Finite Group Databasebeta

← #21433336

Subgroups — #21433336

Order 7552.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11745685C211yesmaximal, characteristic, complemented
2C2×#14273757C211yesmaximal, characteristic, complemented
3C2×#14273752C211yesmaximal, characteristic, complemented
4#16837562C214yesmaximal, complemented
5#16837545C214yesmaximal, complemented
6#16837410C211yesmaximal, characteristic, complemented
7#16837406C211yesmaximal, characteristic, complemented
8#16837366C211yesmaximal, characteristic, complemented
9C59×#525C211yesmaximal, characteristic, complemented
10C2×#11745685—22no—
11C2×#11745685—24no—
12C2×#11745685—24no—
13C2×#11745685C2214yes—
14C22×Dic118C2211yescharacteristic
15C22×D236C2211yescharacteristic
16C22×C22⋊D59—21no—
17C22×C4×D59—21no—
18#14273757—24no—
19#14273752—24no—
20C2×#11745672C2212yescomplemented
21C2×#11745670C2212yescomplemented
22C2×#11745671C2212yescomplemented
23C2×#11745669C2212yescomplemented
24C2×#11745664C2211yescharacteristic
25C2×#11745664C2211yescharacteristic
26#14273650C2214yescomplemented
27#14273647C2214yescomplemented
28C22⋊(C59⋊C8)C2214yescomplemented
29C2×#41×C59C2211yescharacteristic
30C23×C4×C59C2211yescharacteristic
31#123×C59—21no—
32C22⋊Q8×C59C2212yescomplemented
33#121×C59C2212yescomplemented
34#120×C59—21no—
35C4×D4×C59—22no—
36#117×C59C2211yescharacteristic
37C24×C59D411yescharacteristic
38C23×D59—21no—
39#41×C59—22no—
40#41×C59—22no—
41C2×Q8×C59C2311yescharacteristic
42C2×D4×C59—22no—
43C2×D4×C59—21no—
44C2×D4×C59C2311yescharacteristic
45C22×C4×C59—22no—
46C22×C4×C59—22no—
47C22×C4×C59—21no—
48C22×C4×C59D412yes—
49C22×C4×C59D411yescharacteristic
50C22×C4×C59C2312yes—
51C22×C4×C59C2311yescharacteristic
52C2×C22⋊D59—28no—
53C2×C22⋊D59—24no—
54C22×Dic59—21no—
55#11745685—216no—
56#11745685—28no—
57#11745685—28no—
58C2×D236—21no—
59C2×D236—22no—
60C2×D236D414yescomplemented
61C2×D236C2312yes—
62C2×C4×D59—28no—
63C2×C4×D59—24no—
64C2×Dic118—21no—
65C2×Dic118—22no—
66C2×Dic118D414yescomplemented
67C2×Dic118C2312yes—
68C4⋊C4×C59—22no—
69C4⋊C4×C59C2312yes—
70C22⋊C4×C59—21no—
71C22⋊C4×C59—22no—
72C22⋊C4×C59—22no—
73C42×C59—21no—
74#11745672—24no—
75#11745671—24no—
76#11745670—24no—
77#11745669—24no—
78#11745664—22no—
79#11745664—22no—
80C2×C59⋊C8C2312yes—
81C2×C59⋊C8C2312yes—
82C23×C59—22no—
83C23×C59—21no—
84C23×C59—42no—
85C23×C59C2×D412yes—
86C23×C59C2×D411yescharacteristic
87C22×D59—24no—
88C22×D59—22no—
89C22×D59—44no—
90Q8×C59—21no—
91Q8×C59—21no—
92D4×C59—21no—
93D4×C59—21no—
94D4×C59—44no—
95D4×C59—42no—
96C2×C4×C59—22no—
97C2×C4×C59—24no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59—42no—
103C2×C4×C59C2×D412yes—
104C2×C4×C59C2×D412yes—
105C2×C4×C59C2×D412yes—
106C2×C4×C59C2×D411yescharacteristic
107C2×C4×C59C2411yescharacteristic
108C22⋊D59—216no—
109C2×Dic59—24no—
110C2×Dic59—22no—
111D236—24no—
112D236—22no—
113D236C2×D414yes—
114C4×D59—216no—
115Dic118—24no—
116Dic118—22no—
117Dic118C2×D414yes—
118C59⋊C8—22no—
119C59⋊C8—22no—
120C22×C59—22no—
121C22×C59—22no—
122C22×C59—22no—
123C22×C59—22no—
124C22×C59—44no—
125C22×C59—41no—
126C22×C59—42no—
127C22×C59#12012yes—
128C22×C59C22×D411yescharacteristic
129C2×D59—24no—
130C2×D59—48no—
131C2×D59—44no—
132C4×C59—22no—
133C4×C59—41no—
134C4×C59—42no—
135C4×C59#12012yes—
136C4×C59C22×D411yescharacteristic
137C4×C59C22×D411yescharacteristic
138Dic59—24no—
139#3005—591nomaximal, Sylow
140C2×C59—22no—
141C2×C59—42no—
142C2×C59—81no—
143C2×C59C2×#12011yescharacteristic
144C2×C59#57512yes—
145D59—44no—
146C22×#41—591no—
147C2×#137—591no—
148C2×AGL(1,ℤ/8)—591no—
149#525D5911yescharacteristic, complemented
150#452—594no—
151#449—594no—
152#419—591no—
153#418—591no—
154#407—591no—
155C59#300511yescharacteristic, complemented, Sylow
156C2×#41—594no—
157C2×#41—1184no—
158C2×#41—1184no—
159C2×#41—1182no—
160C2×#41C2×D5911yescharacteristic
161C22×Q8—591no—
162C22×D4—591no—
163C22×D4—1181no—
164C23×C4—1181no—
165C23×C4C2×D5911yescharacteristic
166#137—1184no—
167AGL(1,ℤ/8)—1184no—
168C2×Q16—592no—
169C2×SD16—592no—
170C2×SD16—592no—
171C2×D8—592no—
172C2×M16—591no—
173C2×M16—591no—
174#123—21no—
175C22⋊Q8C2×D5912yescomplemented
176#121C2×D5912yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D5911yescharacteristic
180#103—594no—
181#102—594no—
182C22⋊C8—594no—
183C24—1181no—
184C24C22⋊D5911yescharacteristic
185#41—22no—
186#41—22no—
187#41—1188no—
188#41—1188no—
189#41—11816no—
190C2×Q8—594no—
191C2×Q8—592no—
192C2×Q8—1182no—
193C2×Q8—1181no—
194C2×Q8C22×D5911yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—594no—
198C2×D4—592no—
199C2×D4—1188no—
200C2×D4—1184no—
201C2×D4—1182no—
202C2×D4—1181no—
203C2×D4C22×D5911yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—1188no—
208C22×C4—1181no—
209C22×C4—1184no—
210C22×C4C22⋊D5912yes—
211C22×C4C22⋊D5911yescharacteristic
212C22×C4C22×D5912yes—
213C22×C4C22×D5911yescharacteristic
214Q16—1184no—
215SD16—1184no—
216SD16—1184no—
217D8—1184no—
218M16—1182no—
219M16—1182no—
220C2×C8—592no—
221C2×C8—592no—
222C4⋊C4—22no—
223C4⋊C4C22×D5912yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—1184no—
232C23—1182no—
233C23—2364no—
234C23C2×C22⋊D5912yes—
235C23C2×C22⋊D5911yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—594no—
239Q8—1184no—
240Q8—1182no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—594no—
246D4—11816no—
247D4—1184no—
248D4—1182no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—11816no—
257C2×C4—1184no—
258C2×C4—1182no—
259C2×C4D4×D5912yes—
260C2×C4D4×D5912yes—
261C2×C4C2×C22⋊D5912yes—
262C2×C4C2×C22⋊D5911yescharacteristic
263C2×C4C23×D5911yescharacteristic
264C8—1182no—
265C8—1182no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—42no—
272C22—41no—
273C22—1184no—
274C22—2368no—
275C22—2364no—
276C22#1427369512yes—
277C22C22×C22⋊D5911yescharacteristic
278C4—22no—
279C4—42no—
280C4—41no—
281C4—1184no—
282C4#1427369512yes—
283C4C2×D4×D5911yescharacteristic
284C4C2×D4×D5911yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—2364no—
289C2C2×#1427369511yescharacteristic
290C2#1683820012yes—
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#525C2
socleC22×C59C22×D4
hypercenter#525D59
hypocenterC59#3005
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#11745685C2semidirect
2. C2×#14273757C2semidirect
3. C2×#14273752C2semidirect
4. #16837562C2semidirect
5. #16837545C2semidirect
6. #16837410C2semidirect
7. #16837406C2semidirect
8. #16837366C2semidirect
9. C59×#525C2semidirect
20. C2×#11745672C22semidirect
21. C2×#11745670C22semidirect
22. C2×#11745671C22semidirect
23. C2×#11745669C22semidirect
26. #14273650C22semidirect
27. #14273647C22semidirect
28. C22⋊(C59⋊C8)C22semidirect
32. C22⋊Q8×C59C22semidirect
33. #121×C59C22semidirect
60. C2×D236D4semidirect
66. C2×Dic118D4semidirect
149. #525D59semidirect
155. C59#3005semidirect
175. C22⋊Q8C2×D59semidirect
176. #121C2×D59semidirect