Finite Group Databasebeta

← #21434899

Subgroups — #21434899

Order 7552.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22⋊Q8×D59C211yesmaximal, characteristic, complemented
2C2×#14273698C211yesmaximal, characteristic
3#16837738C211yesmaximal, characteristic, complemented
4#16837735C211yesmaximal, characteristic, complemented
5#16837714C212yesmaximal, complemented
6#16837647C212yesmaximal, complemented
7#16837645C212yesmaximal, complemented
8#16837553C211yesmaximal, characteristic, complemented
9#16837539C212yesmaximal, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#450C211yesmaximal, characteristic, complemented
12C2×Q8×D59C2211yescharacteristic, complemented
13C22×C22⋊D59—21no—
14C22×C4×D59C2211yescharacteristic
15C2×#11745670C2212yescomplemented
16C2×#11745659C2212yescomplemented
17#14273707C2211yescharacteristic, complemented
18#14273706C2211yescharacteristic
19#14273705—21no—
20#14273704—21no—
21#14273703C2211yescharacteristic, complemented
22#14273698—21no—
23#14273698—21no—
24#14273698C2212yes—
25#14273698C2212yes—
26C2×C4⋊Dic59C2211yescharacteristic
27C4⋊C4×D59—21no—
28C4⋊C4×D59C2211yescharacteristic, complemented
29C2×C22⋊Dic59—21no—
30C22⋊C4×D59—21no—
31#14273661C2212yescomplemented
32#14273660C2212yescomplemented
33#14273652C2212yescomplemented
34#14273651C2212yescomplemented
35#14273648C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37(C2×D59)⋊C8C2212yescomplemented
38C22×D4×C59C2211yescharacteristic
39C2×SD16×C59C2212yescomplemented
40C22⋊Q8×C59C2211yescharacteristic
41#102×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C24×C59—21no—
44C23×D59D411yescharacteristic
45C2×Q8×C59C2311yescharacteristic
46C2×D4×C59—21no—
47C2×D4×C59—42no—
48C2×D4×C59C2312yes—
49C22×C4×C59C2311yescharacteristic
50C2×C22⋊D59—22no—
51C2×C22⋊D59—22no—
52C2×C22⋊D59—44no—
53C22×Dic59—21no—
54C22×Dic59D411yescharacteristic
55Q8×D59—42no—
56C2×C4×D59—21no—
57C2×C4×D59—21no—
58C2×C4×D59—42no—
59C2×C4×D59D411yescharacteristic
60C2×C4×D59D411yescharacteristic
61C2×C4×D59C2311yescharacteristic
62C2×C4×D59C2311yescharacteristic
63C2×Dic118—21no—
64C2×Dic118C2311yescharacteristic
65SD16×C59—42no—
66C2×C8×C59C2312yes—
67C4⋊C4×C59—21no—
68C4⋊C4×C59C2311yescharacteristic
69C22⋊C4×C59—21no—
70C22⋊Dic59—21no—
71C22⋊Dic59—22no—
72C22⋊Dic59—22no—
73#11745670—42no—
74(C2×D59)⋊C4—21no—
75(C2×D59)⋊C4—21no—
76C4⋊Dic59—21no—
77C4⋊Dic59—21no—
78C4⋊Dic59C2311yescharacteristic
79C4⋊Dic59C2312yes—
80Dic59⋊C4—21no—
81Dic59⋊C4—21no—
82Dic59⋊C4—21no—
83C2×C59⋊C8C2312yes—
84#11745659—42no—
85C23×C59—22no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59C2×D411yescharacteristic
90C22×D59—22no—
91C22×D59—22no—
92C22×D59—41no—
93C22×D59C2×D411yescharacteristic
94C22×D59C2×D411yescharacteristic
95Q8×C59—22no—
96D4×C59—22no—
97D4×C59—22no—
98D4×C59—42no—
99C2×C4×C59—22no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59C2×D411yescharacteristic
103C2×C4×C59C2411yescharacteristic
104C22⋊D59—44no—
105C22⋊D59—44no—
106C2×Dic59—21no—
107C2×Dic59—21no—
108C2×Dic59—22no—
109C2×Dic59—41no—
110C2×Dic59—42no—
111C2×Dic59C2×D411yescharacteristic
112C2×Dic59C2×D411yescharacteristic
113C4×D59—22no—
114C4×D59—22no—
115C4×D59—41no—
116C4×D59—41no—
117C4×D59—42no—
118C4×D59—42no—
119Dic118—22no—
120Dic118—42no—
121C8×C59—22no—
122C59⋊C8—22no—
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59—84no—
129C22×C59#13712yes—
130C22×C59C22×D411yescharacteristic
131C2×D59—21no—
132C2×D59—21no—
133C2×D59—21no—
134C2×D59—21no—
135C2×D59—21no—
136C2×D59—44no—
137C2×D59AGL(1,ℤ/8)12yes—
138C4×C59—21no—
139C4×C59—41no—
140C4×C59—41no—
141C4×C59H(5,𝔽2)12yes—
142Dic59—21no—
143Dic59—21no—
144Dic59—41no—
145Dic59—41no—
146Dic59—42no—
147#3193—591nomaximal, Sylow
148C2×C59—22no—
149C2×C59—82no—
150C2×C59#53411yescharacteristic
151C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C59C2×#13711yescharacteristic
153D59—22no—
154D59—41no—
155C2×C22⋊Q8—591no—
156C2×#121—591no—
157#484—591no—
158#484—591no—
159#480—592no—
160#468—592no—
161#468—592no—
162#450—591no—
163#450D5911yescharacteristic, complemented
164#448—592no—
165#407—591no—
166C59#319311yescharacteristic, complemented, Sylow
167C22×Q8—591no—
168C22×D4—1181no—
169C22×D4C2×D5911yescharacteristic
170C23×C4—591no—
171C2×SD16—592no—
172C2×SD16C2×D5912yescomplemented
173C2×M16—592no—
174C22⋊Q8—591no—
175C22⋊Q8—591no—
176C22⋊Q8—591no—
177C22⋊Q8—1181no—
178C22⋊Q8—1181no—
179C22⋊Q8C2×D5911yescharacteristic
180#121—592no—
181#121—592no—
182#121—1181no—
183#121—1181no—
184C2×C4⋊C4—591no—
185C2×C4⋊C4—591no—
186C2×C4⋊C4—1181no—
187C2×C22⋊C4—1181no—
188C2×C22⋊C4—1181no—
189#107—592no—
190#107—592no—
191#103—592no—
192#103—592no—
193#102—592no—
194#102C2×D5912yescomplemented
195C22⋊C8—591no—
196C22⋊C8—592no—
197C22⋊C8C2×D5911yescharacteristic
198C24—21no—
199C24—591no—
200C2×Q8—591no—
201C2×Q8—1181no—
202C2×Q8—2362no—
203C2×Q8C22×D5911yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—1182no—
207C2×D4—1182no—
208C2×D4—2364no—
209C2×D4C22×D5912yes—
210C22×C4—591no—
211C22×C4—591no—
212C22×C4—591no—
213C22×C4—591no—
214C22×C4—591no—
215C22×C4—1181no—
216C22×C4—1181no—
217C22×C4—1181no—
218C22×C4—2362no—
219C22×C4C22×D5911yescharacteristic
220SD16—42no—
221SD16—2362no—
222M16—2362no—
223C2×C8—592no—
224C2×C8C22×D5912yes—
225C4⋊C4—21no—
226C4⋊C4—591no—
227C4⋊C4—592no—
228C4⋊C4—1181no—
229C4⋊C4—1181no—
230C4⋊C4—1181no—
231C4⋊C4—1181no—
232C4⋊C4—1181no—
233C4⋊C4C22×D5911yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—1181no—
236C22⋊C4—1181no—
237C22⋊C4—1182no—
238C22⋊C4—1182no—
239C22⋊C4—1181no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—591no—
245C23—591no—
246C23—1182no—
247C23—1182no—
248C23—2361no—
249C23D4×D5911yescharacteristic
250Q8—22no—
251Q8—1182no—
252Q8—2362no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2364no—
257D4—2364no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—591no—
262C2×C4—591no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—1181no—
266C2×C4—1182no—
267C2×C4—1181no—
268C2×C4—2361no—
269C2×C4—2361no—
270C2×C4—2362no—
271C2×C4—2362no—
272C2×C4—2361no—
273C2×C4—2362no—
274C2×C4D4×D5911yescharacteristic
275C2×C4C23×D5911yescharacteristic
276C8—22no—
277C8—1182no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—592no—
285C22—1181no—
286C22—1181no—
287C22—1181no—
288C22—1181no—
289C22—1181no—
290C22—2364no—
291C22#1427375612yes—
292C22C2×D4×D5911yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—1181no—
297C4—1181no—
298C4—2362no—
299C4—2361no—
300C4—2361no—
301C4#1427377212yes—
302C2—22no—
303C2—82no—
304C2—1182no—
305C2—2361no—
306C2#1683796211yescharacteristic
307C2AGL(1,ℤ/8)×D5911yescharacteristic
308C2C2×#1427375611yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3193
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. C22⋊Q8×D59C2semidirect
3. #16837738C2semidirect
4. #16837735C2semidirect
5. #16837714C2semidirect
6. #16837647C2semidirect
7. #16837645C2semidirect
8. #16837553C2semidirect
9. #16837539C2semidirect
10. #16837364C2semidirect
11. C59×#450C2semidirect
12. C2×Q8×D59C22semidirect
15. C2×#11745670C22semidirect
16. C2×#11745659C22semidirect
17. #14273707C22semidirect
21. #14273703C22semidirect
28. C4⋊C4×D59C22semidirect
31. #14273661C22semidirect
32. #14273660C22semidirect
33. #14273652C22semidirect
34. #14273651C22semidirect
35. #14273648C22semidirect
37. (C2×D59)⋊C8C22semidirect
39. C2×SD16×C59C22semidirect
41. #102×C59C22semidirect
163. #450D59semidirect
166. C59#3193semidirect
172. C2×SD16C2×D59semidirect
194. #102C2×D59semidirect