Finite Group Databasebeta

← #21435035

Subgroups — #21435035

Order 7552.

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

normal subgroups only

This group has 488 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×#14273723C211yesmaximal, characteristic
3#16837773C212yesmaximal, complemented
4#16837747C212yesmaximal, complemented
5#16837601C212yesmaximal, complemented
6#16837600C212yesmaximal, complemented
7#16837555C211yesmaximal, characteristic, complemented
8#16837553C211yesmaximal, characteristic, complemented
9#16837552C211yesmaximal, characteristic, complemented
10#16837452C211yesmaximal, characteristic, complemented
11C59×#450C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C22×C22⋊D59—21no—
14C22×C22⋊D59—21no—
15C2×#11745660C2212yescomplemented
16C2×#11745670C2212yescomplemented
17C2×#11745671C2212yescomplemented
18#14273726C2211yescharacteristic, complemented
19#14273725C2211yescharacteristic, complemented
20#14273723—21no—
21#14273723—21no—
22#14273723C2212yes—
23#14273723C2212yes—
24#14273717—21no—
25#14273706C2211yescharacteristic
26#14273704—21no—
27Q8×Dic59C2211yescharacteristic, complemented
28#14273692—21no—
29#14273692C2211yescharacteristic, complemented
30#14273683—21no—
31C2×C4×Dic59C2211yescharacteristic
32Dic59⋊C8C2212yescomplemented
33#14273649C2212yescomplemented
34#14273648C2212yescomplemented
35#14273647C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37C8×Dic59C2212yescomplemented
38C22×D4×C59C2211yescharacteristic
39C2×SD16×C59C2212yescomplemented
40C22⋊Q8×C59C2211yescharacteristic
41#102×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C24×C59—21no—
44C23×D59—21no—
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—22no—
53C2×C22⋊D59—22no—
54C2×C22⋊D59—44no—
55C2×C22⋊D59—44no—
56C22×Dic59D411yescharacteristic
57C22×Dic59D411yescharacteristic
58C2×D236—21no—
59C2×D236—42no—
60C2×D236C2312yes—
61C2×Dic118—21no—
62C2×Dic118C2311yescharacteristic
63SD16×C59—42no—
64C2×C8×C59C2312yes—
65C4⋊C4×C59—21no—
66C4⋊C4×C59C2311yescharacteristic
67C22⋊C4×C59—21no—
68C22⋊Dic59—21no—
69#11745671—42no—
70#11745670—42no—
71C4⋊Dic59—21no—
72C4⋊Dic59C2311yescharacteristic
73Dic59⋊C4—21no—
74Dic59⋊C4—21no—
75Dic59⋊C4—21no—
76Dic59⋊C4—21no—
77Dic59⋊C4—21no—
78C4×Dic59—21no—
79C4×Dic59—21no—
80C4×Dic59D411yescharacteristic
81C4×Dic59D411yescharacteristic
82C4×Dic59C2311yescharacteristic
83C4×Dic59C2311yescharacteristic
84C2×C59⋊C8C2312yes—
85#11745660—42no—
86C23×C59—22no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59C2×D411yescharacteristic
91C22×D59—22no—
92C22×D59—42no—
93C22×D59—42no—
94C22×D59—42no—
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—
106C22⋊D59—44no—
107C22⋊D59—44no—
108C2×Dic59—22no—
109C2×Dic59—21no—
110C2×Dic59—21no—
111C2×Dic59SD1614yescomplemented
112C2×Dic59C2×D411yescharacteristic
113C2×Dic59C2×D411yescharacteristic
114C2×Dic59C2×D411yescharacteristic
115C2×Dic59C2×D411yescharacteristic
116D236—22no—
117D236—22no—
118D236—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×C59C2×SD1612yes—
130C22×C59C22×D411yescharacteristic
131C2×D59—42no—
132C2×D59—42no—
133C2×D59—42no—
134C2×D59—84no—
135C4×C59—21no—
136C4×C59—41no—
137C4×C59—41no—
138C4×C59H(5,𝔽2)12yes—
139Dic59—21no—
140Dic59—41no—
141Dic59—41no—
142Dic59C2×SD1612yes—
143Dic59C2×SD1612yes—
144Dic59AGL(1,ℤ/8)12yes—
145#3208—591nomaximal, Sylow
146C2×C59—22no—
147C2×C59—82no—
148C2×C59#53511yescharacteristic
149C2×C59C22×SD1611yescharacteristic
150C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
151D59—82no—
152#533—591no—
153C2×#127—591no—
154#492—592no—
155#486—592no—
156#460—592no—
157#460—592no—
158#450—591no—
159#450—591no—
160#450—591no—
161#450D5911yescharacteristic, complemented
162#431—591no—
163C59#320811yescharacteristic, complemented, Sylow
164C22×D4—591no—
165C22×D4—1181no—
166C22×D4—1181no—
167C22×D4C2×D5911yescharacteristic
168C2×SD16—592no—
169C2×SD16—592no—
170C2×SD16—592no—
171C2×SD16C2×D5912yescomplemented
172C4⋊Q8—591no—
173C4⋊Q8—591no—
174#127—592no—
175#127—592no—
176#127—1181no—
177#127—1181no—
178#125—1181no—
179C22⋊Q8—591no—
180C22⋊Q8—1181no—
181C22⋊Q8C2×D5911yescharacteristic
182C4×Q8—591no—
183C4×Q8—591no—
184C4×Q8—1181no—
185#117—1181no—
186C2×C42—591no—
187C4⋊C8—592no—
188#102—592no—
189#102—592no—
190#102—592no—
191#102C2×D5912yescomplemented
192C22⋊C8—591no—
193C22⋊C8C2×D5911yescharacteristic
194C4×C8—592no—
195C24—21no—
196C24—1181no—
197C2×Q8—591no—
198C2×Q8—1181no—
199C2×Q8C22×D5911yescharacteristic
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—592no—
203C2×D4—1182no—
204C2×D4—1182no—
205C2×D4—1182no—
206C2×D4—1182no—
207C2×D4—1181no—
208C2×D4—2362no—
209C2×D4—2364no—
210C2×D4—2364no—
211C2×D4C22×D5912yes—
212C22×C4—591no—
213C22×C4—591no—
214C22×C4C22×D5911yescharacteristic
215SD16—42no—
216SD16—2362no—
217SD16—2362no—
218SD16—2362no—
219C2×C8—592no—
220C2×C8C22×D5912yes—
221C4⋊C4—21no—
222C4⋊C4—591no—
223C4⋊C4—1181no—
224C4⋊C4—1181no—
225C4⋊C4—1181no—
226C4⋊C4—1181no—
227C4⋊C4—1181no—
228C4⋊C4—1181no—
229C4⋊C4C22×D5911yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—1181no—
232C42—591no—
233C42—591no—
234C42—591no—
235C42—591no—
236C42—1181no—
237C42—1181no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—1182no—
243C23—2362no—
244C23—2362no—
245C23—2362no—
246C23D4×D5911yescharacteristic
247Q8—22no—
248Q8—1182no—
249Q8—2362no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—1182no—
254D4—1182no—
255D4—2362no—
256D4—2364no—
257D4—2364no—
258D4—2364no—
259D4—2364no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—594no—
264C2×C4—591no—
265C2×C4—591no—
266C2×C4—591no—
267C2×C4—591no—
268C2×C4—1182no—
269C2×C4—1181no—
270C2×C4—1181no—
271C2×C4D4×D5911yescharacteristic
272C2×C4C23×D5911yescharacteristic
273C8—22no—
274C8—1182no—
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—84no—
281C22—4724no—
282C22—2362no—
283C22—2362no—
284C22—2362no—
285C22SD16×D5912yes—
286C22C2×D4×D5911yescharacteristic
287C4—21no—
288C4—41no—
289C4—41no—
290C4—592no—
291C4—592no—
292C4—592no—
293C4—1181no—
294C4—2361no—
295C4—2361no—
296C4#1427377212yes—
297C2—22no—
298C2—82no—
299C2—4722no—
300C2#1683796711yescharacteristic
301C2C2×SD16×D5911yescharacteristic
302C2AGL(1,ℤ/8)×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3208
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. #16837773C2semidirect
4. #16837747C2semidirect
5. #16837601C2semidirect
6. #16837600C2semidirect
7. #16837555C2semidirect
8. #16837553C2semidirect
9. #16837552C2semidirect
10. #16837452C2semidirect
11. C59×#450C2semidirect
15. C2×#11745660C22semidirect
16. C2×#11745670C22semidirect
17. C2×#11745671C22semidirect
18. #14273726C22semidirect
19. #14273725C22semidirect
27. Q8×Dic59C22semidirect
29. #14273692C22semidirect
32. Dic59⋊C8C22semidirect
33. #14273649C22semidirect
34. #14273648C22semidirect
35. #14273647C22semidirect
37. C8×Dic59C22semidirect
39. C2×SD16×C59C22semidirect
41. #102×C59C22semidirect
111. C2×Dic59SD16semidirect
161. #450D59semidirect
163. C59#3208semidirect
171. C2×SD16C2×D59semidirect
191. #102C2×D59semidirect