Finite Group Databasebeta

← #21434969

Subgroups — #21434969

Order 7552.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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#16837973C211yesmaximal, characteristic, complemented
2#16837966C211yesmaximal, characteristic
3#16837744C212yesmaximal, complemented
4#16837727C212yesmaximal, complemented
5#16837648C212yesmaximal, complemented
6#16837631C212yesmaximal, complemented
7#16837555C212yesmaximal, complemented
8#16837543C212yesmaximal, complemented
9C59×#408C211yesmaximal, characteristic, complemented
10C22×Dic118C2211yescharacteristic, complemented
11C22×D236C2211yescharacteristic
12C2×#11745660C2212yes—
13C2×#11745660C2212yes—
14#14273725—21no—
15#14273723—21no—
16#14273716—21no—
17#14273714—21no—
18#14273708—21no—
19#14273708—21no—
20#14273706C2212yes—
21#14273706C2212yescomplemented
22#14273701C2212yes—
23#14273701C2212yes—
24#14273695—21no—
25#14273695—21no—
26#14273686C2211yescharacteristic
27#14273686C2211yescharacteristic, complemented
28#14273661C2212yescomplemented
29#14273659C2212yescomplemented
30#14273652C2212yescomplemented
31#14273652C2212yescomplemented
32#14273649C2212yes—
33#14273649C2212yes—
34C2×#41×C59C2211yescharacteristic
35C2×M16×C59C2211yescharacteristic, complemented
36C22×C8×C59C2211yescharacteristic, complemented
37C22⋊C8×C59C2212yes—
38C22⋊C8×C59C2212yescomplemented
39C23×D59—21no—
40#41×C59—41no—
41#41×C59—41no—
42C2×Q8×C59D411yescharacteristic
43C2×D4×C59D411yescharacteristic
44C2×D4×C59D411yescharacteristic
45C2×D4×C59D411yescharacteristic
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2311yescharacteristic
48C22×C4×C59C2311yescharacteristic
49C2×C22⋊D59—22no—
50C2×C22⋊D59—22no—
51C22×Dic59—21no—
52C2×D236—21no—
53C2×D236—41no—
54C2×D236—41no—
55C2×D236C2312yes—
56C2×Dic118—21no—
57C2×Dic118—41no—
58C2×Dic118—41no—
59C2×Dic118C2312yes—
60M16×C59—41no—
61C2×C8×C59—41no—
62C2×C8×C59C2312yes—
63C2×C8×C59C2312yes—
64C22⋊Dic59—21no—
65C22⋊Dic59—21no—
66C22⋊Dic59—22no—
67C22⋊Dic59—22no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—22no—
70C4⋊Dic59C2312yes—
71C4⋊Dic59C2312yes—
72Dic59⋊C4—22no—
73Dic59⋊C4—22no—
74C4×Dic59—21no—
75C4×Dic59—21no—
76#11745660—42no—
77#11745660—42no—
78C23×C59C2×D411yescharacteristic
79C23×C59C2×D411yescharacteristic
80C23×C59C2×D411yescharacteristic
81C22×D59—22no—
82C22×D59—42no—
83C22×D59—82no—
84Q8×C59—41no—
85D4×C59—41no—
86D4×C59—41no—
87D4×C59—41no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59C2×D411yescharacteristic
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59C2×D411yescharacteristic
97C2×C4×C59C2411yescharacteristic
98C22⋊D59—82no—
99C22⋊D59—82no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—42no—
104D236—22no—
105D236—22no—
106D236—42no—
107Dic118—22no—
108Dic118—22no—
109Dic118—42no—
110C8×C59—22no—
111C8×C59—22no—
112C22×C59—21no—
113C22×C59—21no—
114C22×C59—21no—
115C22×C59—22no—
116C22×C59—22no—
117C22×C59—22no—
118C22×C59C22×D411yescharacteristic
119C2×D59—42no—
120C2×D59—42no—
121C2×D59—42no—
122C2×D59—84no—
123C4×C59—21no—
124C4×C59—21no—
125C4×C59—21no—
126C4×C59H(5,𝔽2)11yescharacteristic
127C4×C59H(5,𝔽2)11yescharacteristic
128Dic59—42no—
129Dic59—42no—
130Dic59—42no—
131#3200—591nomaximal, Sylow
132C2×C59—41no—
133C2×C59—41no—
134C2×C59—41no—
135C2×C59#53411yescharacteristic
136C2×C59#57712yes—
137D59—82no—
138#536—591no—
139#535—591no—
140#485—592no—
141#482—592no—
142#468—592no—
143#465—592no—
144#450—592no—
145#448—592no—
146#408D5911yescharacteristic, complemented
147C59#320011yescharacteristic, complemented, Sylow
148C2×#41C2×D5911yescharacteristic
149C22×Q8—591no—
150C22×D4—591no—
151C2×SD16—592no—
152C2×SD16—592no—
153C2×M16C2×D5911yescharacteristic, complemented
154C22×C8C2×D5911yescharacteristic, complemented
155C4⋊Q8—1181no—
156#127—1181no—
157#124—1181no—
158#124—1181no—
159#123—1181no—
160#123—1181no—
161C22⋊Q8—592no—
162C22⋊Q8—592no—
163#121—592no—
164#121—592no—
165#120—1181no—
166#120—1181no—
167#117—591no—
168#117—591no—
169#107—592no—
170#106—592no—
171#103—592no—
172#103—592no—
173#102—592no—
174#102—592no—
175C22⋊C8C2×D5912yes—
176C22⋊C8C2×D5912yescomplemented
177C24—1181no—
178#41—41no—
179#41—41no—
180C2×Q8—592no—
181C2×Q8—1181no—
182C2×Q8—2361no—
183C2×Q8—2361no—
184C2×Q8D23611yescharacteristic
185C2×D4—592no—
186C2×D4—1182no—
187C2×D4—1182no—
188C2×D4—1181no—
189C2×D4—2361no—
190C2×D4—2361no—
191C2×D4D23611yescharacteristic
192C2×D4D23611yescharacteristic
193C2×D4D23611yescharacteristic
194C22×C4—1181no—
195C22×C4C22×D5911yescharacteristic
196C22×C4C22×D5911yescharacteristic
197C22×C4C22×D5911yescharacteristic
198SD16—2362no—
199SD16—2362no—
200M16—41no—
201C2×C8—41no—
202C2×C8C22×D5912yes—
203C2×C8C22×D5912yes—
204C4⋊C4—592no—
205C4⋊C4—592no—
206C4⋊C4—1182no—
207C4⋊C4—1182no—
208C22⋊C4—1181no—
209C22⋊C4—1181no—
210C22⋊C4—1182no—
211C22⋊C4—1182no—
212C22⋊C4—1182no—
213C22⋊C4—1182no—
214C42—1181no—
215C42—1181no—
216C23—1182no—
217C23—4722no—
218C23—2362no—
219C23C2×D23611yescharacteristic
220C23C2×D23611yescharacteristic
221C23C2×D23611yescharacteristic
222Q8—41no—
223Q8—1182no—
224Q8—1182no—
225Q8—2362no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—1182no—
230D4—1182no—
231D4—4722no—
232D4—4722no—
233D4—2362no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—1182no—
241C2×C4—1182no—
242C2×C4—1182no—
243C2×C4—2362no—
244C2×C4C2×D23611yescharacteristic
245C2×C4C2×D23611yescharacteristic
246C2×C4C2×D23611yescharacteristic
247C2×C4C23×D5911yescharacteristic
248C8—22no—
249C8—22no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—4724no—
257C22—2362no—
258C22—2362no—
259C22—2362no—
260C22C22×D23611yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—2362no—
265C4—2362no—
266C4—2362no—
267C4#1427377211yescharacteristic
268C4#1427377211yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—4722no—
273C2#1683796411yescharacteristic
274C2#1683821412yes—
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#408C2
socleC22×C59C22×D4
hypercenter#408D59
hypocenterC59#3200
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #16837973C2semidirect
3. #16837744C2semidirect
4. #16837727C2semidirect
5. #16837648C2semidirect
6. #16837631C2semidirect
7. #16837555C2semidirect
8. #16837543C2semidirect
9. C59×#408C2semidirect
10. C22×Dic118C22semidirect
21. #14273706C22semidirect
27. #14273686C22semidirect
28. #14273661C22semidirect
29. #14273659C22semidirect
30. #14273652C22semidirect
31. #14273652C22semidirect
35. C2×M16×C59C22semidirect
36. C22×C8×C59C22semidirect
38. C22⋊C8×C59C22semidirect
146. #408D59semidirect
147. C59#3200semidirect
153. C2×M16C2×D59semidirect
154. C22×C8C2×D59semidirect
176. C22⋊C8C2×D59semidirect