Finite Group Databasebeta

← #21434932

Subgroups — #21434932

Order 7552.

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

normal subgroups only

This group has 418 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#16837912C212yesmaximal, complemented
2#16837746C212yesmaximal, complemented
3#16837725C212yesmaximal, complemented
4#16837660C211yesmaximal, characteristic, complemented
5#16837646C212yesmaximal, complemented
6#16837565C212yesmaximal, complemented
7#16837544C212yesmaximal, complemented
8#16837365C211yesmaximal, characteristic, complemented
9C59×#469C211yesmaximal, characteristic, complemented
10C2×#11745687C2212yes—
11C22×C4×D59C2211yescharacteristic
12C2×Dic236C2211yescharacteristic, complemented
13C2×#11745670C2212yescomplemented
14C2×#11745664C2211yescharacteristic
15#14273709—22no—
16#14273707C2212yes—
17#14273706C2212yes—
18#14273698C2212yes—
19#14273695—22no—
20D4×Dic59—22no—
21(C22⋊D59)⋊C4—22no—
22#14273685C2212yes—
23#14273660C2212yescomplemented
24#14273659C2211yescharacteristic, complemented
25#14273652C2212yescomplemented
26#14273650C2212yescomplemented
27#14273648C2212yescomplemented
28(C2×D59)⋊C8C2212yes—
29(C2×D59)⋊C8C2212yes—
30C2×D8×C59C2211yescharacteristic, complemented
31C2×M16×C59C2211yescharacteristic
32#121×C59C2212yes—
33#106×C59C2211yescharacteristic, complemented
34#102×C59C2212yescomplemented
35C23×D59D411yescharacteristic
36C2×D4×C59—22no—
37C2×D4×C59C2312yes—
38C22×C4×C59C2311yescharacteristic
39C2×C22⋊D59—22no—
40C2×C22⋊D59—22no—
41C22×Dic59—22no—
42C22×Dic59D411yescharacteristic
43#11745687—42no—
44#11745687—42no—
45C2×C4×D59—41no—
46C2×C4×D59—41no—
47C2×C4×D59D412yes—
48C2×C4×D59C2312yes—
49C2×Dic118C2312yes—
50D8×C59—41no—
51M16×C59—41no—
52C2×C8×C59C2311yescharacteristic
53C2×C8×C59C2311yescharacteristic
54C4⋊C4×C59C2312yes—
55C22⋊C4×C59—22no—
56C22⋊Dic59—22no—
57C22⋊Dic59—22no—
58#11745670—42no—
59(C2×D59)⋊C4—22no—
60(C2×D59)⋊C4—22no—
61C4⋊Dic59—22no—
62C4⋊Dic59C2312yes—
63Dic59⋊C4—22no—
64C4×Dic59—22no—
65#11745664—41no—
66C2×C59⋊C8C2312yes—
67Dic236—41no—
68C23×C59—22no—
69C23×C59C2×D411yescharacteristic
70C22×D59—41no—
71C22×D59—42no—
72C22×D59C2×D412yes—
73D4×C59—22no—
74D4×C59—22no—
75D4×C59—44no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—22no—
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59C2411yescharacteristic
81C22⋊D59—44no—
82C22⋊D59—44no—
83C2×Dic59—22no—
84C2×Dic59—22no—
85C2×Dic59—41no—
86C2×Dic59—44no—
87C2×Dic59C2×D412yes—
88C4×D59—22no—
89C4×D59—22no—
90C4×D59—42no—
91Dic118—22no—
92Dic118—22no—
93C8×C59—21no—
94C8×C59—21no—
95C59⋊C8—22no—
96C22×C59—21no—
97C22×C59—22no—
98C22×C59—42no—
99C22×C59—42no—
100C22×C59—42no—
101C22×C59C22×D411yescharacteristic
102C2×D59—22no—
103C2×D59—22no—
104C2×D59—22no—
105C2×D59—44no—
106C4×C59—21no—
107C4×C59—42no—
108C4×C59H(5,𝔽2)11yescharacteristic
109C4×C59H(5,𝔽2)11yescharacteristic
110Dic59—22no—
111Dic59—42no—
112Dic59—42no—
113#3197—591nomaximal, Sylow
114C2×C59—41no—
115C2×C59—82no—
116C2×C59#53411yescharacteristic
117C2×C59#57512yes—
118D59—42no—
119#525—592no—
120#485—592no—
121#482—592no—
122#470—591no—
123#469D5911yescharacteristic, complemented
124#468—592no—
125#452—592no—
126#449—592no—
127#407—591no—
128C59#319711yescharacteristic, complemented, Sylow
129C2×#41—592no—
130C23×C4—591no—
131C2×Q16—591no—
132C2×SD16—592no—
133C2×D8C2×D5911yescharacteristic, complemented
134C2×M16—591no—
135C2×M16C2×D5911yescharacteristic
136#123—1182no—
137C22⋊Q8—592no—
138C22⋊Q8—592no—
139#121—592no—
140#121C2×D5912yes—
141#120—1182no—
142C4×D4—1182no—
143C4×D4—1182no—
144#117—592no—
145#107—592no—
146#106—591no—
147#106C2×D5911yescharacteristic, complemented
148#103—592no—
149#103—592no—
150#102—592no—
151#102C2×D5912yescomplemented
152C22⋊C8—592no—
153C22⋊C8—592no—
154C24—591no—
155#41—2362no—
156#41—2362no—
157C2×Q8—592no—
158C2×D4—22no—
159C2×D4—1182no—
160C2×D4—1182no—
161C2×D4C22×D5912yes—
162C22×C4—591no—
163C22×C4—592no—
164C22×C4—592no—
165C22×C4—1182no—
166C22×C4—2361no—
167C22×C4—2361no—
168C22×C4C22×D5911yescharacteristic
169Q16—2361no—
170SD16—2362no—
171D8—41no—
172M16—41no—
173M16—2361no—
174C2×C8—592no—
175C2×C8C22×D5911yescharacteristic
176C2×C8C22×D5911yescharacteristic
177C4⋊C4—592no—
178C4⋊C4—1182no—
179C4⋊C4—1182no—
180C4⋊C4C22×D5912yes—
181C22⋊C4—22no—
182C22⋊C4—1182no—
183C22⋊C4—1182no—
184C22⋊C4—1182no—
185C22⋊C4—1182no—
186C42—1182no—
187C23—22no—
188C23—592no—
189C23—2361no—
190C23—2362no—
191C23D4×D5911yescharacteristic
192Q8—1182no—
193Q8—1182no—
194D4—22no—
195D4—22no—
196D4—44no—
197D4—2364no—
198D4—2364no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—592no—
203C2×C4—1182no—
204C2×C4—1182no—
205C2×C4—1182no—
206C2×C4—1182no—
207C2×C4—2362no—
208C2×C4—2361no—
209C2×C4—2364no—
210C2×C4D4×D5911yescharacteristic
211C2×C4C23×D5911yescharacteristic
212C8—21no—
213C8—21no—
214C8—1182no—
215C22—22no—
216C22—21no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—1182no—
221C22—1182no—
222C22—1182no—
223C22—2364no—
224C22C2×D4×D5911yescharacteristic
225C4—21no—
226C4—42no—
227C4—1182no—
228C4—2362no—
229C4—2362no—
230C4#1427377211yescharacteristic
231C4#1427377311yescharacteristic
232C2—41no—
233C2—82no—
234C2—2362no—
235C2#1683796011yescharacteristic
236C2#1683819812yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#469C2
socleC22×C59C22×D4
hypercenter#469D59
hypocenterC59#3197
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. #16837912C2semidirect
2. #16837746C2semidirect
3. #16837725C2semidirect
4. #16837660C2semidirect
5. #16837646C2semidirect
6. #16837565C2semidirect
7. #16837544C2semidirect
8. #16837365C2semidirect
9. C59×#469C2semidirect
12. C2×Dic236C22semidirect
13. C2×#11745670C22semidirect
23. #14273660C22semidirect
24. #14273659C22semidirect
25. #14273652C22semidirect
26. #14273650C22semidirect
27. #14273648C22semidirect
30. C2×D8×C59C22semidirect
33. #106×C59C22semidirect
34. #102×C59C22semidirect
123. #469D59semidirect
128. C59#3197semidirect
133. C2×D8C2×D59semidirect
147. #106C2×D59semidirect
151. #102C2×D59semidirect