Finite Group Databasebeta

← #21434190

Subgroups — #21434190

Order 7552.

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

normal subgroups only

This group has 444 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#16837954C211yesmaximal, characteristic, complemented
2C2×D4×Dic59C211yesmaximal, characteristic
3#16837738C211yesmaximal, characteristic, complemented
4#16837735C211yesmaximal, characteristic, complemented
5#16837680C212yesmaximal, complemented
6#16837605C212yesmaximal, complemented
7#16837553C211yesmaximal, characteristic
8#16837503C212yesmaximal, complemented
9#16837499C212yesmaximal, complemented
10#16837456C211yesmaximal, characteristic, complemented
11C59×#450C211yesmaximal, characteristic
12C23×Dic59—21no—
13C2×#11745670C2212yes—
14#14273726C2211yescharacteristic, complemented
15#14273725C2211yescharacteristic, complemented
16#14273717—21no—
17#14273706C2211yescharacteristic
18#14273704—21no—
19Q8×Dic59C2211yescharacteristic, complemented
20#14273692—21no—
21#14273692C2211yescharacteristic, complemented
22D4×Dic59—21no—
23D4×Dic59—21no—
24D4×Dic59C2212yes—
25D4×Dic59C2212yes—
26#14273683—21no—
27C2×C4⋊Dic59C2211yescharacteristic
28C2×C22⋊Dic59—21no—
29C2×C4×Dic59C2211yescharacteristic
30#14273661C2212yescomplemented
31#14273660C2212yescomplemented
32Dic59⋊C8C2212yescomplemented
33#14273652C2212yescomplemented
34#14273651C2212yescomplemented
35#14273648C2212yes—
36C22⋊(C59⋊C8)C2211yescharacteristic
37#14273637C2212yescomplemented
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
49C22×Dic59—21no—
50C22×Dic59—22no—
51C22×Dic59—22no—
52C22×Dic59—44no—
53C22×Dic59D411yescharacteristic
54C22×Dic59D411yescharacteristic
55C2×Dic118—21no—
56C2×Dic118C2311yescharacteristic
57SD16×C59—24no—
58C2×C8×C59C2312yes—
59C4⋊C4×C59—21no—
60C4⋊C4×C59C2311yescharacteristic
61C22⋊C4×C59—21no—
62C22⋊Dic59—21no—
63C22⋊Dic59—22no—
64C22⋊Dic59—22no—
65#11745670—24no—
66C4⋊Dic59—21no—
67C4⋊Dic59—21no—
68C4⋊Dic59C2311yescharacteristic
69C4⋊Dic59C2312yes—
70Dic59⋊C4—21no—
71Dic59⋊C4—21no—
72Dic59⋊C4—21no—
73Dic59⋊C4—21no—
74Dic59⋊C4—21no—
75C4×Dic59—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
87Q8×C59—22no—
88D4×C59—24no—
89D4×C59—22no—
90D4×C59#4114yes—
91C2×C4×C59—22no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59C2×D411yescharacteristic
95C2×C4×C59C2411yescharacteristic
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59—21no—
99C2×Dic59—21no—
100C2×Dic59—22no—
101C2×Dic59—44no—
102C2×Dic59—42no—
103C2×Dic59—44no—
104C2×Dic59C2×D411yescharacteristic
105C2×Dic59C2×D411yescharacteristic
106C2×Dic59C2×D411yescharacteristic
107C2×Dic59C2×D411yescharacteristic
108Dic118—22no—
109Dic118—42no—
110C8×C59—22no—
111C59⋊C8—22no—
112C22×C59—21no—
113C22×C59—21no—
114C22×C59—24no—
115C22×C59—48no—
116C22×C59—42no—
117C22×C59—42no—
118C22×C59#13712yes—
119C22×C59C22×D411yescharacteristic
120C4×C59—21no—
121C4×C59—41no—
122C4×C59—41no—
123C4×C59C2×#4112yes—
124Dic59—21no—
125Dic59—21no—
126Dic59—21no—
127Dic59—41no—
128Dic59—41no—
129Dic59—42no—
130Dic59AGL(1,ℤ/8)12yes—
131#3110—591nomaximal, Sylow
132C2×C59—22no—
133C2×C59—44no—
134C2×C59#52511yescharacteristic
135C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C59C2×#13711yescharacteristic
137#533—591no—
138C2×C4×D4—591no—
139#484—591no—
140#484—591no—
141#474—592no—
142#461—592no—
143#450—591no—
144#450D5911yescharacteristic
145#440—592no—
146#440—592no—
147#432—591no—
148C59#311011yescharacteristic, complemented, Sylow
149C22×D4C2×D5911yescharacteristic
150C23×C4—1181no—
151C2×SD16—592no—
152C2×SD16C2×D5912yes—
153C4⋊Q8—591no—
154C4⋊Q8—591no—
155#125—1181no—
156C22⋊Q8—591no—
157C22⋊Q8—1181no—
158C22⋊Q8C2×D5911yescharacteristic
159C4×Q8—591no—
160C4×Q8—591no—
161C4×Q8—1181no—
162C4×D4—592no—
163C4×D4—592no—
164C4×D4—1181no—
165C4×D4—1181no—
166#117—1181no—
167C2×C4⋊C4—591no—
168C2×C22⋊C4—1181no—
169C2×C42—591no—
170#107—592no—
171#107—592no—
172C4⋊C8—592no—
173#103—592no—
174#103—592no—
175#102—592no—
176#102C2×D5912yes—
177C22⋊C8—591no—
178C22⋊C8C2×D5911yescharacteristic
179#97—592no—
180C24—21no—
181C2×Q8—591no—
182C2×Q8—1181no—
183C2×Q8C22×D5911yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D5912yes—
187C22×C4—591no—
188C22×C4—591no—
189C22×C4—1182no—
190C22×C4—1182no—
191C22×C4—1181no—
192C22×C4—2364no—
193C22×C4C22×D5911yescharacteristic
194SD16—24no—
195SD16—1184no—
196C2×C8—592no—
197C2×C8C22×D5912yes—
198C4⋊C4—21no—
199C4⋊C4—591no—
200C4⋊C4—592no—
201C4⋊C4—1181no—
202C4⋊C4—1181no—
203C4⋊C4—1181no—
204C4⋊C4—1181no—
205C4⋊C4—1181no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C4⋊C4C22×D5911yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—1182no—
211C22⋊C4—1182no—
212C22⋊C4—1181no—
213C42—591no—
214C42—591no—
215C42—591no—
216C42—591no—
217C42—1181no—
218C42—1181no—
219C23—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D5911yescharacteristic
224Q8—22no—
225Q8—1182no—
226Q8—2362no—
227D4—24no—
228D4—22no—
229D4#1174568714yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—591no—
234C2×C4—591no—
235C2×C4—591no—
236C2×C4—591no—
237C2×C4—1182no—
238C2×C4—1182no—
239C2×C4—1181no—
240C2×C4—1181no—
241C2×C4—1182no—
242C2×C4—2364no—
243C2×C4—2364no—
244C2×C4—2362no—
245C2×C4D4×D5911yescharacteristic
246C2×C4C23×D5911yescharacteristic
247C8—22no—
248C8—1182no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22#1427375612yes—
256C22C2×D4×D5911yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—592no—
261C4—1181no—
262C4—1181no—
263C4—1181no—
264C4—2361no—
265C4—2362no—
266C4—2361no—
267C4C2×#1174568712yes—
268C2—22no—
269C2—44no—
270C2#1683790811yescharacteristic
271C2AGL(1,ℤ/8)×D5911yescharacteristic
272C2C2×#1427375611yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3110
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. #16837954C2semidirect
3. #16837738C2semidirect
4. #16837735C2semidirect
5. #16837680C2semidirect
6. #16837605C2semidirect
8. #16837503C2semidirect
9. #16837499C2semidirect
10. #16837456C2semidirect
14. #14273726C22semidirect
15. #14273725C22semidirect
19. Q8×Dic59C22semidirect
21. #14273692C22semidirect
30. #14273661C22semidirect
31. #14273660C22semidirect
32. Dic59⋊C8C22semidirect
33. #14273652C22semidirect
34. #14273651C22semidirect
37. #14273637C22semidirect
148. C59#3110semidirect