Finite Group Databasebeta

← #21434963

Subgroups — #21434963

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#16837966C211yesmaximal, characteristic
2#16837740C212yesmaximal, complemented
3#16837726C212yesmaximal, complemented
4#16837650C212yesmaximal, complemented
5#16837633C212yesmaximal, complemented
6#16837552C212yesmaximal, complemented
7#16837541C212yesmaximal, complemented
8#16837369C211yesmaximal, characteristic, complemented
9C59×#536C211yesmaximal, characteristic, complemented
10C22×D236C2211yescharacteristic
11C2×#11745671C2212yes—
12C2×#11745671C2212yes—
13C2×#11745664C2211yescharacteristic, complemented
14C22×C59⋊C8C2211yescharacteristic, complemented
15#14273723—21no—
16#14273716—21no—
17#14273701C2212yes—
18#14273701C2212yes—
19#14273695—21no—
20#14273695—21no—
21#14273686C2211yescharacteristic
22#14273660C2212yescomplemented
23#14273658C2212yescomplemented
24#14273651C2212yescomplemented
25#14273651C2212yescomplemented
26#14273647C2212yes—
27#14273647C2212yes—
28C22⋊(C59⋊C8)C2212yes—
29C22⋊(C59⋊C8)C2212yescomplemented
30C2×#41×C59C2211yescharacteristic
31C22×Q8×C59C2211yescharacteristic, complemented
32C4⋊Q8×C59—21no—
33#124×C59—21no—
34#123×C59—21no—
35#123×C59—21no—
36C22⋊Q8×C59C2212yes—
37C22⋊Q8×C59C2212yescomplemented
38#117×C59C2211yescharacteristic, complemented
39C23×D59—21no—
40#41×C59—41no—
41#41×C59—41no—
42C2×Q8×C59—21no—
43C2×Q8×C59—41no—
44C2×Q8×C59—41no—
45C2×Q8×C59D411yescharacteristic
46C2×Q8×C59C2312yes—
47C2×D4×C59D411yescharacteristic
48C2×D4×C59D411yescharacteristic
49C2×D4×C59D411yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59C2311yescharacteristic
52C22×C4×C59C2311yescharacteristic
53C22×C4×C59C2311yescharacteristic
54C2×C22⋊D59—22no—
55C2×C22⋊D59—22no—
56C2×D236—21no—
57C2×D236—41no—
58C2×D236—41no—
59C2×D236C2312yes—
60C4⋊C4×C59—22no—
61C4⋊C4×C59—22no—
62C4⋊C4×C59C2312yes—
63C22⋊C4×C59—21no—
64C22⋊C4×C59—22no—
65C22⋊C4×C59—22no—
66C42×C59—21no—
67C22⋊Dic59—21no—
68#11745671—42no—
69#11745671—42no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—22no—
72C4⋊Dic59C2312yes—
73C4×Dic59—21no—
74#11745664—41no—
75C2×C59⋊C8—41no—
76C2×C59⋊C8C2312yes—
77C2×C59⋊C8C2312yes—
78C23×C59C2×D411yescharacteristic
79C23×C59C2×D411yescharacteristic
80C23×C59C2×D411yescharacteristic
81C22×D59—22no—
82C22×D59—42no—
83C22×D59—82no—
84Q8×C59—22no—
85Q8×C59—22no—
86Q8×C59—41no—
87Q8×C59—42no—
88D4×C59—41no—
89D4×C59—41no—
90D4×C59—41no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—42no—
100C2×C4×C59C2×D411yescharacteristic
101C2×C4×C59C2×D411yescharacteristic
102C2×C4×C59C2×D411yescharacteristic
103C2×C4×C59C2411yescharacteristic
104C22⋊D59—82no—
105C22⋊D59—82no—
106C2×Dic59—22no—
107D236—22no—
108D236—22no—
109D236—42no—
110C59⋊C8—22no—
111C59⋊C8—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×C59—42no—
127C4×C59—42no—
128C4×C59H(5,𝔽2)11yescharacteristic
129C4×C59H(5,𝔽2)11yescharacteristic
130Dic59—42no—
131#3200—591nomaximal, Sylow
132C2×C59—41no—
133C2×C59—41no—
134C2×C59—41no—
135C2×C59#53411yescharacteristic
136C2×C59#57712yes—
137D59—82no—
138#536D5911yescharacteristic, complemented
139#535—591no—
140#485—592no—
141#482—592no—
142#468—592no—
143#465—592no—
144#450—592no—
145#448—592no—
146#408—591no—
147C59#320011yescharacteristic, complemented, Sylow
148C2×#41C2×D5911yescharacteristic
149C22×Q8C2×D5911yescharacteristic, complemented
150C22×D4—591no—
151C2×SD16—592no—
152C2×SD16—592no—
153C2×M16—591no—
154C22×C8—591no—
155C4⋊Q8—21no—
156#127—1181no—
157#124—21no—
158#124—1181no—
159#123—21no—
160#123—21no—
161C22⋊Q8C2×D5912yes—
162C22⋊Q8C2×D5912yescomplemented
163#121—592no—
164#121—592no—
165#120—1181no—
166#120—1181no—
167#117—591no—
168#117C2×D5911yescharacteristic, complemented
169#107—592no—
170#106—592no—
171#103—592no—
172#103—592no—
173#102—592no—
174#102—592no—
175C22⋊C8—592no—
176C22⋊C8—592no—
177C24—1181no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8C22⋊D5911yescharacteristic
184C2×Q8C22×D5912yes—
185C2×D4—592no—
186C2×D4—1182no—
187C2×D4—1182no—
188C2×D4—1181no—
189C2×D4—2361no—
190C2×D4—2361no—
191C2×D4C22⋊D5911yescharacteristic
192C2×D4C22⋊D5911yescharacteristic
193C2×D4C22⋊D5911yescharacteristic
194C22×C4—21no—
195C22×C4C22×D5911yescharacteristic
196C22×C4C22×D5911yescharacteristic
197C22×C4C22×D5911yescharacteristic
198SD16—2362no—
199SD16—2362no—
200M16—2361no—
201C2×C8—592no—
202C2×C8—592no—
203C2×C8—2361no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—592no—
207C4⋊C4C22×D5912yes—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—1181no—
212C22⋊C4—1182no—
213C22⋊C4—1182no—
214C42—21no—
215C42—1181no—
216C23—1182no—
217C23—4722no—
218C23—2362no—
219C23C2×C22⋊D5911yescharacteristic
220C23C2×C22⋊D5911yescharacteristic
221C23C2×C22⋊D5911yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—42no—
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—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—1182no—
244C2×C4C2×C22⋊D5911yescharacteristic
245C2×C4C2×C22⋊D5911yescharacteristic
246C2×C4C2×C22⋊D5911yescharacteristic
247C2×C4C23×D5911yescharacteristic
248C8—1182no—
249C8—1182no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—4724no—
257C22—2362no—
258C22—2362no—
259C22—2362no—
260C22C22×C22⋊D5911yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—2362no—
267C4#1427377211yescharacteristic
268C4#1427377211yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—4722no—
273C2#1683796111yescharacteristic
274C2#1683821012yes—
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#536C2
socleC22×C59C22×D4
hypercenter#536D59
hypocenterC59#3200
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #16837740C2semidirect
3. #16837726C2semidirect
4. #16837650C2semidirect
5. #16837633C2semidirect
6. #16837552C2semidirect
7. #16837541C2semidirect
8. #16837369C2semidirect
9. C59×#536C2semidirect
13. C2×#11745664C22semidirect
14. C22×C59⋊C8C22semidirect
22. #14273660C22semidirect
23. #14273658C22semidirect
24. #14273651C22semidirect
25. #14273651C22semidirect
29. C22⋊(C59⋊C8)C22semidirect
31. C22×Q8×C59C22semidirect
37. C22⋊Q8×C59C22semidirect
38. #117×C59C22semidirect
138. #536D59semidirect
147. C59#3200semidirect
149. C22×Q8C2×D59semidirect
162. C22⋊Q8C2×D59semidirect
168. #117C2×D59semidirect