Finite Group Databasebeta

← #21435719

Subgroups — #21435719

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
3D4×D236C211yesmaximal, characteristic, complemented
4#16837778C211yesmaximal, characteristic, complemented
5#16837654C212yesmaximal, complemented
6#16837603C211yesmaximal, characteristic, complemented
7#16837548C212yesmaximal, complemented
8#16837534C212yesmaximal, complemented
9C4×#11745670C211yesmaximal, characteristic, complemented
10#16837479C211yesmaximal, characteristic, complemented
11C59×#459C211yesmaximal, characteristic, complemented
12C2×#11745687C2212yescomplemented
13C22×D236—21no—
14C2×D4×D59C2212yes—
15#14273751—28no—
16C2×#11745660C2212yescomplemented
17C2×#11745670C2211yescharacteristic
18C2×#11745669C2212yescomplemented
19C2×#11745669C2212yescomplemented
20C2×#11745659C2212yescomplemented
21#14273724C2211yescharacteristic
22#14273723—21no—
23#14273702C2212yes—
24#14273701—21no—
25#14273699—22no—
26#14273698C2212yescomplemented
27#14273697—22no—
28C4×Dic118C2211yescharacteristic, complemented
29C4×D236C2211yescharacteristic
30D4×Dic59—21no—
31#14273658C2211yescharacteristic, complemented
32#14273650C2211yescharacteristic, complemented
33#14273649C2212yescomplemented
34#14273648C2211yescharacteristic
35(C2×D59)⋊C8C2212yescomplemented
36C4×C59⋊C8C2211yescharacteristic, complemented
37C2×D8×C59C2212yes—
38#127×C59C2211yescharacteristic, complemented
39C4×D4×C59C2211yescharacteristic
40C4⋊C8×C59C2211yescharacteristic, complemented
41#102×C59C2212yescomplemented
42C23×D59—22no—
43C2×D4×C59—21no—
44C2×D4×C59—21no—
45C2×D4×C59C2312yes—
46C2×D4×C59C2311yescharacteristic
47C22×C4×C59—21no—
48C2×C22⋊D59—22no—
49C2×C22⋊D59—22no—
50C2×C22⋊D59—22no—
51C22×Dic59—22no—
52#11745687—24no—
53#11745687—42no—
54D4×D59—24no—
55D4×D59—42no—
56C2×D236—22no—
57C2×D236—21no—
58C2×D236—21no—
59C2×D236—44no—
60C2×D236D411yescharacteristic
61C2×D236C2312yes—
62C2×C4×D59C2312yes—
63C2×Dic118C2311yescharacteristic
64D8×C59—24no—
65C2×C8×C59C2312yes—
66C4⋊C4×C59C2311yescharacteristic
67C22⋊C4×C59—21no—
68C42×C59C2311yescharacteristic
69C22⋊Dic59—22no—
70#11745670D414yescomplemented
71#11745669—24no—
72#11745669—42no—
73(C2×D59)⋊C4—22no—
74(C2×D59)⋊C4D412yes—
75C4⋊Dic59D411yescharacteristic
76C4⋊Dic59C2311yescharacteristic
77Dic59⋊C4—21no—
78C4×Dic59—21no—
79C2×C59⋊C8C2311yescharacteristic
80C2×C59⋊C8C2311yescharacteristic
81#11745660—24no—
82#11745659—24no—
83C23×C59—21no—
84C23×C59—22no—
85C22×D59—22no—
86C22×D59—44no—
87C22×D59—44no—
88C22×D59—82no—
89C22×D59C2×D412yes—
90D4×C59—22no—
91D4×C59—21no—
92D4×C59—22no—
93D4×C59—42no—
94D4×C59—81no—
95D4×C59C2×D412yes—
96C2×C4×C59—21no—
97C2×C4×C59—42no—
98C2×C4×C59C2×D411yescharacteristic
99C2×C4×C59C2×D411yescharacteristic
100C2×C4×C59C2411yescharacteristic
101C22⋊D59—44no—
102C22⋊D59—44no—
103C22⋊D59—82no—
104C2×Dic59—21no—
105C2×Dic59—21no—
106C2×Dic59—44no—
107C2×Dic59C2×D412yes—
108D236—22no—
109D236—22no—
110D236—22no—
111D236—44no—
112D236—42no—
113D236—81no—
114C4×D59—22no—
115C4×D59—22no—
116Dic118—21no—
117Dic118C2×D412yescomplemented
118C8×C59—22no—
119C59⋊C8—21no—
120C59⋊C8C2×D412yescomplemented
121C22×C59—22no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—42no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59C22×D411yescharacteristic
128C2×D59—22no—
129C2×D59—22no—
130C2×D59—22no—
131C2×D59—42no—
132C2×D59—42no—
133C2×D59—42no—
134C2×D59—88no—
135C4×C59—21no—
136C4×C59—41no—
137C4×C59AGL(1,ℤ/8)12yescomplemented
138C4×C59C22×D411yescharacteristic
139C4×C59H(5,𝔽2)11yescharacteristic
140Dic59—22no—
141Dic59—22no—
142Dic59—41no—
143#3274—591nomaximal, Sylow
144C2×C59—42no—
145C2×C59—82no—
146C2×C59D4211yescharacteristic
147C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C59#57611yescharacteristic
149D59—42no—
150D59—82no—
151C2×AGL(1,ℤ/8)—592no—
152#550—591no—
153D42—591no—
154#493—591no—
155#469—592no—
156#460—591no—
157#459D5911yescharacteristic, complemented
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×SD16—592no—
169C2×D8—592no—
170C2×D8—592no—
171C2×D8C2×D5912yes—
172C2×M16—592no—
173#127—591no—
174#127—1181no—
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—1181no—
184C4×D4C2×D5911yescharacteristic
185#106—591no—
186C4⋊C8C2×D5911yescharacteristic, complemented
187#103—591no—
188#102—591no—
189#102—592no—
190#102C2×D5912yescomplemented
191C22⋊C8—592no—
192C4×C8—591no—
193C24—1182no—
194#41—1184no—
195#41—2362no—
196C2×Q8—591no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—591no—
200C2×D4—592no—
201C2×D4—1182no—
202C2×D4—1182no—
203C2×D4—1184no—
204C2×D4—1182no—
205C2×D4—1182no—
206C2×D4—1181no—
207C2×D4—1181no—
208C2×D4—2364no—
209C2×D4—2362no—
210C2×D4C22×D5912yes—
211C2×D4C22×D5911yescharacteristic
212C22×C4—21no—
213C22×C4—592no—
214C22×C4—1182no—
215SD16—594no—
216SD16—1184no—
217D8—24no—
218D8—1184no—
219D8—2362no—
220M16—1184no—
221C2×C8—591no—
222C2×C8—591no—
223C2×C8C22×D5912yes—
224C4⋊C4—591no—
225C4⋊C4—591no—
226C4⋊C4—1181no—
227C4⋊C4C22×D5911yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—592no—
230C22⋊C4—1182no—
231C22⋊C4—1182no—
232C42—1181no—
233C42C22×D5911yescharacteristic
234C23—21no—
235C23—22no—
236C23—592no—
237C23—1182no—
238C23—4722no—
239C23—2364no—
240C23—2364no—
241Q8—592no—
242Q8—1181no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—1182no—
249D4—1182no—
250D4—1182no—
251D4—4721no—
252D4—4722no—
253D4—2364no—
254D4—2364no—
255D4—2364no—
256D4—2362no—
257D4D4×D5912yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—592no—
261C2×C4—1181no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1181no—
265C2×C4—2364no—
266C2×C4D4×D5911yescharacteristic
267C2×C4D4×D5911yescharacteristic
268C2×C4C23×D5911yescharacteristic
269C8—22no—
270C8—592no—
271C8—1181no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—1182no—
279C22—1182no—
280C22—1182no—
281C22—4728no—
282C22—2362no—
283C22—2362no—
284C22—2362no—
285C22C2×D4×D5911yescharacteristic
286C4—21no—
287C4—41no—
288C4—1182no—
289C4—1182no—
290C4—2361no—
291C4#1427375112yescomplemented
292C4C2×D4×D5911yescharacteristic
293C4#1427377311yescharacteristic
294C2—42no—
295C2—82no—
296C2—4722no—
297C2—2362no—
298C2#1683803611yescharacteristic
299C2C2×#1427375111yescharacteristic
300C2#1683820611yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#459C2
socleC22×C59C22×D4
hypercenter#459D59
hypocenterC59#3274
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#14273751C2semidirect
2. #16838072C2semidirect
3. D4×D236C2semidirect
4. #16837778C2semidirect
5. #16837654C2semidirect
6. #16837603C2semidirect
7. #16837548C2semidirect
8. #16837534C2semidirect
9. C4×#11745670C2semidirect
10. #16837479C2semidirect
11. C59×#459C2semidirect
12. C2×#11745687C22semidirect
16. C2×#11745660C22semidirect
18. C2×#11745669C22semidirect
19. C2×#11745669C22semidirect
20. C2×#11745659C22semidirect
26. #14273698C22semidirect
28. C4×Dic118C22semidirect
31. #14273658C22semidirect
32. #14273650C22semidirect
33. #14273649C22semidirect
35. (C2×D59)⋊C8C22semidirect
36. C4×C59⋊C8C22semidirect
38. #127×C59C22semidirect
40. C4⋊C8×C59C22semidirect
41. #102×C59C22semidirect
70. #11745670D4semidirect
117. Dic118C2×D4semidirect
120. C59⋊C8C2×D4semidirect
137. C4×C59AGL(1,ℤ/8)semidirect
157. #459D59semidirect
162. C59#3274semidirect
175. #127C2×D59semidirect
186. C4⋊C8C2×D59semidirect
190. #102C2×D59semidirect
291. C4#14273751semidirect