Finite Group Databasebeta

← #21434194

Subgroups — #21434194

Order 7552.

444 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 121 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#16837957C211yesmaximal, characteristic, complemented
2C4×D4×D59C211yesmaximal, characteristic
3#16837736C212yesmaximal, complemented
4#16837683C211yesmaximal, characteristic, complemented
5#16837682C211yesmaximal, characteristic, complemented
6#16837604C211yesmaximal, characteristic, complemented
7#16837551C212yesmaximal
8#16837505C212yesmaximal, complemented
9#16837503C212yesmaximal, complemented
10#16837458C211yesmaximal, characteristic, complemented
11C59×#461C211yesmaximal, characteristic, complemented
12C2×D4×D59C2211yescharacteristic
13C22×C4×D59—21no—
14C2×#11745671C2212yes—
15#14273727C2211yescharacteristic, complemented
16#14273718—21no—
17#14273707—21no—
18#14273703C2212yes—
19Q8×Dic59C2212yescomplemented
20#14273692—21no—
21C4×D236C2211yescharacteristic
22D4×Dic59C2211yescharacteristic
23#14273689C2211yescharacteristic
24C4×C22⋊D59—21no—
25(C22⋊D59)⋊C4—21no—
26#14273685—21no—
27C4⋊C4×D59C2211yescharacteristic
28C22⋊C4×D59—21no—
29C42×D59C2211yescharacteristic
30#14273661C2212yescomplemented
31#14273660C2212yescomplemented
32C4⋊(C59⋊C8)C2211yescharacteristic, complemented
33#14273651C2212yescomplemented
34#14273649C2212yes—
35#14273648C2212yes—
36(C2×D59)⋊C8C2212yes—
37#14273637C2212yescomplemented
38C2×SD16×C59C2212yes—
39C4⋊Q8×C59C2211yescharacteristic, complemented
40C4×D4×C59C2211yescharacteristic
41C4⋊C8×C59C2211yescharacteristic, complemented
42#103×C59C2212yescomplemented
43C23×D59—21no—
44C2×Q8×C59C2312yes—
45C2×D4×C59C2311yescharacteristic
46C22×C4×C59—21no—
47C2×C22⋊D59—21no—
48C22×Dic59—21no—
49D4×D59—22no—
50D4×D59—22no—
51C2×D236C2311yescharacteristic
52C2×C4×D59—21no—
53C2×C4×D59—21no—
54C2×C4×D59—21no—
55C2×C4×D59—44no—
56C2×C4×D59D411yescharacteristic
57C2×C4×D59D411yescharacteristic
58C2×C4×D59C2311yescharacteristic
59C2×Dic118—21no—
60SD16×C59—24no—
61C2×C8×C59C2312yes—
62C4⋊C4×C59—21no—
63C4⋊C4×C59—21no—
64C4⋊C4×C59C2311yescharacteristic
65C22⋊C4×C59—21no—
66C42×C59C2311yescharacteristic
67C22⋊Dic59—21no—
68#11745671—24no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—21no—
71(C2×D59)⋊C4—21no—
72C4⋊Dic59—21no—
73C4⋊Dic59—22no—
74C4⋊Dic59C2312yes—
75C4⋊Dic59C2311yescharacteristic
76Dic59⋊C4—21no—
77Dic59⋊C4—22no—
78C4×Dic59—22no—
79C4×Dic59D411yescharacteristic
80C4×Dic59D411yescharacteristic
81C4×Dic59C2311yescharacteristic
82C2×C59⋊C8C2312yes—
83C23×C59—21no—
84C22×D59—24no—
85C22×D59—21no—
86C22×D59—42no—
87C22×D59—42no—
88C22×D59C2×D411yescharacteristic
89Q8×C59—22no—
90Q8×C59—22no—
91D4×C59—21no—
92D4×C59#4112yes—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—42no—
96C2×C4×C59C2×D411yescharacteristic
97C2×C4×C59C2×D411yescharacteristic
98C2×C4×C59C2411yescharacteristic
99C22⋊D59—24no—
100C2×Dic59—22no—
101C2×Dic59—21no—
102C2×Dic59—42no—
103C2×Dic59C2×D411yescharacteristic
104C2×Dic59C2×D411yescharacteristic
105C2×Dic59C2×D411yescharacteristic
106D236—21no—
107D236#4112yes—
108C4×D59—22no—
109C4×D59—22no—
110C4×D59—21no—
111C4×D59—21no—
112C4×D59—42no—
113C4×D59—42no—
114C4×D59—42no—
115Dic118—42no—
116C8×C59—22no—
117C59⋊C8—22no—
118C22×C59—22no—
119C22×C59—41no—
120C22×C59—41no—
121C22×C59C22×D411yescharacteristic
122C2×D59—21no—
123C2×D59—21no—
124C2×D59—22no—
125C2×D59—48no—
126C2×D59—41no—
127C2×D59—41no—
128C2×D59#13712yes—
129C4×C59—21no—
130C4×C59—42no—
131C4×C59—41no—
132C4×C59AGL(1,ℤ/8)12yes—
133C4×C59C2×#4111yescharacteristic
134C4×C59C2×#4111yescharacteristic
135Dic59—21no—
136Dic59—21no—
137Dic59—21no—
138Dic59—42no—
139Dic59—41no—
140#3110—591nomaximal, Sylow
141C2×C59—42no—
142C2×C59#52511yescharacteristic
143C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C59C2×#13711yescharacteristic
145D59—22no—
146D59—42no—
147#533—591no—
148C2×C4×D4—591no—
149#484—592no—
150#474—591no—
151#474—591no—
152#461—591no—
153#461D5911yescharacteristic, complemented
154#450—592no—
155#440—592no—
156#440—592no—
157#432—591no—
158C59#311011yescharacteristic, complemented, Sylow
159C22×D4—591no—
160C23×C4—1181no—
161C2×SD16—592no—
162C2×SD16C2×D5912yes—
163C4⋊Q8—591no—
164C4⋊Q8C2×D5911yescharacteristic, complemented
165#125—1181no—
166C22⋊Q8—592no—
167C22⋊Q8—1181no—
168C4×Q8—592no—
169C4×Q8—1181no—
170C4×D4—591no—
171C4×D4—591no—
172C4×D4—591no—
173C4×D4—1181no—
174C4×D4—1181no—
175C4×D4C2×D5911yescharacteristic
176#117—1181no—
177C2×C4⋊C4—591no—
178C2×C22⋊C4—1181no—
179C2×C42—591no—
180#107—592no—
181#107—592no—
182C4⋊C8—591no—
183C4⋊C8C2×D5911yescharacteristic, complemented
184#103—592no—
185#103C2×D5912yescomplemented
186#102—592no—
187#102—592no—
188C22⋊C8—592no—
189#97—592no—
190C24—1181no—
191C2×Q8—1181no—
192C2×Q8C22×D5912yes—
193C2×D4—591no—
194C2×D4—1181no—
195C2×D4—1182no—
196C2×D4—1182no—
197C2×D4C22×D5911yescharacteristic
198C22×C4—21no—
199C22×C4—591no—
200C22×C4—591no—
201C22×C4—591no—
202C22×C4—1181no—
203C22×C4—1181no—
204C22×C4—1181no—
205C22×C4—1181no—
206C22×C4—2364no—
207SD16—24no—
208SD16—1184no—
209C2×C8—592no—
210C2×C8C22×D5912yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—592no—
214C4⋊C4—591no—
215C4⋊C4—1181no—
216C4⋊C4—1182no—
217C4⋊C4—1182no—
218C4⋊C4—1181no—
219C4⋊C4C22×D5911yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—1181no—
222C22⋊C4—1182no—
223C22⋊C4—1181no—
224C22⋊C4—1181no—
225C42—591no—
226C42—591no—
227C42—591no—
228C42—1182no—
229C42C22×D5911yescharacteristic
230C23—21no—
231C23—591no—
232C23—1184no—
233C23—1181no—
234C23—2362no—
235C23—2362no—
236Q8—22no—
237Q8—22no—
238Q8—2362no—
239D4—21no—
240D4—592no—
241D4—1184no—
242D4—1181no—
243D4#1174568712yes—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—591no—
248C2×C4—591no—
249C2×C4—591no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1181no—
253C2×C4—1181no—
254C2×C4—1182no—
255C2×C4—1181no—
256C2×C4—2362no—
257C2×C4—2362no—
258C2×C4—2362no—
259C2×C4—2362no—
260C2×C4D4×D5911yescharacteristic
261C2×C4D4×D5911yescharacteristic
262C2×C4C23×D5911yescharacteristic
263C8—22no—
264C8—1182no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—592no—
269C22—1181no—
270C22—1181no—
271C22—1182no—
272C22—2368no—
273C22—2361no—
274C22—2361no—
275C22C2×D4×D5911yescharacteristic
276C4—21no—
277C4—42no—
278C4—41no—
279C4—1181no—
280C4—1181no—
281C4—1181no—
282C4—2361no—
283C4—2362no—
284C4#1427375412yes—
285C4#41×D5911yescharacteristic
286C4C2×#1174568711yescharacteristic
287C2—42no—
288C2—1182no—
289C2—2362no—
290C2#1683791211yescharacteristic
291C2C2×#1427375411yescharacteristic
292C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#461C2
socleC22×C59C22×D4
hypercenter#461D59
hypocenterC59#3110
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #16837957C2semidirect
3. #16837736C2semidirect
4. #16837683C2semidirect
5. #16837682C2semidirect
6. #16837604C2semidirect
8. #16837505C2semidirect
9. #16837503C2semidirect
10. #16837458C2semidirect
11. C59×#461C2semidirect
15. #14273727C22semidirect
19. Q8×Dic59C22semidirect
30. #14273661C22semidirect
31. #14273660C22semidirect
32. C4⋊(C59⋊C8)C22semidirect
33. #14273651C22semidirect
37. #14273637C22semidirect
39. C4⋊Q8×C59C22semidirect
41. C4⋊C8×C59C22semidirect
42. #103×C59C22semidirect
153. #461D59semidirect
158. C59#3110semidirect
164. C4⋊Q8C2×D59semidirect
183. C4⋊C8C2×D59semidirect
185. #103C2×D59semidirect