Finite Group Databasebeta

← #21434192

Subgroups — #21434192

Order 7552.

444 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14273689C211yesmaximal, characteristic
3#16837737C211yesmaximal, characteristic, complemented
4#16837681C212yesmaximal, complemented
5#16837606C212yesmaximal, complemented
6#16837555C211yesmaximal, characteristic
7#16837552C211yesmaximal, characteristic
8#16837505C212yesmaximal, complemented
9#16837501C212yesmaximal, complemented
10#16837456C211yesmaximal, characteristic, complemented
11C59×#484C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C4×D59—21no—
14C2×#11745660C2212yes—
15C2×#11745671C2212yes—
16#14273726C2211yescharacteristic, complemented
17#14273725C2211yescharacteristic, complemented
18#14273717—21no—
19#14273706C2211yescharacteristic
20#14273704—21no—
21Q8×Dic59C2211yescharacteristic, complemented
22#14273692—21no—
23#14273692C2211yescharacteristic, complemented
24#14273689—21no—
25#14273689—21no—
26#14273689C2212yes—
27#14273689C2212yes—
28#14273683—21no—
29C2×(C2×D59)⋊C4—21no—
30C2×C4×Dic59C2211yescharacteristic
31#14273660C2212yescomplemented
32Dic59⋊C8C2212yescomplemented
33#14273650C2212yescomplemented
34#14273649C2212yes—
35#14273647C2212yes—
36C22⋊(C59⋊C8)C2211yescharacteristic
37#14273637C2212yescomplemented
38C22⋊Q8×C59C2211yescharacteristic
39C2×C4⋊C4×C59C2211yescharacteristic
40#107×C59C2212yescomplemented
41#103×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C23×D59—21no—
44C2×Q8×C59C2311yescharacteristic
45C22×C4×C59—21no—
46C22×C4×C59C2311yescharacteristic
47C22×Dic59D411yescharacteristic
48C22×Dic59D411yescharacteristic
49C2×D236—21no—
50C2×D236—24no—
51C2×D236C2312yes—
52C2×C4×D59—22no—
53C2×C4×D59—22no—
54C2×C4×D59—44no—
55C2×Dic118—21no—
56C2×Dic118C2311yescharacteristic
57C2×C8×C59C2312yes—
58C4⋊C4×C59—21no—
59C4⋊C4×C59—21no—
60C4⋊C4×C59C2312yes—
61C4⋊C4×C59C2311yescharacteristic
62C22⋊C4×C59—21no—
63C22⋊Dic59—21no—
64#11745671—24no—
65(C2×D59)⋊C4—22no—
66(C2×D59)⋊C4—22no—
67C4⋊Dic59—21no—
68C4⋊Dic59C2311yescharacteristic
69Dic59⋊C4—21no—
70Dic59⋊C4—21no—
71Dic59⋊C4—21no—
72Dic59⋊C4—21no—
73Dic59⋊C4—21no—
74C4×Dic59—21no—
75C4×Dic59—21no—
76C4×Dic59D411yescharacteristic
77C4×Dic59D411yescharacteristic
78C4×Dic59C2311yescharacteristic
79C4×Dic59C2311yescharacteristic
80C2×C59⋊C8C2312yes—
81#11745660—24no—
82C23×C59C2×D411yescharacteristic
83C22×D59—24no—
84C22×D59—22no—
85C22×D59—42no—
86C22×D59—42no—
87Q8×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—22no—
92C2×C4×C59—42no—
93C2×C4×C59C2×D411yescharacteristic
94C2×C4×C59C2411yescharacteristic
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—21no—
98C2×Dic59—21no—
99C2×Dic59C2×D411yescharacteristic
100C2×Dic59C2×D411yescharacteristic
101C2×Dic59C2×D411yescharacteristic
102C2×Dic59C2×D411yescharacteristic
103D236—24no—
104D236—22no—
105D236#4114yes—
106C4×D59—44no—
107C4×D59—44no—
108Dic118—22no—
109Dic118—42no—
110C8×C59—22no—
111C59⋊C8—22no—
112C22×C59—21no—
113C22×C59—21no—
114C22×C59#13712yes—
115C22×C59C22×D411yescharacteristic
116C2×D59—24no—
117C2×D59—48no—
118C2×D59—42no—
119C2×D59—42no—
120C4×C59—21no—
121C4×C59—41no—
122C4×C59—41no—
123C4×C59—42no—
124C4×C59C2×#4112yes—
125Dic59—21no—
126Dic59—21no—
127Dic59—21no—
128Dic59—41no—
129Dic59—41no—
130Dic59AGL(1,ℤ/8)12yes—
131#3110—591nomaximal, Sylow
132C2×C59—22no—
133C2×C59#52511yescharacteristic
134C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C59C2×#13711yescharacteristic
136D59—44no—
137#533—591no—
138C2×C4×D4—591no—
139#484—591no—
140#484D5911yescharacteristic, complemented
141#474—592no—
142#461—592no—
143#450—591no—
144#450—591no—
145#440—592no—
146#440—592no—
147#432—591no—
148C59#311011yescharacteristic, complemented, Sylow
149C22×D4—591no—
150C23×C4—1181no—
151C2×SD16—592no—
152C2×SD16—592no—
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⋊C4C2×D5911yescharacteristic
168C2×C22⋊C4—1181no—
169C2×C42—591no—
170#107—592no—
171#107C2×D5912yescomplemented
172C4⋊C8—592no—
173#103—592no—
174#103C2×D5912yescomplemented
175#102—592no—
176#102—592no—
177C22⋊C8—591no—
178C22⋊C8C2×D5911yescharacteristic
179#97—592no—
180C24—1181no—
181C2×Q8—591no—
182C2×Q8—1181no—
183C2×Q8C22×D5911yescharacteristic
184C2×D4—592no—
185C2×D4—1184no—
186C2×D4—1181no—
187C22×C4—21no—
188C22×C4—591no—
189C22×C4—591no—
190C22×C4—1182no—
191C22×C4—1182no—
192C22×C4—2364no—
193C22×C4C22×D5911yescharacteristic
194SD16—1184no—
195SD16—1184no—
196C2×C8—592no—
197C2×C8C22×D5912yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—591no—
201C4⋊C4—1181no—
202C4⋊C4—1181no—
203C4⋊C4—1181no—
204C4⋊C4—1181no—
205C4⋊C4—1181no—
206C4⋊C4—1181no—
207C4⋊C4C22×D5911yescharacteristic
208C4⋊C4C22×D5912yes—
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—1184no—
220C23—1182no—
221C23—2362no—
222C23—2362no—
223C23D4×D5911yescharacteristic
224Q8—22no—
225Q8—1182no—
226Q8—2362no—
227D4—594no—
228D4—1184no—
229D4—1182no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—591no—
236C2×C4—591no—
237C2×C4—591no—
238C2×C4—591no—
239C2×C4—1182no—
240C2×C4—1182no—
241C2×C4—1181no—
242C2×C4—1181no—
243C2×C4—2364no—
244C2×C4—2364no—
245C2×C4D4×D5911yescharacteristic
246C2×C4C23×D5911yescharacteristic
247C8—22no—
248C8—1182no—
249C22—21no—
250C22—21no—
251C22—1184no—
252C22—2368no—
253C22—2362no—
254C22—2362no—
255C22#1427375912yes—
256C22C2×D4×D5911yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—592no—
262C4—1181no—
263C4—1181no—
264C4—1181no—
265C4—2361no—
266C4—2361no—
267C4#41×D5912yes—
268C2—22no—
269C2—2364no—
270C2#1683791011yescharacteristic
271C2AGL(1,ℤ/8)×D5911yescharacteristic
272C2C2×#1427375911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#484C2
socleC22×C59C22×D4
hypercenter#484D59
hypocenterC59#3110
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #16837954C2semidirect
3. #16837737C2semidirect
4. #16837681C2semidirect
5. #16837606C2semidirect
8. #16837505C2semidirect
9. #16837501C2semidirect
10. #16837456C2semidirect
11. C59×#484C2semidirect
16. #14273726C22semidirect
17. #14273725C22semidirect
21. Q8×Dic59C22semidirect
23. #14273692C22semidirect
31. #14273660C22semidirect
32. Dic59⋊C8C22semidirect
33. #14273650C22semidirect
37. #14273637C22semidirect
40. #107×C59C22semidirect
41. #103×C59C22semidirect
140. #484D59semidirect
148. C59#3110semidirect
171. #107C2×D59semidirect
174. #103C2×D59semidirect