
54C2×D236—22no—
55C2×D236—21no—
56C2×D236—22no—
57C2×D236—21no—
58C2×D236D411yescharacteristic
59C2×D236C2312yes—
60C2×D236C2311yescharacteristic
61C2×C4×D59—21no—
62C2×C4×D59—22no—
63C2×C4×D59C2312yes—
64SD16×C59—24no—
65C2×C8×C59C2312yes—
66C4⋊C4×C59—21no—
67C4⋊C4×C59C2311yescharacteristic
68C42×C59—21no—
69C42×C59C2311yescharacteristic
70C22⋊Dic59—22no—
71#11745671D414yescomplemented
72#11745669—24no—
73#11745669—42no—
74(C2×D59)⋊C4—21no—
75(C2×D59)⋊C4—22no—
76(C2×D59)⋊C4D412yes—
77C4⋊Dic59D411yescharacteristic
78C4⋊Dic59C2311yescharacteristic
79C2×C59⋊C8C2311yescharacteristic
80C2×C59⋊C8C2311yescharacteristic
81D472—24no—
82#11745659—24no—
83C23×C59—22no—
84C22×D59—21no—
85C22×D59—22no—
86C22×D59—44no—
87C22×D59—44no—
88C22×D59—82no—
89C22×D59C2×D412yes—
90Q8×C59—21no—
91Q8×C59C2×D412yescomplemented
92D4×C59—22no—
93D4×C59—22no—
94D4×C59—42no—
95D4×C59—81no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59C2×D411yescharacteristic
99C2×C4×C59C2×D411yescharacteristic
100C2×C4×C59C2411yescharacteristic
101C22⋊D59—44no—
102C22⋊D59—44no—
103C2×Dic59—21no—
104C2×Dic59C2×D412yes—
105D236—22no—
106D236—22no—
107D236—21no—
108D236—22no—
109D236—44no—
110D236—42no—
111D236—81no—
112D236—82no—
113D236C2×D412yes—
114C4×D59—22no—
115C4×D59—22no—
116C4×D59—44no—
117C4×D59—42no—
118C8×C59—22no—
119C59⋊C8—21no—
120C59⋊C8C2×D412yescomplemented
121C22×C59—42no—
122C22×C59—42no—
123C22×C59—42no—
124C22×C59C22×D411yescharacteristic
125C2×D59—22no—
126C2×D59—22no—
127C2×D59—22no—
128C2×D59—22no—
129C2×D59—41no—
130C2×D59—41no—
131C2×D59—42no—
132C2×D59—42no—
133C2×D59—42no—
134C2×D59—88no—
135C4×C59—22no—
136C4×C59—21no—
137C4×C59—41no—
138C4×C59AGL(1,ℤ/8)12yescomplemented
139C4×C59C22×D411yescharacteristic
140C4×C59H(5,𝔽2)11yescharacteristic
141Dic59—22no—
142Dic59—41no—
143#3274—591nomaximal, Sylow
144C2×C59—82no—
145C2×C59D4211yescharacteristic
146C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C59#57611yescharacteristic
148D59—42no—
149D59—42no—
150D59—82no—
151C2×AGL(1,ℤ/8)—592no—
152#550—591no—
153D42—591no—
154#493—591no—
155#469—592no—
156#460D5911yescharacteristic, complemented
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—591no—
168C2×SD16C2×D5912yescomplemented
169C2×D8—592no—
170C2×D8—592no—
171C2×D8—592no—
172C2×M16—592no—
173#127—591no—
174#127—1181no—
175#127C2×D5911yescharacteristic
176#121—592no—
177#121—592no—
178#121—1182no—
179#121—1181no—
180#120—1182no—
181C4×Q8C2×D5911yescharacteristic, complemented
182C4×D4—591no—
183C4×D4—591no—
184C4×D4—1181no—
185#106—591no—
186C4⋊C8C2×D5911yescharacteristic, complemented
187#103—591no—
188#102—592no—
189#102—591no—
190#102C2×D5912yescomplemented
191C22⋊C8—592no—
192C4×C8—591no—
193C24—1182no—
194#41—1184no—
195#41—2362no—
196C2×Q8C22×D5911yescharacteristic
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—591no—
200C2×D4—591no—
201C2×D4—592no—
202C2×D4—1182no—
203C2×D4—1182no—
204C2×D4—1182no—
205C2×D4—1184no—
206C2×D4—1182no—
207C2×D4—1181no—
208C2×D4—1181no—
209C2×D4—2362no—
210C2×D4—2364no—
211C2×D4C22×D5912yes—
212C22×C4—592no—
213C22×C4—1182no—
214C22×C4—1181no—
215SD16—24no—
216SD16—594no—
217D8—1184no—
218D8—1184no—
219D8—2362no—
220M16—1184no—
221C2×C8—591no—
222C2×C8—591no—
223C2×C8C22×D5912yes—
224C4⋊C4—21no—
225C4⋊C4—591no—
226C4⋊C4—591no—
227C4⋊C4C22×D5911yescharacteristic
228C22⋊C4—592no—
229C22⋊C4—1182no—
230C22⋊C4—1182no—
231C22⋊C4—1181no—
232C42—21no—
233C42C22×D5911yescharacteristic
234C23—22no—
235C23—592no—
236C23—1181no—
237C23—1182no—
238C23—4722no—
239C23—2364no—
240C23—2364no—
241Q8—21no—
242Q8D4×D5912yescomplemented
243D4—22no—
244D4—22no—
245D4—42no—
246D4—81no—
247D4—592no—
248D4—1182no—
249D4—1182no—
250D4—1181no—
251D4—1182no—
252D4—4721no—
253D4—4722no—
254D4—2364no—
255D4—2364no—
256D4—2364no—
257D4—2362no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—592no—
261C2×C4—1182no—
262C2×C4—1182no—
263C2×C4—1181no—
264C2×C4—2364no—
265C2×C4—2362no—
266C2×C4D4×D5911yescharacteristic
267C2×C4D4×D5911yescharacteristic
268C2×C4C23×D5911yescharacteristic
269C8—22no—
270C8—592no—
271C8—1181no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—1182no—
276C22—1182no—
277C22—1182no—
278C22—1182no—
279C22—4728no—
280C22—2362no—
281C22—2362no—
282C22—2362no—
283C22—2361no—
284C22—2361no—
285C22C2×D4×D5911yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—1182no—
290C4—2361no—
291C4#1427375412yescomplemented
292C4C2×D4×D5911yescharacteristic
293C4#1427377311yescharacteristic
294C2—82no—
295C2—4722no—
296C2—2362no—
297C2—2362no—
298C2#1683803611yescharacteristic
299C2C2×#1427375411yescharacteristic
300C2#1683820611yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#460C2
socleC22×C59C22×D4
hypercenter#460D59
hypocenterC59#3274
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#14273754C2semidirect
2. #16838074C2semidirect
3. #16838037C2semidirect
4. #16837778C2semidirect
5. #16837654C2semidirect
6. #16837598C2semidirect
7. #16837546C2semidirect
8. #16837537C2semidirect
9. C4×#11745671C2semidirect
10. #16837479C2semidirect
11. C59×#460C2semidirect
12. C2×#11745689C22semidirect
18. C2×#11745669C22semidirect
19. C2×#11745669C22semidirect
20. C2×#11745659C22semidirect
22. #14273724C22semidirect
24. #14273702C22semidirect
31. #14273658C22semidirect
32. #14273651C22semidirect
33. #14273649C22semidirect
35. (C2×D59)⋊C8C22semidirect
36. C4×C59⋊C8C22semidirect
37. C2×SD16×C59C22semidirect
39. C4×Q8×C59C22semidirect
40. C4⋊C8×C59C22semidirect
41. #102×C59C22semidirect
71. #11745671D4semidirect
91. Q8×C59C2×D4semidirect
120. C59⋊C8C2×D4semidirect
138. C4×C59AGL(1,ℤ/8)semidirect
156. #460D59semidirect
162. C59#3274semidirect
168. C2×SD16C2×D59semidirect
181. C4×Q8C2×D59semidirect
186. C4⋊C8C2×D59semidirect
190. #102C2×D59semidirect
242. Q8D4×D59semidirect
291. C4#14273754semidirect