Finite Group Databasebeta

← #21433445

Subgroups — #21433445

Order 7552.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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#16838200C212yesmaximal, complemented
2#16837948C211yesmaximal, characteristic, complemented
3#16837585C212yesmaximal, complemented
4#16837580C212yesmaximal, complemented
5#16837575C212yesmaximal, complemented
6#16837570C212yesmaximal, complemented
7C2×#14273654C212yesmaximal, complemented
8#16837386C211yesmaximal, characteristic, complemented
9C59×#585C211yesmaximal, characteristic, complemented
10C2×#11745685C2211yescharacteristic, complemented
11#14273758—22no—
12#14273757C2214yescomplemented
13#14273753—22no—
14#14273752C2214yescomplemented
15#14273747—24no—
16C2×#11745664C2212yescomplemented
17#14273725C2211yescharacteristic, complemented
18#14273723C2211yescharacteristic, complemented
19#14273716C2211yescharacteristic, complemented
20#14273714C2211yescharacteristic, complemented
21#14273699—22no—
22C4×C22⋊D59—22no—
23C2×C4×Dic59C2211yescharacteristic, complemented
24#14273654C2214yescomplemented
25#14273654C2214yescomplemented
26#14273646C2212yescomplemented
27#14273644C2212yescomplemented
282−1+4×C59—22no—
292−1+4×C59C2212yes—
30H(5,𝔽2)×C59—22no—
31H(5,𝔽2)×C59—22no—
32H(5,𝔽2)×C59C2212yes—
33C2×#41×C59—24no—
34C2×#41×C59—22no—
35C2×#41×C59C2212yes—
36C2×#41×C59C2211yescharacteristic
37#41×C59—24no—
38#41×C59—28no—
39#41×C59—24no—
40#41×C59—24no—
41#41×C59—24no—
42#41×C59—22no—
43#41×C59—22no—
44#41×C59—24no—
45#41×C59—22no—
46#41×C59—22no—
47#41×C59D414yescomplemented
48#41×C59C2314yes—
49C2×Q8×C59—22no—
50C2×Q8×C59—24no—
51C2×Q8×C59D412yescomplemented
52C2×Q8×C59C2311yescharacteristic
53C2×D4×C59—28no—
54C2×D4×C59—22no—
55C2×D4×C59—24no—
56C2×D4×C59—22no—
57C2×D4×C59—24no—
58C2×D4×C59D412yes—
59C2×D4×C59D412yescomplemented
60C2×D4×C59C2311yescharacteristic
61C22×C4×C59—24no—
62C22×C4×C59—22no—
63C22×C4×C59D412yes—
64C22×C4×C59C2311yescharacteristic
65C2×C22⋊D59—22no—
66C2×C22⋊D59—21no—
67C22×Dic59—21no—
68#11745685—22no—
69#11745685—22no—
70C2×D236C2311yescharacteristic
71C2×C4×D59—21no—
72C2×Dic118C2311yescharacteristic
73C22⋊Dic59—22no—
74#11745672—24no—
75#11745671—24no—
76#11745670—24no—
77#11745669—24no—
78(C2×D59)⋊C4—22no—
79Dic59⋊C4—22no—
80C4×Dic59—21no—
81C4×Dic59C2311yescharacteristic
82C4×Dic59C2311yescharacteristic
83#11745664—22no—
84#11745664C2314yes—
85C2×C59⋊C8—22no—
86C23×C59—24no—
87C23×C59—22no—
88C23×C59C2×D412yes—
89C23×C59C2×D411yescharacteristic
90C22×D59—21no—
91Q8×C59—24no—
92Q8×C59—22no—
93Q8×C59—22no—
94Q8×C59C2×D414yes—
95D4×C59—28no—
96D4×C59—24no—
97D4×C59—24no—
98D4×C59—24no—
99D4×C59—24no—
100D4×C59—22no—
101D4×C59—22no—
102D4×C59C2×D414yes—
103C2×C4×C59—24no—
104C2×C4×C59—28no—
105C2×C4×C59—22no—
106C2×C4×C59—24no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—24no—
110C2×C4×C59C2×D412yes—
111C2×C4×C59C2×D411yescharacteristic
112C2×C4×C59C2×D412yes—
113C2×C4×C59C2×D412yes—
114C2×C4×C59C2411yescharacteristic
115C22⋊D59—44no—
116C22⋊D59—42no—
117C2×Dic59—21no—
118C2×Dic59—21no—
119C2×Dic59—21no—
120C2×Dic59—42no—
121D236—22no—
122C4×D59—41no—
123C4×D59—41no—
124Dic118—22no—
125C59⋊C8—24no—
126C22×C59—22no—
127C22×C59—24no—
128C22×C59—41no—
129C22×C59—42no—
130C22×C59—82no—
131C22×C59—84no—
132C22×C59#12012yes—
133C22×C59C22×D411yescharacteristic
134C2×D59—42no—
135C2×D59—41no—
136C4×C59—24no—
137C4×C59—22no—
138C4×C59#12011yescharacteristic
139C4×C59#12011yescharacteristic
140C4×C59C22×D412yes—
141Dic59—22no—
142Dic59—41no—
143Dic59—41no—
144#3018—591nomaximal, Sylow
145C2×C59—21no—
146C2×C59—22no—
147C2×C59—42no—
148C2×C59—44no—
149C2×C59C2×#12011yescharacteristic
150D59—81no—
151#585D5911yescharacteristic, complemented
152#575—592no—
153#532—591no—
154#456—592no—
155#455—592no—
156#454—592no—
157#453—592no—
158C2×#104—592no—
159#413—591no—
160C59#301811yescharacteristic, complemented, Sylow
1612−1+4—22no—
1622−1+4C2×D5912yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D5912yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—591no—
169C2×#41C2×D5912yes—
170C2×#41C2×D5911yescharacteristic
171#137—594no—
172#137—1182no—
173AGL(1,ℤ/8)—594no—
174AGL(1,ℤ/8)—1182no—
175#135—1184no—
176C2×M16—592no—
177C4⋊Q8—591no—
178#127—591no—
179#124—591no—
180#124—591no—
181#121—1182no—
182C4×D4—1182no—
183C2×C42—591no—
184#104—594no—
185#104—594no—
186#101—592no—
187#100—592no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—1182no—
199#41—1182no—
200#41C22⋊D5914yescomplemented
201#41C22×D5914yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—591no—
205C2×Q8C22⋊D5912yescomplemented
206C2×Q8C22×D5911yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—591no—
213C2×D4—1182no—
214C2×D4—1181no—
215C2×D4C22⋊D5912yes—
216C2×D4C22⋊D5912yescomplemented
217C2×D4C22×D5911yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—1181no—
221C22×C4—1181no—
222C22×C4C22⋊D5912yes—
223C22×C4C22×D5911yescharacteristic
224Q16—1184no—
225SD16—1184no—
226SD16—1184no—
227D8—1184no—
228M16—594no—
229M16—1182no—
230C2×C8—1182no—
231C4⋊C4—1182no—
232C22⋊C4—1182no—
233C22⋊C4—1182no—
234C42—591no—
235C42—591no—
236C42—1181no—
237C23—24no—
238C23—22no—
239C23—1181no—
240C23C2×C22⋊D5912yes—
241C23C2×C22⋊D5911yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—1182no—
246Q8C2×C22⋊D5914yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—1182no—
255D4—2364no—
256D4—2362no—
257D4C2×C22⋊D5914yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—1181no—
266C2×C4—1181no—
267C2×C4—1181no—
268C2×C4—2362no—
269C2×C4—2361no—
270C2×C4—2361no—
271C2×C4C2×C22⋊D5912yes—
272C2×C4C2×C22⋊D5911yescharacteristic
273C2×C4C2×C22⋊D5912yes—
274C2×C4C2×C22⋊D5912yes—
275C2×C4C23×D5911yescharacteristic
276C8—1184no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—2362no—
284C22—2361no—
285C22#1427369612yes—
286C22C22×C22⋊D5911yescharacteristic
287C4—24no—
288C4—22no—
289C4—1182no—
290C4—2361no—
291C4—2361no—
292C4#1427369611yescharacteristic
293C4#1427369611yescharacteristic
294C4C22×C22⋊D5912yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—4721no—
300C2C2×#1427369611yescharacteristic
subgroupG/H
centerC4#14273696
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#585C2
socleC2×C59C2×#120
hypercenter#585D59
hypocenterC59#3018
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #16838200C2semidirect
2. #16837948C2semidirect
3. #16837585C2semidirect
4. #16837580C2semidirect
5. #16837575C2semidirect
6. #16837570C2semidirect
7. C2×#14273654C2semidirect
8. #16837386C2semidirect
9. C59×#585C2semidirect
10. C2×#11745685C22semidirect
12. #14273757C22semidirect
14. #14273752C22semidirect
16. C2×#11745664C22semidirect
17. #14273725C22semidirect
18. #14273723C22semidirect
19. #14273716C22semidirect
20. #14273714C22semidirect
23. C2×C4×Dic59C22semidirect
24. #14273654C22semidirect
25. #14273654C22semidirect
26. #14273646C22semidirect
27. #14273644C22semidirect
47. #41×C59D4semidirect
51. C2×Q8×C59D4semidirect
59. C2×D4×C59D4semidirect
151. #585D59semidirect
160. C59#3018semidirect
200. #41C22⋊D59semidirect
205. C2×Q8C22⋊D59semidirect
216. C2×D4C22⋊D59semidirect