Finite Group Databasebeta

← #21434144

Subgroups — #21434144

Order 7552.

454 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 123 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#16837938C211yesmaximal, characteristic, complemented
2C2×D4×Dic59C211yesmaximal, characteristic, complemented
3#16837735C211yesmaximal, characteristic, complemented
4#16837721C211yesmaximal, characteristic, complemented
5#16837704C212yesmaximal, complemented
6#16837619C212yesmaximal, complemented
7#16837536C211yesmaximal, characteristic
8SD16×Dic59C212yesmaximal, complemented
9#16837482C212yesmaximal, complemented
10#16837452C211yesmaximal, characteristic, complemented
11C59×#450C211yesmaximal, characteristic, complemented
12C23×Dic59—21no—
13C2×#11745669C2212yes—
14#14273722—21no—
15#14273718C2211yescharacteristic, complemented
16#14273716C2211yescharacteristic, complemented
17#14273711—21no—
18#14273701C2211yescharacteristic
19Q8×Dic59C2211yescharacteristic, complemented
20D4×Dic59—21no—
21D4×Dic59—21no—
22D4×Dic59C2212yes—
23D4×Dic59C2212yes—
24#14273689C2211yescharacteristic, complemented
25(C22⋊D59)⋊C4—21no—
26#14273685—21no—
27C2×C4⋊Dic59C2211yescharacteristic
28C2×C22⋊Dic59—21no—
29C2×C4×Dic59C2211yescharacteristic
30#14273660C2212yescomplemented
31#14273659C2212yescomplemented
32Dic59⋊C8C2212yescomplemented
33#14273651C2212yescomplemented
34#14273649C2212yescomplemented
35#14273648C2212yes—
36C22⋊(C59⋊C8)C2211yescharacteristic
37C8×Dic59C2212yescomplemented
38C22×D4×C59C2211yescharacteristic
39C2×SD16×C59C2212yes—
40C22⋊Q8×C59C2211yescharacteristic
41#102×C59C2212yes—
42C22⋊C8×C59C2211yescharacteristic
43C24×C59—21no—
44C2×Q8×C59C2311yescharacteristic
45C2×D4×C59—24no—
46C2×D4×C59—21no—
47C2×D4×C59C2312yes—
48C22×C4×C59C2311yescharacteristic
49C2×C22⋊D59—21no—
50C22×Dic59—21no—
51C22×Dic59—22no—
52C22×Dic59—22no—
53C22×Dic59—44no—
54C22×Dic59D411yescharacteristic
55C22×Dic59D411yescharacteristic
56C2×D236C2311yescharacteristic
57C2×C4×D59—21no—
58SD16×C59—24no—
59C2×C8×C59C2312yes—
60C4⋊C4×C59—21no—
61C4⋊C4×C59C2311yescharacteristic
62C22⋊C4×C59—21no—
63C22⋊Dic59—22no—
64C22⋊Dic59—22no—
65#11745669—24no—
66(C2×D59)⋊C4—21no—
67(C2×D59)⋊C4—21no—
68(C2×D59)⋊C4—21no—
69(C2×D59)⋊C4—21no—
70C4⋊Dic59—21no—
71C4⋊Dic59—21no—
72C4⋊Dic59—21no—
73C4⋊Dic59C2311yescharacteristic
74C4⋊Dic59C2312yes—
75Dic59⋊C4—21no—
76C4×Dic59—21no—
77C4×Dic59D411yescharacteristic
78C4×Dic59D411yescharacteristic
79C4×Dic59C2311yescharacteristic
80C4×Dic59C2311yescharacteristic
81C2×C59⋊C8C2312yes—
82C23×C59—24no—
83C23×C59—22no—
84C23×C59—42no—
85C23×C59—42no—
86C23×C59C2×D411yescharacteristic
87C22×D59—21no—
88Q8×C59—22no—
89D4×C59—24no—
90D4×C59—22no—
91D4×C59#4114yes—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59C2411yescharacteristic
97C22⋊D59—42no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—21no—
101C2×Dic59—22no—
102C2×Dic59—44no—
103C2×Dic59—42no—
104C2×Dic59—44no—
105C2×Dic59C2×D411yescharacteristic
106C2×Dic59C2×D411yescharacteristic
107C2×Dic59C2×D411yescharacteristic
108C2×Dic59C2×D411yescharacteristic
109D236—22no—
110C4×D59—42no—
111C8×C59—22no—
112C59⋊C8—22no—
113C22×C59—21no—
114C22×C59—21no—
115C22×C59—24no—
116C22×C59—48no—
117C22×C59—42no—
118C22×C59—42no—
119C22×C59#13512yes—
120C22×C59C22×D411yescharacteristic
121C2×D59—41no—
122C2×D59—41no—
123C2×D59—41no—
124C4×C59—21no—
125C4×C59—41no—
126C4×C59—41no—
127C4×C59C2×#4112yes—
128Dic59—21no—
129Dic59—21no—
130Dic59—21no—
131Dic59—41no—
132Dic59—42no—
133Dic59AGL(1,ℤ/8)12yes—
134#3105—591nomaximal, Sylow
135C2×C59—22no—
136C2×C59—44no—
137C2×C59#52511yescharacteristic
138C2×C59C2×#13511yescharacteristic
139C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
140D59—81no—
141#529—591no—
142C2×C4×D4—591no—
143#484—591no—
144#481—591no—
145#478—592no—
146#463—592no—
147#450D5911yescharacteristic, complemented
148#447—591no—
149C4×SD16—592no—
150C4×D8—592no—
151#431—591no—
152C59#310511yescharacteristic, complemented, Sylow
153C22×D4C2×D5911yescharacteristic
154C23×C4—1181no—
155C2×SD16C2×D5912yes—
156C2×D8—592no—
157#126—1181no—
158#125—591no—
159#124—591no—
160#123—1181no—
161C22⋊Q8C2×D5911yescharacteristic
162#121—591no—
163C4×Q8—591no—
164C4×D4—592no—
165C4×D4—592no—
166C4×D4—591no—
167C4×D4—1181no—
168C4×D4—1181no—
169C4×D4—1181no—
170#117—1181no—
171C2×C4⋊C4—591no—
172C2×C22⋊C4—1181no—
173C2×C42—591no—
174#107—592no—
175#106—592no—
176C4⋊C8—592no—
177#103—592no—
178#102—592no—
179#102—592no—
180#102C2×D5912yes—
181C22⋊C8—591no—
182C22⋊C8C2×D5911yescharacteristic
183C4×C8—592no—
184C24—21no—
185C2×Q8C22×D5911yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—591no—
189C2×D4—1181no—
190C2×D4C22×D5912yes—
191C22×C4—591no—
192C22×C4—591no—
193C22×C4—1181no—
194C22×C4—1182no—
195C22×C4—1182no—
196C22×C4—1181no—
197C22×C4—2364no—
198C22×C4C22×D5911yescharacteristic
199SD16—24no—
200D8—1184no—
201C2×C8—592no—
202C2×C8C22×D5912yes—
203C4⋊C4—21no—
204C4⋊C4—591no—
205C4⋊C4—592no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4—1181no—
210C4⋊C4C22×D5911yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—1181no—
213C22⋊C4—1181no—
214C22⋊C4—1181no—
215C22⋊C4—1182no—
216C22⋊C4—1182no—
217C22⋊C4—1181no—
218C42—591no—
219C42—591no—
220C42—591no—
221C42—591no—
222C42—1181no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—1181no—
228C23D4×D5911yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—1182no—
233D4—2362no—
234D4#1174568714yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—591no—
239C2×C4—591no—
240C2×C4—591no—
241C2×C4—591no—
242C2×C4—1182no—
243C2×C4—1182no—
244C2×C4—1181no—
245C2×C4—1182no—
246C2×C4—2362no—
247C2×C4—2364no—
248C2×C4—2364no—
249C2×C4—2362no—
250C2×C4D4×D5911yescharacteristic
251C2×C4C23×D5911yescharacteristic
252C8—22no—
253C8—1182no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—2361no—
261C22—2361no—
262C22—2361no—
263C22#1427374812yes—
264C22C2×D4×D5911yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—592no—
269C4—1181no—
270C4—1181no—
271C4—1181no—
272C4—2361no—
273C4—2362no—
274C4C2×#1174568712yes—
275C2—22no—
276C2—44no—
277C2—4721no—
278C2#1683790811yescharacteristic
279C2C2×#1427374811yescharacteristic
280C2AGL(1,ℤ/8)×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3105
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16837938C2semidirect
2. C2×D4×Dic59C2semidirect
3. #16837735C2semidirect
4. #16837721C2semidirect
5. #16837704C2semidirect
6. #16837619C2semidirect
8. SD16×Dic59C2semidirect
9. #16837482C2semidirect
10. #16837452C2semidirect
11. C59×#450C2semidirect
15. #14273718C22semidirect
16. #14273716C22semidirect
19. Q8×Dic59C22semidirect
24. #14273689C22semidirect
30. #14273660C22semidirect
31. #14273659C22semidirect
32. Dic59⋊C8C22semidirect
33. #14273651C22semidirect
34. #14273649C22semidirect
37. C8×Dic59C22semidirect
147. #450D59semidirect
152. C59#3105semidirect