Finite Group Databasebeta

← #21434182

Subgroups — #21434182

Order 7552.

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

normal subgroups only

This group has 488 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#16837947C211yesmaximal, characteristic, complemented
2C2×D4×Dic59C211yesmaximal, characteristic, complemented
3#16837721C211yesmaximal, characteristic, complemented
4#16837718C211yesmaximal, characteristic, complemented
5#16837690C212yesmaximal, complemented
6#16837595C212yesmaximal, complemented
7#16837536C211yesmaximal, characteristic, complemented
8#16837514C212yesmaximal, complemented
9#16837511C212yesmaximal, complemented
10#16837456C211yesmaximal, characteristic, complemented
11C59×#447C211yesmaximal, characteristic, complemented
12C2×#11745687—21no—
13C23×Dic59—21no—
14C2×#11745669C2212yes—
15#14273726C2211yescharacteristic, complemented
16#14273723C2211yescharacteristic, complemented
17#14273713—21no—
18#14273701C2211yescharacteristic
19#14273699—21no—
20D4×Dic59—21no—
21D4×Dic59—21no—
22D4×Dic59C2212yes—
23D4×Dic59C2212yes—
24D4×Dic59C2211yescharacteristic, complemented
25#14273689C2211yescharacteristic, complemented
26(C22⋊D59)⋊C4—21no—
27C2×C4⋊Dic59C2211yescharacteristic
28C2×C22⋊Dic59—21no—
29C2×C4×Dic59C2211yescharacteristic
30#14273659C2212yescomplemented
31#14273658C2212yescomplemented
32Dic59⋊C8C2212yescomplemented
33#14273649C2212yescomplemented
34#14273648C2212yes—
35#14273648C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37#14273637C2212yescomplemented
38C22×D4×C59C2211yescharacteristic
39C2×D8×C59C2212yes—
40#121×C59C2211yescharacteristic
41#102×C59C2212yes—
42C22⋊C8×C59C2211yescharacteristic
43C24×C59—21no—
44C2×D4×C59—24no—
45C2×D4×C59—21no—
46C2×D4×C59—21no—
47C2×D4×C59C2311yescharacteristic
48C2×D4×C59C2312yes—
49C22×C4×C59C2311yescharacteristic
50C2×C22⋊D59—21no—
51C2×C22⋊D59—21no—
52C2×C22⋊D59—21no—
53C22×Dic59—21no—
54C22×Dic59—21no—
55C22×Dic59—22no—
56C22×Dic59—22no—
57C22×Dic59—44no—
58C22×Dic59D411yescharacteristic
59C22×Dic59D411yescharacteristic
60#11745687—44no—
61C2×D236C2311yescharacteristic
62C2×C4×D59—21no—
63C2×Dic118—21no—
64D8×C59—24no—
65C2×C8×C59C2312yes—
66C4⋊C4×C59C2311yescharacteristic
67C22⋊C4×C59—21no—
68C22⋊Dic59—22no—
69C22⋊Dic59—22no—
70C22⋊Dic59—21no—
71#11745669—24no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74C4⋊Dic59—21no—
75C4⋊Dic59C2311yescharacteristic
76C4⋊Dic59C2312yes—
77Dic59⋊C4—21no—
78C4×Dic59D411yescharacteristic
79C4×Dic59D411yescharacteristic
80C4×Dic59C2311yescharacteristic
81C4×Dic59C2311yescharacteristic
82C2×C59⋊C8C2312yes—
83C23×C59—24no—
84C23×C59—22no—
85C23×C59—21no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59C2×D411yescharacteristic
89C22×D59—21no—
90D4×C59—24no—
91D4×C59—22no—
92D4×C59—22no—
93D4×C59—42no—
94D4×C59#4114yescomplemented
95C2×C4×C59—22no—
96C2×C4×C59—21no—
97C2×C4×C59C2×D411yescharacteristic
98C2×C4×C59C2411yescharacteristic
99C22⋊D59—42no—
100C22⋊D59—42no—
101C22⋊D59—42no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—21no—
105C2×Dic59—22no—
106C2×Dic59—42no—
107C2×Dic59—44no—
108C2×Dic59—42no—
109C2×Dic59—44no—
110C2×Dic59C2×D411yescharacteristic
111C2×Dic59C2×D411yescharacteristic
112C2×Dic59C2×D411yescharacteristic
113C2×Dic59C2×D411yescharacteristic
114D236—22no—
115C4×D59—42no—
116Dic118—42no—
117C8×C59—22no—
118C59⋊C8—22no—
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—24no—
122C22×C59—48no—
123C22×C59—42no—
124C22×C59—42no—
125C22×C59—41no—
126C22×C59—41no—
127C22×C59—41no—
128C22×C59AGL(1,ℤ/8)12yes—
129C22×C59C22×D411yescharacteristic
130C2×D59—41no—
131C2×D59—41no—
132C2×D59—41no—
133C4×C59—21no—
134C4×C59—41no—
135C4×C59C2×#4112yes—
136Dic59—21no—
137Dic59—21no—
138Dic59—21no—
139Dic59—41no—
140Dic59—42no—
141Dic59AGL(1,ℤ/8)12yes—
142#3109—591nomaximal, Sylow
143C2×C59—22no—
144C2×C59—44no—
145C2×C59—81no—
146C2×C59#52511yescharacteristic
147C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
149D59—81no—
150#532—591no—
151C2×C4×D4—591no—
152#481—591no—
153#481—591no—
154#476—592no—
155#459—592no—
156#447—591no—
157#447D5911yescharacteristic, complemented
158#442—592no—
159#442—592no—
160#432—591no—
161C59#310911yescharacteristic, complemented, Sylow
162C2×#41—1181no—
163C22×D4C2×D5911yescharacteristic
164C23×C4—1181no—
165C2×D8—592no—
166C2×D8C2×D5912yes—
167C4⋊Q8—591no—
168#127—591no—
169#124—1181no—
170#121—591no—
171#121—1181no—
172#121C2×D5911yescharacteristic
173C4×D4—592no—
174C4×D4—592no—
175C4×D4—591no—
176C4×D4—591no—
177C4×D4—1181no—
178C4×D4—1181no—
179C4×D4—1181no—
180C2×C4⋊C4—591no—
181C2×C22⋊C4—1181no—
182C2×C42—591no—
183#106—592no—
184#106—592no—
185C4⋊C8—592no—
186#102—592no—
187#102—592no—
188#102—592no—
189#102C2×D5912yes—
190C22⋊C8—591no—
191C22⋊C8C2×D5911yescharacteristic
192#97—592no—
193C24—21no—
194#41—2364no—
195C2×Q8—1181no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—591no—
200C2×D4—1181no—
201C2×D4—1181no—
202C2×D4—1181no—
203C2×D4C22×D5911yescharacteristic
204C2×D4C22×D5912yes—
205C22×C4—591no—
206C22×C4—591no—
207C22×C4—1181no—
208C22×C4—1182no—
209C22×C4—1182no—
210C22×C4—1181no—
211C22×C4—1181no—
212C22×C4—2364no—
213C22×C4C22×D5911yescharacteristic
214D8—24no—
215D8—1184no—
216C2×C8—592no—
217C2×C8C22×D5912yes—
218C4⋊C4—592no—
219C4⋊C4—591no—
220C4⋊C4—1181no—
221C4⋊C4—1181no—
222C4⋊C4C22×D5911yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—1181no—
225C22⋊C4—1182no—
226C22⋊C4—1182no—
227C22⋊C4—1181no—
228C22⋊C4—1181no—
229C42—591no—
230C42—591no—
231C42—591no—
232C42—591no—
233C23—24no—
234C23—22no—
235C23—21no—
236C23—42no—
237C23—42no—
238C23—1181no—
239C23D4×D5911yescharacteristic
240Q8—2362no—
241D4—24no—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—1182no—
246D4—2362no—
247D4—2362no—
248D4—2362no—
249D4#1174568714yescomplemented
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—591no—
253C2×C4—591no—
254C2×C4—591no—
255C2×C4—591no—
256C2×C4—1182no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—1181no—
260C2×C4—2362no—
261C2×C4—2364no—
262C2×C4—2364no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4D4×D5911yescharacteristic
266C2×C4C23×D5911yescharacteristic
267C8—22no—
268C8—1182no—
269C22—21no—
270C22—21no—
271C22—24no—
272C22—48no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—2361no—
279C22—2361no—
280C22—2361no—
281C22#1427375112yes—
282C22C2×D4×D5911yescharacteristic
283C4—21no—
284C4—41no—
285C4—592no—
286C4—1181no—
287C4—1181no—
288C4—1181no—
289C4—2362no—
290C4—2361no—
291C4C2×#1174568712yes—
292C2—22no—
293C2—44no—
294C2—81no—
295C2—4721no—
296C2#1683790811yescharacteristic
297C2AGL(1,ℤ/8)×D5911yescharacteristic
298C2C2×#1427375111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#447C2
socleC22×C59C22×D4
hypercenter#447D59
hypocenterC59#3109
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #16837947C2semidirect
2. C2×D4×Dic59C2semidirect
3. #16837721C2semidirect
4. #16837718C2semidirect
5. #16837690C2semidirect
6. #16837595C2semidirect
7. #16837536C2semidirect
8. #16837514C2semidirect
9. #16837511C2semidirect
10. #16837456C2semidirect
11. C59×#447C2semidirect
15. #14273726C22semidirect
16. #14273723C22semidirect
24. D4×Dic59C22semidirect
25. #14273689C22semidirect
30. #14273659C22semidirect
31. #14273658C22semidirect
32. Dic59⋊C8C22semidirect
33. #14273649C22semidirect
35. #14273648C22semidirect
37. #14273637C22semidirect
94. D4×C59#41semidirect
157. #447D59semidirect
161. C59#3109semidirect
249. D4#11745687semidirect