Finite Group Databasebeta

← #21434237

Subgroups — #21434237

Order 7552.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14273689C211yesmaximal, characteristic, complemented
3#16837715C211yesmaximal, characteristic, complemented
4#16837701C212yesmaximal, complemented
5#16837620C212yesmaximal, complemented
6#16837552C211yesmaximal, characteristic, complemented
7#16837538C211yesmaximal, characteristic
8#16837513C212yesmaximal, complemented
9#16837505C212yesmaximal, complemented
10#16837456C211yesmaximal, characteristic, complemented
11C59×#483C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C4×D59—21no—
14C2×#11745671C2212yes—
15C2×D472C2212yes—
16#14273722—21no—
17#14273718C2211yescharacteristic, complemented
18#14273716C2211yescharacteristic, complemented
19#14273711—21no—
20#14273701C2211yescharacteristic
21Q8×Dic59C2211yescharacteristic, complemented
22#14273689—21no—
23#14273689—21no—
24#14273689C2212yes—
25#14273689C2212yes—
26#14273689C2211yescharacteristic, complemented
27(C22⋊D59)⋊C4—21no—
28#14273685—21no—
29C2×(C2×D59)⋊C4—21no—
30C2×C4×Dic59C2211yescharacteristic
31#14273660C2212yescomplemented
32Dic59⋊C8C2212yescomplemented
33#14273649C2212yes—
34#14273647C2212yes—
35#14273647C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37#14273637C2212yescomplemented
38C22⋊Q8×C59C2211yescharacteristic
39C2×C4⋊C4×C59C2211yescharacteristic
40#106×C59C2212yescomplemented
41#103×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C23×D59—21no—
44C2×Q8×C59C2311yescharacteristic
45C22×C4×C59—21no—
46C22×C4×C59C2311yescharacteristic
47C2×C22⋊D59—21no—
48C22×Dic59D411yescharacteristic
49C22×Dic59D411yescharacteristic
50C2×D236—21no—
51C2×D236—24no—
52C2×D236C2311yescharacteristic
53C2×D236C2312yes—
54C2×C4×D59—21no—
55C2×C4×D59—22no—
56C2×C4×D59—22no—
57C2×C4×D59—44no—
58C2×C8×C59C2312yes—
59C4⋊C4×C59—21no—
60C4⋊C4×C59—21no—
61C4⋊C4×C59C2312yes—
62C4⋊C4×C59C2311yescharacteristic
63C22⋊C4×C59—21no—
64#11745671—24no—
65(C2×D59)⋊C4—21no—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—21no—
69(C2×D59)⋊C4—21no—
70(C2×D59)⋊C4—21no—
71C4⋊Dic59—21no—
72C4⋊Dic59—21no—
73C4⋊Dic59C2311yescharacteristic
74Dic59⋊C4—21no—
75C4×Dic59—21no—
76C4×Dic59D411yescharacteristic
77C4×Dic59D411yescharacteristic
78C4×Dic59C2311yescharacteristic
79C4×Dic59C2311yescharacteristic
80C2×C59⋊C8C2312yes—
81D472—24no—
82C23×C59C2×D411yescharacteristic
83C22×D59—24no—
84C22×D59—22no—
85C22×D59—21no—
86C22×D59—42no—
87C22×D59—42no—
88Q8×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—22no—
93C2×C4×C59—42no—
94C2×C4×C59C2×D411yescharacteristic
95C2×C4×C59C2411yescharacteristic
96C22⋊D59—42no—
97C2×Dic59—22no—
98C2×Dic59—21no—
99C2×Dic59—41no—
100C2×Dic59C2×D411yescharacteristic
101C2×Dic59C2×D411yescharacteristic
102C2×Dic59C2×D411yescharacteristic
103C2×Dic59C2×D411yescharacteristic
104D236—24no—
105D236—22no—
106D236—22no—
107D236#4114yes—
108C4×D59—42no—
109C4×D59—44no—
110C4×D59—44no—
111C8×C59—22no—
112C59⋊C8—22no—
113C22×C59—21no—
114C22×C59—21no—
115C22×C59AGL(1,ℤ/8)12yes—
116C22×C59C22×D411yescharacteristic
117C2×D59—24no—
118C2×D59—48no—
119C2×D59—42no—
120C2×D59—42no—
121C2×D59—41no—
122C2×D59—41no—
123C2×D59—41no—
124C4×C59—21no—
125C4×C59—41no—
126C4×C59—42no—
127C4×C59—41no—
128C4×C59C2×#4112yes—
129Dic59—21no—
130Dic59—21no—
131Dic59—21no—
132Dic59—21no—
133Dic59—41no—
134#3114—591nomaximal, Sylow
135C2×C59—22no—
136C2×C59#52511yescharacteristic
137C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C59#57511yescharacteristic
139D59—44no—
140D59—81no—
141#529—591no—
142C2×C4×D4—591no—
143#483D5911yescharacteristic, complemented
144#480—591no—
145#478—592no—
146#463—592no—
147#450—591no—
148#447—591no—
149#442—592no—
150#440—592no—
151#432—591no—
152C59#311411yescharacteristic, complemented, Sylow
153C22×D4—591no—
154C23×C4—1181no—
155C2×SD16—592no—
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⋊C4C2×D5911yescharacteristic
172C2×C22⋊C4—1181no—
173C2×C42—591no—
174#107—592no—
175#106C2×D5912yescomplemented
176C4⋊C8—592no—
177#103C2×D5912yescomplemented
178#102—592no—
179#102—592no—
180#102—592no—
181C22⋊C8—591no—
182C22⋊C8C2×D5911yescharacteristic
183#97—592no—
184C24—1181no—
185C2×Q8C22×D5911yescharacteristic
186C2×D4—592no—
187C2×D4—591no—
188C2×D4—1184no—
189C2×D4—1181no—
190C2×D4—1181no—
191C22×C4—21no—
192C22×C4—591no—
193C22×C4—591no—
194C22×C4—1181no—
195C22×C4—1182no—
196C22×C4—1182no—
197C22×C4—2364no—
198C22×C4C22×D5911yescharacteristic
199SD16—1184no—
200D8—1184no—
201C2×C8—592no—
202C2×C8C22×D5912yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—591no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4C22×D5911yescharacteristic
210C4⋊C4C22×D5912yes—
211C22⋊C4—21no—
212C22⋊C4—1182no—
213C22⋊C4—1182no—
214C22⋊C4—1181no—
215C22⋊C4—1181no—
216C22⋊C4—1181no—
217C22⋊C4—1181no—
218C42—591no—
219C42—591no—
220C42—591no—
221C42—591no—
222C42—1181no—
223C23—1184no—
224C23—1182no—
225C23—1181no—
226C23—2362no—
227C23—2362no—
228C23D4×D5911yescharacteristic
229Q8—22no—
230D4—594no—
231D4—1184no—
232D4—1182no—
233D4—1182no—
234D4—2362no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—591no—
241C2×C4—591no—
242C2×C4—591no—
243C2×C4—591no—
244C2×C4—1182no—
245C2×C4—1181no—
246C2×C4—2362no—
247C2×C4—2364no—
248C2×C4—2364no—
249C2×C4—2361no—
250C2×C4D4×D5911yescharacteristic
251C2×C4C23×D5911yescharacteristic
252C8—22no—
253C8—1182no—
254C22—21no—
255C22—21no—
256C22—1184no—
257C22—2368no—
258C22—2361no—
259C22—2361no—
260C22—2361no—
261C22—2362no—
262C22—2362no—
263C22#1427375412yes—
264C22C2×D4×D5911yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—1181no—
270C4—1181no—
271C4—1181no—
272C4—1181no—
273C4—2361no—
274C4#41×D5912yes—
275C2—22no—
276C2—4721no—
277C2—2364no—
278C2#1683791011yescharacteristic
279C2C2×#1427375411yescharacteristic
280C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#483C2
socleC22×C59C22×D4
hypercenter#483D59
hypocenterC59#3114
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #16837938C2semidirect
2. C2×#14273689C2semidirect
3. #16837715C2semidirect
4. #16837701C2semidirect
5. #16837620C2semidirect
6. #16837552C2semidirect
8. #16837513C2semidirect
9. #16837505C2semidirect
10. #16837456C2semidirect
11. C59×#483C2semidirect
17. #14273718C22semidirect
18. #14273716C22semidirect
21. Q8×Dic59C22semidirect
26. #14273689C22semidirect
31. #14273660C22semidirect
32. Dic59⋊C8C22semidirect
35. #14273647C22semidirect
37. #14273637C22semidirect
40. #106×C59C22semidirect
41. #103×C59C22semidirect
143. #483D59semidirect
152. C59#3114semidirect
175. #106C2×D59semidirect
177. #103C2×D59semidirect