Finite Group Databasebeta

← #21434958

Subgroups — #21434958

Order 7552.

442 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 109 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#16837968C212yesmaximal, complemented
2#16837728C212yesmaximal, complemented
3#16837725C212yesmaximal, complemented
4#16837655C211yesmaximal, characteristic, complemented
5#16837636C212yesmaximal, complemented
6#16837537C212yesmaximal, complemented
7#16837534C212yesmaximal, complemented
8#16837370C211yesmaximal, characteristic, complemented
9C59×#469C211yesmaximal, characteristic, complemented
10C2×#11745685C2211yescharacteristic
11C2×D4×D59C2212yescomplemented
12C2×D472C2211yescharacteristic
13C2×#11745669C2212yes—
14C22×C59⋊C8C2211yescharacteristic, complemented
15#14273723—22no—
16#14273713—22no—
17#14273702C2212yes—
18#14273701C2212yescomplemented
19#14273698C2212yes—
20#14273697—22no—
21#14273695—22no—
22#14273685C2212yescomplemented
23#14273660C2212yescomplemented
24#14273659C2211yescharacteristic, complemented
25#14273649C2212yescomplemented
26#14273648C2212yescomplemented
27#14273647C2212yescomplemented
28(C2×D59)⋊C8C2212yescomplemented
29(C2×D59)⋊C8C2212yescomplemented
30C2×D8×C59C2211yescharacteristic
31C2×M16×C59C2211yescharacteristic, complemented
32#121×C59C2212yescomplemented
33#106×C59C2211yescharacteristic, complemented
34#102×C59C2212yescomplemented
35C23×D59—22no—
36C2×D4×C59—22no—
37C2×D4×C59C2312yes—
38C22×C4×C59C2311yescharacteristic
39C2×C22⋊D59—22no—
40C2×C22⋊D59—22no—
41C2×C22⋊D59—22no—
42C2×C22⋊D59D412yes—
43D4×D59—42no—
44D4×D59—42no—
45#11745685—41no—
46#11745685—41no—
47C2×D236—22no—
48C2×D236D411yescharacteristic
49C2×D236C2312yes—
50C2×C4×D59C2312yes—
51C2×Dic118D411yescharacteristic
52D8×C59—41no—
53M16×C59—41no—
54C2×C8×C59C2311yescharacteristic
55C2×C8×C59C2311yescharacteristic
56C4⋊C4×C59C2312yes—
57C22⋊C4×C59—22no—
58C22⋊Dic59—22no—
59#11745669—42no—
60(C2×D59)⋊C4—22no—
61(C2×D59)⋊C4—22no—
62(C2×D59)⋊C4—22no—
63C4⋊Dic59C2312yes—
64C4×Dic59—22no—
65C2×C59⋊C8—41no—
66C2×C59⋊C8C2312yes—
67D472—41no—
68C23×C59—22no—
69C23×C59C2×D411yescharacteristic
70C22×D59—22no—
71C22×D59—44no—
72C22×D59—82no—
73C22×D59—82no—
74C22×D59C2×D412yes—
75D4×C59—22no—
76D4×C59—22no—
77D4×C59—82no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—22no—
81C2×C4×C59C2×D411yescharacteristic
82C2×C4×C59C2411yescharacteristic
83C22⋊D59—42no—
84C22⋊D59—44no—
85C22⋊D59—82no—
86C22⋊D59—82no—
87C2×Dic59—22no—
88C2×Dic59C2×D412yes—
89D236—22no—
90D236—22no—
91D236—41no—
92D236—82no—
93C4×D59—22no—
94C4×D59—22no—
95Dic118—41no—
96C8×C59—21no—
97C8×C59—21no—
98C59⋊C8—22no—
99C22×C59—21no—
100C22×C59—22no—
101C22×C59—42no—
102C22×C59—42no—
103C22×C59—42no—
104C22×C59C22×D411yescharacteristic
105C2×D59—22no—
106C2×D59—22no—
107C2×D59—22no—
108C2×D59—42no—
109C2×D59—42no—
110C2×D59—42no—
111C2×D59—88no—
112C4×C59—21no—
113C4×C59—42no—
114C4×C59H(5,𝔽2)11yescharacteristic
115C4×C59H(5,𝔽2)11yescharacteristic
116Dic59—22no—
117Dic59—42no—
118#3199—591nomaximal, Sylow
119C2×C59—41no—
120C2×C59—82no—
121C2×C59#53411yescharacteristic
122C2×C59#57612yes—
123D59—42no—
124D59—82no—
125#535—592no—
126#482—592no—
127#482—592no—
128#469—591no—
129#469D5911yescharacteristic, complemented
130#466—592no—
131#447—592no—
132#447—592no—
133#408—591no—
134C59#319911yescharacteristic, complemented, Sylow
135C2×#41—591no—
136C22×D4—592no—
137C2×D8—592no—
138C2×D8—591no—
139C2×D8C2×D5911yescharacteristic
140C2×M16C2×D5911yescharacteristic, complemented
141C22×C8—591no—
142#127—1182no—
143#124—1182no—
144#121—592no—
145#121—592no—
146#121—592no—
147#121C2×D5912yescomplemented
148#120—1182no—
149#120—1182no—
150#117—592no—
151#107—592no—
152#106—591no—
153#106C2×D5911yescharacteristic, complemented
154#102—592no—
155#102—592no—
156#102—592no—
157#102C2×D5912yescomplemented
158C22⋊C8—592no—
159C22⋊C8—592no—
160C24—1182no—
161#41—2361no—
162#41—2361no—
163C2×Q8—591no—
164C2×D4—22no—
165C2×D4—592no—
166C2×D4—591no—
167C2×D4—592no—
168C2×D4—1182no—
169C2×D4—1182no—
170C2×D4—1182no—
171C2×D4—1182no—
172C2×D4—2362no—
173C2×D4—2362no—
174C2×D4C22×D5912yes—
175C22×C4—592no—
176C22×C4C22×D5911yescharacteristic
177D8—41no—
178D8—2361no—
179D8—2362no—
180M16—41no—
181C2×C8—592no—
182C2×C8—2361no—
183C2×C8C22×D5911yescharacteristic
184C2×C8C22×D5911yescharacteristic
185C4⋊C4—592no—
186C4⋊C4C22×D5912yes—
187C22⋊C4—22no—
188C22⋊C4—1182no—
189C22⋊C4—1182no—
190C22⋊C4—1182no—
191C22⋊C4—1182no—
192C42—1182no—
193C23—22no—
194C23—592no—
195C23—1182no—
196C23—4722no—
197C23—4722no—
198C23—2364no—
199C23D4×D5911yescharacteristic
200Q8—2361no—
201D4—22no—
202D4—22no—
203D4—82no—
204D4—1182no—
205D4—1182no—
206D4—4722no—
207D4—4722no—
208D4—4722no—
209D4—2361no—
210D4—2362no—
211D4—2364no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—592no—
216C2×C4—1182no—
217C2×C4—1182no—
218C2×C4—1182no—
219C2×C4D4×D5911yescharacteristic
220C2×C4C23×D5911yescharacteristic
221C8—21no—
222C8—21no—
223C8—1182no—
224C22—22no—
225C22—21no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—1182no—
230C22—1182no—
231C22—1182no—
232C22—4728no—
233C22—2362no—
234C22—2362no—
235C22—2362no—
236C22C2×D4×D5911yescharacteristic
237C4—21no—
238C4—42no—
239C4—1182no—
240C4—2362no—
241C4#1427377211yescharacteristic
242C4#1427377311yescharacteristic
243C2—41no—
244C2—82no—
245C2—4722no—
246C2—2362no—
247C2#1683796011yescharacteristic
248C2#1683820612yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#469C2
socleC22×C59C22×D4
hypercenter#469D59
hypocenterC59#3199
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #16837968C2semidirect
2. #16837728C2semidirect
3. #16837725C2semidirect
4. #16837655C2semidirect
5. #16837636C2semidirect
6. #16837537C2semidirect
7. #16837534C2semidirect
8. #16837370C2semidirect
9. C59×#469C2semidirect
11. C2×D4×D59C22semidirect
14. C22×C59⋊C8C22semidirect
18. #14273701C22semidirect
22. #14273685C22semidirect
23. #14273660C22semidirect
24. #14273659C22semidirect
25. #14273649C22semidirect
26. #14273648C22semidirect
27. #14273647C22semidirect
28. (C2×D59)⋊C8C22semidirect
29. (C2×D59)⋊C8C22semidirect
31. C2×M16×C59C22semidirect
32. #121×C59C22semidirect
33. #106×C59C22semidirect
34. #102×C59C22semidirect
129. #469D59semidirect
134. C59#3199semidirect
140. C2×M16C2×D59semidirect
147. #121C2×D59semidirect
153. #106C2×D59semidirect
157. #102C2×D59semidirect