Finite Group Databasebeta

← #21434903

Subgroups — #21434903

Order 7552.

462 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#14273706C211yesmaximal, characteristic, complemented
2C2×#14273701C211yesmaximal, characteristic
3#16837738C212yesmaximal, complemented
4#16837716C212yesmaximal, complemented
5#16837648C214yesmaximal, complemented
6#16837555C212yesmaximal, complemented
7#16837543C212yesmaximal, complemented
8C59×#407C211yesmaximal, characteristic, complemented
9C22×Dic118C2211yescharacteristic, complemented
10C22×D236C2211yescharacteristic
11C22×C22⋊D59—21no—
12C2×#11745660C2214yescomplemented
13#14273706—21no—
14#14273706—21no—
15#14273706C2212yes—
16#14273706C2212yescomplemented
17#14273701—21no—
18#14273701—21no—
19#14273701C2212yes—
20#14273701C2212yes—
21C2×C4⋊Dic59C2211yescharacteristic
22C2×C4⋊Dic59C2211yescharacteristic, complemented
23C2×Dic59⋊C4—21no—
24C2×C22⋊Dic59—21no—
25C2×(C2×D59)⋊C4—21no—
26#14273661C2214yescomplemented
27#14273652C2214yescomplemented
28#14273649C2214yescomplemented
29C23×C4×C59C2211yescharacteristic
30C2×M16×C59C2212yescomplemented
31C22⋊C8×C59C2212yes—
32C22⋊C8×C59C2212yescomplemented
33C24×C59D411yescharacteristic
34C23×D59—21no—
35C22×C4×C59—42no—
36C22×C4×C59D411yescharacteristic
37C22×C4×C59D411yescharacteristic
38C22×C4×C59D411yescharacteristic
39C22×C4×C59C2311yescharacteristic
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2311yescharacteristic
42C2×C22⋊D59—22no—
43C2×C22⋊D59—22no—
44C2×C22⋊D59—44no—
45C22×Dic59—21no—
46C22×Dic59—21no—
47C22×Dic59—21no—
48C2×D236—21no—
49C2×D236—42no—
50C2×D236C2312yes—
51C2×Dic118—21no—
52C2×Dic118—42no—
53C2×Dic118C2312yes—
54M16×C59—42no—
55C2×C8×C59C2314yes—
56C22⋊Dic59—22no—
57C22⋊Dic59—22no—
58(C2×D59)⋊C4—22no—
59(C2×D59)⋊C4—22no—
60C4⋊Dic59—21no—
61C4⋊Dic59—21no—
62C4⋊Dic59C2312yes—
63C4⋊Dic59C2312yes—
64Dic59⋊C4—22no—
65Dic59⋊C4—22no—
66#11745660—44no—
67C23×C59—22no—
68C23×C59—22no—
69C23×C59—41no—
70C23×C59C2×D411yescharacteristic
71C23×C59C2×D411yescharacteristic
72C23×C59C2×D411yescharacteristic
73C22×D59—22no—
74C22×D59—42no—
75C22×D59—42no—
76C22×D59—42no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—41no—
81C2×C4×C59—41no—
82C2×C4×C59—41no—
83C2×C4×C59C2×D411yescharacteristic
84C2×C4×C59C2×D411yescharacteristic
85C2×C4×C59C2×D411yescharacteristic
86C2×C4×C59C2411yescharacteristic
87C22⋊D59—44no—
88C22⋊D59—44no—
89C2×Dic59—22no—
90C2×Dic59—22no—
91C2×Dic59—22no—
92C2×Dic59—42no—
93C2×Dic59—42no—
94C2×Dic59—42no—
95D236—24no—
96D236—42no—
97Dic118—24no—
98Dic118—42no—
99C8×C59—24no—
100C22×C59—21no—
101C22×C59—21no—
102C22×C59—21no—
103C22×C59—21no—
104C22×C59—21no—
105C22×C59—21no—
106C22×C59—21no—
107C22×C59—44no—
108C22×C59AGL(1,ℤ/8)12yes—
109C22×C59#13712yes—
110C22×C59C22×D411yescharacteristic
111C2×D59—42no—
112C2×D59—42no—
113C2×D59—42no—
114C2×D59—84no—
115C4×C59—21no—
116C4×C59—21no—
117C4×C59—21no—
118C4×C59H(5,𝔽2)12yes—
119Dic59—42no—
120Dic59—42no—
121Dic59—42no—
122#3193—591nomaximal, Sylow
123C2×C59—22no—
124C2×C59—22no—
125C2×C59—41no—
126C2×C59#53411yescharacteristic
127C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C59C2×#13711yescharacteristic
129D59—82no—
130C2×C22⋊Q8—591no—
131C2×#121—591no—
132#484—592no—
133#480—592no—
134#468—594no—
135#450—592no—
136#448—592no—
137#407D5911yescharacteristic, complemented
138C59#319311yescharacteristic, complemented, Sylow
139C22×Q8—591no—
140C22×D4—591no—
141C22×D4—1181no—
142C23×C4C2×D5911yescharacteristic
143C2×SD16—594no—
144C2×M16C2×D5912yescomplemented
145C22⋊Q8—592no—
146C22⋊Q8—592no—
147C22⋊Q8—1181no—
148C22⋊Q8—1181no—
149#121—592no—
150#121—592no—
151#121—1181no—
152#121—1181no—
153C2×C4⋊C4—591no—
154C2×C4⋊C4—591no—
155C2×C4⋊C4—1181no—
156C2×C22⋊C4—1181no—
157C2×C22⋊C4—1181no—
158#107—594no—
159#103—594no—
160#102—594no—
161C22⋊C8C2×D5912yes—
162C22⋊C8C2×D5912yescomplemented
163C24—1181no—
164C24D23611yescharacteristic
165C2×Q8—592no—
166C2×Q8—1181no—
167C2×Q8—2362no—
168C2×D4—592no—
169C2×D4—1182no—
170C2×D4—1182no—
171C2×D4—1181no—
172C2×D4—2362no—
173C2×D4—2364no—
174C22×C4—42no—
175C22×C4—1181no—
176C22×C4—1181no—
177C22×C4—1181no—
178C22×C4D23611yescharacteristic
179C22×C4D23611yescharacteristic
180C22×C4D23611yescharacteristic
181C22×C4C22×D5911yescharacteristic
182C22×C4C22×D5911yescharacteristic
183C22×C4C22×D5911yescharacteristic
184SD16—2364no—
185M16—42no—
186C2×C8C22×D5914yes—
187C4⋊C4—592no—
188C4⋊C4—592no—
189C4⋊C4—1181no—
190C4⋊C4—1181no—
191C4⋊C4—1182no—
192C4⋊C4—1182no—
193C22⋊C4—1182no—
194C22⋊C4—1182no—
195C22⋊C4—1182no—
196C22⋊C4—1182no—
197C23—22no—
198C23—22no—
199C23—41no—
200C23—1182no—
201C23—2362no—
202C23—2362no—
203C23—2362no—
204C23C2×D23611yescharacteristic
205C23C2×D23611yescharacteristic
206C23C2×D23611yescharacteristic
207Q8—1184no—
208Q8—2362no—
209D4—1184no—
210D4—2362no—
211D4—2364no—
212D4—2364no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—1182no—
220C2×C4—1182no—
221C2×C4—1182no—
222C2×C4—2362no—
223C2×C4—2362no—
224C2×C4—2362no—
225C2×C4C2×D23611yescharacteristic
226C2×C4C2×D23611yescharacteristic
227C2×C4C2×D23611yescharacteristic
228C2×C4C23×D5911yescharacteristic
229C8—24no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—44no—
238C22—4724no—
239C22—2362no—
240C22—2362no—
241C22—2362no—
242C22#1427375512yes—
243C22#1427376012yes—
244C22C22×D23611yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—2362no—
249C4—2362no—
250C4—2362no—
251C4#1427377212yes—
252C2—22no—
253C2—22no—
254C2—41no—
255C2—4722no—
256C2#1683796411yescharacteristic
257C2C2×#1427375511yescharacteristic
258C2C2×#1427376011yescharacteristic
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#407C2
socleC22×C59C22×D4
hypercenter#407D59
hypocenterC59#3193
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C2×#14273706C2semidirect
3. #16837738C2semidirect
4. #16837716C2semidirect
5. #16837648C2semidirect
6. #16837555C2semidirect
7. #16837543C2semidirect
8. C59×#407C2semidirect
9. C22×Dic118C22semidirect
12. C2×#11745660C22semidirect
16. #14273706C22semidirect
22. C2×C4⋊Dic59C22semidirect
26. #14273661C22semidirect
27. #14273652C22semidirect
28. #14273649C22semidirect
30. C2×M16×C59C22semidirect
32. C22⋊C8×C59C22semidirect
137. #407D59semidirect
138. C59#3193semidirect
144. C2×M16C2×D59semidirect
162. C22⋊C8C2×D59semidirect