Finite Group Databasebeta

← #21433468

Subgroups — #21433468

Order 7552.

736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14273752C212yesmaximal, complemented
2#16837966C211yesmaximal, characteristic, complemented
3#16837575C214yesmaximal, complemented
4#16837570C214yesmaximal, complemented
5#16837419C212yesmaximal, complemented
6#16837386C211yesmaximal, characteristic, complemented
7C59×#585C211yesmaximal, characteristic, complemented
8C22×D236C2211yescharacteristic, complemented
9#14273752—24no—
10#14273752C2218yescomplemented
11C2×#11745671—22no—
12C2×#11745669—22no—
13C2×#11745664C2212yescomplemented
14#14273723C2212yescomplemented
15#14273716C2212yescomplemented
16#14273701—22no—
17#14273695—22no—
18#14273686C2211yescharacteristic, complemented
19#14273654C2218yescomplemented
20#14273646C2212yescomplemented
21#14273644C2212yescomplemented
222−1+4×C59—22no—
232−1+4×C59C2212yescomplemented
24H(5,𝔽2)×C59—22no—
25H(5,𝔽2)×C59—22no—
26H(5,𝔽2)×C59C2212yescomplemented
27C2×#41×C59—24no—
28C2×#41×C59—22no—
29C2×#41×C59C2212yes—
30C2×#41×C59C2211yescharacteristic
31C23×D59—21no—
32#41×C59—24no—
33#41×C59—28no—
34#41×C59—24no—
35#41×C59—24no—
36#41×C59—24no—
37#41×C59—22no—
38#41×C59—22no—
39#41×C59—24no—
40#41×C59—22no—
41#41×C59—22no—
42#41×C59D414yescomplemented
43#41×C59C2314yes—
44C2×Q8×C59—22no—
45C2×Q8×C59—24no—
46C2×Q8×C59D412yescomplemented
47C2×Q8×C59C2311yescharacteristic
48C2×D4×C59—28no—
49C2×D4×C59—22no—
50C2×D4×C59—24no—
51C2×D4×C59—22no—
52C2×D4×C59—24no—
53C2×D4×C59D412yes—
54C2×D4×C59D412yescomplemented
55C2×D4×C59C2311yescharacteristic
56C22×C4×C59—24no—
57C22×C4×C59—22no—
58C22×C4×C59D412yes—
59C22×C4×C59C2311yescharacteristic
60C2×C22⋊D59—24no—
61C2×D236—22no—
62C2×D236—22no—
63C2×D236—21no—
64C2×D236C2312yes—
65C22⋊Dic59—21no—
66#11745671—28no—
67#11745669—28no—
68(C2×D59)⋊C4—24no—
69C4⋊Dic59—21no—
70C4×Dic59C2312yes—
71#11745664—22no—
72#11745664C2314yes—
73C2×C59⋊C8—22no—
74C23×C59—24no—
75C23×C59—22no—
76C23×C59C2×D412yes—
77C23×C59C2×D411yescharacteristic
78C22×D59—22no—
79C22×D59—42no—
80C22×D59—82no—
81Q8×C59—24no—
82Q8×C59—22no—
83Q8×C59—22no—
84Q8×C59C2×D414yes—
85D4×C59—28no—
86D4×C59—24no—
87D4×C59—24no—
88D4×C59—24no—
89D4×C59—24no—
90D4×C59—22no—
91D4×C59—22no—
92D4×C59C2×D414yes—
93C2×C4×C59—24no—
94C2×C4×C59—28no—
95C2×C4×C59—22no—
96C2×C4×C59—24no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—24no—
100C2×C4×C59C2×D412yes—
101C2×C4×C59C2×D411yescharacteristic
102C2×C4×C59C2×D412yes—
103C2×C4×C59C2×D412yes—
104C2×C4×C59C2411yescharacteristic
105C22⋊D59—84no—
106C2×Dic59—22no—
107D236—24no—
108D236—41no—
109D236—41no—
110C59⋊C8—24no—
111C22×C59—22no—
112C22×C59—24no—
113C22×C59—41no—
114C22×C59—42no—
115C22×C59—82no—
116C22×C59—84no—
117C22×C59#12012yes—
118C22×C59C22×D411yescharacteristic
119C2×D59—44no—
120C2×D59—42no—
121C2×D59—84no—
122C4×C59—24no—
123C4×C59—22no—
124C4×C59#12011yescharacteristic
125C4×C59#12011yescharacteristic
126C4×C59C22×D412yes—
127Dic59—42no—
128#3021—591nomaximal, Sylow
129C2×C59—21no—
130C2×C59—22no—
131C2×C59—42no—
132C2×C59—44no—
133C2×C59C2×#12011yescharacteristic
134D59—82no—
135#585D5911yescharacteristic, complemented
136C2×AGL(1,ℤ/8)—592no—
137#535—591no—
138#454—594no—
139#453—594no—
140#421—592no—
141#413—591no—
142C59#302111yescharacteristic, complemented, Sylow
1432−1+4—22no—
1442−1+4C2×D5912yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D5912yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D5912yes—
151C2×#41C2×D5911yescharacteristic
152C22×D4—591no—
153AGL(1,ℤ/8)—598no—
154AGL(1,ℤ/8)—1184no—
155C2×SD16—1182no—
156C2×D8—1182no—
157C2×M16—592no—
158#127—592no—
159#124—592no—
160#121—1182no—
161#120—1182no—
162#117—591no—
163#104—598no—
164#101—592no—
165#100—592no—
166C24—1181no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D5914yescomplemented
178#41C22×D5914yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D5912yescomplemented
182C2×Q8C22×D5911yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—592no—
189C2×D4—1184no—
190C2×D4—1182no—
191C2×D4—1182no—
192C2×D4—1181no—
193C2×D4C22⋊D5912yes—
194C2×D4C22⋊D5912yescomplemented
195C2×D4C22×D5911yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D5912yes—
199C22×C4C22×D5911yescharacteristic
200SD16—1188no—
201D8—1188no—
202M16—594no—
203M16—1182no—
204C2×C8—1182no—
205C4⋊C4—1181no—
206C22⋊C4—1181no—
207C22⋊C4—1184no—
208C42—592no—
209C23—24no—
210C23—22no—
211C23—1182no—
212C23—4722no—
213C23—2362no—
214C23C2×C22⋊D5912yes—
215C23C2×C22⋊D5911yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D5914yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—1184no—
228D4—4724no—
229D4—2361no—
230D4—2361no—
231D4C2×C22⋊D5914yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—1182no—
240C2×C4C2×C22⋊D5912yes—
241C2×C4C2×C22⋊D5911yescharacteristic
242C2×C4C2×C22⋊D5912yes—
243C2×C4C2×C22⋊D5912yes—
244C2×C4C23×D5911yescharacteristic
245C8—1184no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—4724no—
253C22—2364no—
254C22—2362no—
255C22#1427369612yes—
256C22C22×C22⋊D5911yescharacteristic
257C4—24no—
258C4—22no—
259C4—2362no—
260C4#1427369611yescharacteristic
261C4#1427369611yescharacteristic
262C4C22×C22⋊D5912yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—4722no—
268C2C2×#1427369611yescharacteristic
subgroupG/H
centerC2C2×#14273696
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#585C2
socleC2×C59C2×#120
hypercenter#585D59
hypocenterC59#3021
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#14273752C2semidirect
2. #16837966C2semidirect
3. #16837575C2semidirect
4. #16837570C2semidirect
5. #16837419C2semidirect
6. #16837386C2semidirect
7. C59×#585C2semidirect
8. C22×D236C22semidirect
10. #14273752C22semidirect
13. C2×#11745664C22semidirect
14. #14273723C22semidirect
15. #14273716C22semidirect
18. #14273686C22semidirect
19. #14273654C22semidirect
20. #14273646C22semidirect
21. #14273644C22semidirect
23. 2−1+4×C59C22semidirect
26. H(5,𝔽2)×C59C22semidirect
42. #41×C59D4semidirect
46. C2×Q8×C59D4semidirect
54. C2×D4×C59D4semidirect
135. #585D59semidirect
142. C59#3021semidirect
144. 2−1+4C2×D59semidirect
147. H(5,𝔽2)C2×D59semidirect
177. #41C22⋊D59semidirect
181. C2×Q8C22⋊D59semidirect
194. C2×D4C22⋊D59semidirect