Finite Group Databasebeta

← #21434197

Subgroups — #21434197

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#16837739C212yesmaximal, complemented
4#16837679C211yesmaximal, characteristic, complemented
5#16837608C211yesmaximal, characteristic, complemented
6#16837607C211yesmaximal, characteristic, complemented
7#16837554C212yesmaximal
8#16837501C212yesmaximal, complemented
9#16837499C212yesmaximal, complemented
10#16837458C211yesmaximal, characteristic, complemented
11C59×#474C211yesmaximal, characteristic, complemented
12C2×D4×D59C2211yescharacteristic
13C22×C4×D59—21no—
14C2×#11745660C2212yes—
15C2×#11745670C2212yes—
16#14273727C2211yescharacteristic, complemented
17#14273718—21no—
18#14273707C2212yes—
19#14273703—21no—
20Q8×Dic59—21no—
21#14273692C2212yescomplemented
22C4×D236C2211yescharacteristic
23D4×Dic59C2211yescharacteristic
24#14273689C2211yescharacteristic
25C4×C22⋊D59—21no—
26(C22⋊D59)⋊C4—21no—
27#14273685—21no—
28C4⋊C4×D59C2211yescharacteristic
29C22⋊C4×D59—21no—
30C42×D59C2211yescharacteristic
31#14273660C2212yescomplemented
32C4⋊(C59⋊C8)C2211yescharacteristic, complemented
33#14273652C2212yescomplemented
34#14273650C2212yescomplemented
35#14273647C2212yes—
36(C2×D59)⋊C8C2212yes—
37#14273637C2212yescomplemented
38C4⋊Q8×C59C2211yescharacteristic, complemented
39C4×D4×C59C2211yescharacteristic
40#107×C59C2212yescomplemented
41C4⋊C8×C59C2211yescharacteristic, complemented
42#102×C59C2212yes—
43C23×D59—21no—
44C2×Q8×C59—21no—
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×Dic118C2312yes—
60C2×C8×C59C2312yes—
61C4⋊C4×C59—21no—
62C4⋊C4×C59C2312yes—
63C4⋊C4×C59C2311yescharacteristic
64C22⋊C4×C59—21no—
65C42×C59C2311yescharacteristic
66C22⋊Dic59—21no—
67#11745670—24no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—21no—
70(C2×D59)⋊C4—21no—
71C4⋊Dic59—21no—
72C4⋊Dic59—21no—
73C4⋊Dic59—22no—
74C4⋊Dic59C2311yescharacteristic
75Dic59⋊C4—21no—
76Dic59⋊C4—22no—
77C4×Dic59—22no—
78C4×Dic59D411yescharacteristic
79C4×Dic59D411yescharacteristic
80C4×Dic59C2311yescharacteristic
81C2×C59⋊C8C2312yes—
82#11745660—24no—
83C23×C59—21no—
84C22×D59—24no—
85C22×D59—21no—
86C22×D59—42no—
87C22×D59—42no—
88C22×D59C2×D411yescharacteristic
89Q8×C59—42no—
90D4×C59—21no—
91D4×C59#4112yes—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—42no—
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59C2×D411yescharacteristic
97C2×C4×C59C2411yescharacteristic
98C22⋊D59—24no—
99C2×Dic59—22no—
100C2×Dic59—21no—
101C2×Dic59—42no—
102C2×Dic59C2×D411yescharacteristic
103C2×Dic59C2×D411yescharacteristic
104C2×Dic59C2×D411yescharacteristic
105D236—21no—
106D236#4112yes—
107C4×D59—22no—
108C4×D59—22no—
109C4×D59—21no—
110C4×D59—21no—
111C4×D59—42no—
112C4×D59—42no—
113C4×D59—42no—
114Dic118—22no—
115Dic118—22no—
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—41no—
131C4×C59—42no—
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#474D5911yescharacteristic, complemented
152#461—591no—
153#461—591no—
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×SD16—592no—
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#107C2×D5912yescomplemented
182C4⋊C8—591no—
183C4⋊C8C2×D5911yescharacteristic, complemented
184#103—592no—
185#103—592no—
186#102—592no—
187#102C2×D5912yes—
188C22⋊C8—592no—
189#97—592no—
190C24—1181no—
191C2×Q8—21no—
192C2×Q8—592no—
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—1184no—
208SD16—1184no—
209C2×C8—592no—
210C2×C8C22×D5912yes—
211C4⋊C4—21no—
212C4⋊C4—591no—
213C4⋊C4—1181no—
214C4⋊C4—1182no—
215C4⋊C4—1182no—
216C4⋊C4—1181no—
217C4⋊C4—1181no—
218C4⋊C4C22×D5912yes—
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—42no—
237Q8—1182no—
238Q8—1182no—
239D4—21no—
240D4—592no—
241D4—1184no—
242D4—1181no—
243D4#1174568912yes—
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—41no—
278C4—42no—
279C4—1181no—
280C4—1181no—
281C4—1181no—
282C4—2362no—
283C4—2361no—
284C4#1427375112yes—
285C4#41×D5911yescharacteristic
286C4C2×#1174568911yescharacteristic
287C2—42no—
288C2—1182no—
289C2—2362no—
290C2#1683791511yescharacteristic
291C2C2×#1427375111yescharacteristic
292C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#474C2
socleC22×C59C22×D4
hypercenter#474D59
hypocenterC59#3110
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #16837957C2semidirect
3. #16837739C2semidirect
4. #16837679C2semidirect
5. #16837608C2semidirect
6. #16837607C2semidirect
8. #16837501C2semidirect
9. #16837499C2semidirect
10. #16837458C2semidirect
11. C59×#474C2semidirect
16. #14273727C22semidirect
21. #14273692C22semidirect
31. #14273660C22semidirect
32. C4⋊(C59⋊C8)C22semidirect
33. #14273652C22semidirect
34. #14273650C22semidirect
37. #14273637C22semidirect
38. C4⋊Q8×C59C22semidirect
40. #107×C59C22semidirect
41. C4⋊C8×C59C22semidirect
151. #474D59semidirect
158. C59#3110semidirect
164. C4⋊Q8C2×D59semidirect
181. #107C2×D59semidirect
183. C4⋊C8C2×D59semidirect