Finite Group Databasebeta

← #21429907

Subgroups — #21429907

Order 7552.

668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14273704C211yesmaximal, characteristic, complemented
2C22×(C2×D59)⋊C4C211yesmaximal, characteristic
3#16837337C212yesmaximal, complemented
4#16837326C212yesmaximal, complemented
5#16837324C212yesmaximal, complemented
6#16837315C212yesmaximal, complemented
7#16837275C211yesmaximal, characteristic, complemented
8#16837260C211yesmaximal, characteristic, complemented
9#16837257C211yesmaximal, characteristic
10#16837257C211yesmaximal, characteristic
11C59×#385C211yesmaximal, characteristic, complemented
12C24×D59C2211yescharacteristic
13C22×Dic118C2211yescharacteristic, complemented
14C23×Dic59C2211yescharacteristic
15#14273704—24no—
16C2×C4⋊Dic59C2211yescharacteristic, complemented
17C2×Dic59⋊C4C2211yescharacteristic, complemented
18C2×Dic59⋊C4C2211yescharacteristic, complemented
19C2×Dic59⋊C4C2212yescomplemented
20C2×Dic59⋊C4C2212yescomplemented
21C2×C22⋊Dic59C2211yescharacteristic
22C2×C22⋊Dic59C2211yescharacteristic
23C2×(C2×D59)⋊C4—24no—
24C2×(C2×D59)⋊C4C2212yes—
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C2×(C2×D59)⋊C4C2212yes—
30C59⋊#95C2212yescomplemented
31C59⋊#95C2212yescomplemented
32C23×C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2212yescomplemented
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2211yescharacteristic
36#95×C59C2212yescomplemented
37C24×C59C2311yescharacteristic
38C23×D59—22no—
39C23×D59—22no—
40C23×D59—42no—
41C23×D59—42no—
42C23×D59Q814yescomplemented
43C23×D59C2312yes—
44C22×C4×C59—22no—
45C22×C4×C59—42no—
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2312yes—
48C22×C4×C59C2312yes—
49C22×C4×C59C2311yescharacteristic
50C22×Dic59—42no—
51C22×Dic59D414yes—
52C22×Dic59C2312yes—
53C22×Dic59C2311yescharacteristic
54C22×Dic59C2311yescharacteristic
55C22×Dic59C2311yescharacteristic
56C22×Dic59C2311yescharacteristic
57C2×Dic118—22no—
58C2×Dic118—42no—
59C4⋊C4×C59—42no—
60C22⋊C4×C59—22no—
61C22⋊C4×C59—22no—
62C22⋊Dic59—22no—
63C22⋊Dic59—22no—
64(C2×D59)⋊C4—24no—
65(C2×D59)⋊C4—24no—
66(C2×D59)⋊C4—42no—
67(C2×D59)⋊C4—42no—
68(C2×D59)⋊C4—42no—
69(C2×D59)⋊C4—42no—
70C4⋊Dic59—22no—
71Dic59⋊C4—22no—
72Dic59⋊C4—22no—
73Dic59⋊C4—42no—
74Dic59⋊C4—42no—
75C23×C59—22no—
76C23×C59—22no—
77C23×C59C2×D412yes—
78C23×C59C2×Q812yes—
79C23×C59#4112yes—
80C23×C59C2411yescharacteristic
81C22×D59—28no—
82C22×D59—22no—
83C22×D59—22no—
84C22×D59—22no—
85C22×D59—22no—
86C22×D59—22no—
87C22×D59—22no—
88C22×D59—48no—
89C22×D59—44no—
90C22×D59—44no—
91C22×D59—48no—
92C22×D59C2×Q814yes—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—44no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—21no—
107C2×Dic59—21no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—44no—
114C2×Dic59C2×D412yes—
115C2×Dic59C2×D412yes—
116Dic118—44no—
117C22×C59—22no—
118C22×C59—22no—
119C22×C59—24no—
120C22×C59—24no—
121C22×C59C22⋊Q814yes—
122C22×C59C22×D411yescharacteristic
123C22×C59C22×Q811yescharacteristic
124C22×C59C2×#4111yescharacteristic
125C22×C59H(5,𝔽2)12yes—
126C22×C59H(5,𝔽2)12yes—
127C2×D59—24no—
128C2×D59—24no—
129C2×D59—24no—
130C2×D59—416no—
131C2×D59—42no—
132C2×D59—42no—
133C2×D59—42no—
134C2×D59—42no—
135C4×C59—41no—
136C4×C59—41no—
137C4×C59—42no—
138C4×C59—42no—
139Dic59—22no—
140Dic59—22no—
141Dic59—41no—
142Dic59—41no—
143Dic59—42no—
144#2626—591nomaximal, Sylow
145C2×C59—24no—
146C2×C59C2×C22⋊Q811yescharacteristic
147C2×C59#53411yescharacteristic
148C2×C59#53511yescharacteristic
149C2×C59#53812yes—
150C2×C59#54312yes—
151D59—44no—
152C2×C22⋊Q8—591no—
153C22×C22⋊C4—591no—
154#399—592no—
155#396—592no—
156#396—592no—
157#393—592no—
158#385—591no—
159#385D5911yescharacteristic, complemented
160#380—591no—
161#379—591no—
162#379—591no—
163C59#262611yescharacteristic, complemented, Sylow
164C25—591no—
165C22×Q8—591no—
166C23×C4—591no—
167C23×C4C2×D5911yescharacteristic
168C22⋊Q8—1184no—
169C2×C4⋊C4—591no—
170C2×C4⋊C4—591no—
171C2×C4⋊C4—591no—
172C2×C4⋊C4—592no—
173C2×C4⋊C4—592no—
174C2×C4⋊C4C2×D5912yescomplemented
175C2×C22⋊C4—591no—
176C2×C22⋊C4—591no—
177C2×C22⋊C4—592no—
178C2×C22⋊C4—592no—
179C2×C22⋊C4—592no—
180C2×C22⋊C4—592no—
181C2×C22⋊C4—592no—
182C2×C22⋊C4—592no—
183C2×C22⋊C4—1184no—
184C2×C22⋊C4C2×D5911yescharacteristic
185C2×C22⋊C4C2×D5911yescharacteristic
186#95—592no—
187#95—592no—
188#95C2×D5912yescomplemented
189C24—592no—
190C24—594no—
191C24—1182no—
192C24—1182no—
193C24—2362no—
194C24—2362no—
195C24C22×D5911yescharacteristic
196C2×Q8—1182no—
197C2×Q8—2362no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—591no—
201C22×C4—591no—
202C22×C4—594no—
203C22×C4—592no—
204C22×C4—591no—
205C22×C4—591no—
206C22×C4—2362no—
207C22×C4C22×D5912yes—
208C22×C4C22×D5912yes—
209C22×C4C22×D5911yescharacteristic
210C22×C4C22×D5911yescharacteristic
211C4⋊C4—42no—
212C4⋊C4—1182no—
213C4⋊C4—1182no—
214C4⋊C4—1182no—
215C4⋊C4—2362no—
216C4⋊C4—2362no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—1184no—
220C22⋊C4—1184no—
221C22⋊C4—1182no—
222C22⋊C4—1182no—
223C22⋊C4—2362no—
224C22⋊C4—2362no—
225C22⋊C4—2362no—
226C22⋊C4—2362no—
227C23—22no—
228C23—22no—
229C23—594no—
230C23—1188no—
231C23—1182no—
232C23—1182no—
233C23—1182no—
234C23—1182no—
235C23—1182no—
236C23—1182no—
237C23—2368no—
238C23—2364no—
239C23—2364no—
240C23—2368no—
241C23#1174568512yes—
242C23D4×D5912yes—
243C23Q8×D5912yes—
244C23C23×D5911yescharacteristic
245Q8—2364no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—44no—
257C2×C4—592no—
258C2×C4—592no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4—1181no—
262C2×C4—1181no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—1182no—
268C2×C4—2364no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—1184no—
274C22—1184no—
275C22—1184no—
276C22—23616no—
277C22—2362no—
278C22—2362no—
279C22—2362no—
280C22—2362no—
281C22#1427370514yes—
282C22C2×D4×D5911yescharacteristic
283C22C2×Q8×D5911yescharacteristic
284C22C2×#1174568511yescharacteristic
285C22#1427377212yes—
286C22#1427377312yes—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—42no—
291C4—1182no—
292C4—1182no—
293C4—2361no—
294C4—2361no—
295C4—2362no—
296C2—24no—
297C2—2364no—
298C2C2×#1427370511yescharacteristic
299C2#1683796311yescharacteristic
300C2#1683796711yescharacteristic
301C2#1683798512yes—
302C2#1683802712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#385C2
hypercenter#385D59
hypocenterC59#2626
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#14273704C2semidirect
3. #16837337C2semidirect
4. #16837326C2semidirect
5. #16837324C2semidirect
6. #16837315C2semidirect
7. #16837275C2semidirect
8. #16837260C2semidirect
11. C59×#385C2semidirect
13. C22×Dic118C22semidirect
16. C2×C4⋊Dic59C22semidirect
17. C2×Dic59⋊C4C22semidirect
18. C2×Dic59⋊C4C22semidirect
19. C2×Dic59⋊C4C22semidirect
20. C2×Dic59⋊C4C22semidirect
30. C59⋊#95C22semidirect
31. C59⋊#95C22semidirect
33. C2×C4⋊C4×C59C22semidirect
36. #95×C59C22semidirect
42. C23×D59Q8semidirect
159. #385D59semidirect
163. C59#2626semidirect
174. C2×C4⋊C4C2×D59semidirect
188. #95C2×D59semidirect