Finite Group Databasebeta

← #21433322

Subgroups — #21433322

Order 7552.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D236C211yesmaximal, characteristic, complemented
2C2×#14273752C212yesmaximal, complemented
3#16837552C214yesmaximal, complemented
4#16837535C214yesmaximal, complemented
5#16837406C212yesmaximal, complemented
6#16837366C211yesmaximal, characteristic, complemented
7C59×#525C211yesmaximal, characteristic, complemented
8C24×D59—21no—
9C22×D236—22no—
10C22×D236—21no—
11C22×D236—28no—
12C22×D236C2214yes—
13C22×D236C2212yes—
14#14273752—28no—
15C2×#11745671C2214yescomplemented
16C2×#11745669C2214yescomplemented
17C2×#11745664C2212yescomplemented
18#14273647C2218yescomplemented
19C22⋊(C59⋊C8)C2214yescomplemented
20C2×#41×C59C2211yescharacteristic, complemented
21C23×C4×C59C2211yescharacteristic
22#123×C59—21no—
23C22⋊Q8×C59C2212yescomplemented
24#121×C59C2212yescomplemented
25#120×C59—21no—
26C4×D4×C59—22no—
27#117×C59C2211yescharacteristic, complemented
28C24×C59D411yescharacteristic
29C23×D59—28no—
30C23×D59—24no—
31C23×D59—22no—
32C23×D59—48no—
33#41×C59—24no—
34C2×Q8×C59C2311yescharacteristic, complemented
35C2×D4×C59—22no—
36C2×D4×C59—21no—
37C2×D4×C59C2311yescharacteristic, complemented
38C22×C4×C59—24no—
39C22×C4×C59—21no—
40C22×C4×C59D412yes—
41C22×C4×C59D411yescharacteristic
42C22×C4×C59C2312yes—
43C22×C4×C59C2311yescharacteristic
44C2×D236—28no—
45C2×D236—216no—
46C2×D236—24no—
47C2×D236—22no—
48C2×D236—24no—
49C2×D236—216no—
50C2×D236D418yescomplemented
51C2×D236C2314yes—
52C4⋊C4×C59—22no—
53C4⋊C4×C59C2312yescomplemented
54C22⋊C4×C59—21no—
55C22⋊C4×C59—22no—
56C22⋊C4×C59—22no—
57C42×C59—21no—
58#11745671—28no—
59#11745669—28no—
60#11745664—24no—
61C2×C59⋊C8C2314yescomplemented
62C23×C59—22no—
63C23×C59—24no—
64C23×C59—21no—
65C23×C59C2×D411yescharacteristic
66C23×C59C2×D412yes—
67C22×D59—216no—
68C22×D59—28no—
69C22×D59—24no—
70C22×D59—432no—
71C22×D59—48no—
72C22×D59—48no—
73C22×D59—48no—
74Q8×C59—22no—
75D4×C59—22no—
76D4×C59—42no—
77D4×C59—44no—
78C2×C4×C59—22no—
79C2×C4×C59—24no—
80C2×C4×C59—24no—
81C2×C4×C59—22no—
82C2×C4×C59—21no—
83C2×C4×C59—42no—
84C2×C4×C59C2×D414yes—
85C2×C4×C59C2×D412yes—
86C2×C4×C59C2×D411yescharacteristic
87C2×C4×C59C2411yescharacteristic
88D236—216no—
89D236—28no—
90D236—24no—
91D236C2×D418yes—
92C59⋊C8—24no—
93C22×C59—28no—
94C22×C59—22no—
95C22×C59—22no—
96C22×C59—22no—
97C22×C59—41no—
98C22×C59—42no—
99C22×C59#12012yes—
100C22×C59AGL(1,ℤ/8)14yescomplemented
101C22×C59C22×D411yescharacteristic
102C2×D59—28no—
103C2×D59—416no—
104C2×D59—432no—
105C2×D59—48no—
106C4×C59—22no—
107C4×C59—41no—
108C4×C59—42no—
109C4×C59#12012yes—
110C4×C59C22×D412yes—
111#3003—591nomaximal, Sylow
112C2×C59—22no—
113C2×C59—24no—
114C2×C59—81no—
115C2×C59C2×#12011yescharacteristic
116C2×C59C2×AGL(1,ℤ/8)12yes—
117D59—48no—
118C23×D4—591no—
119C2×AGL(1,ℤ/8)—592no—
120#525D5911yescharacteristic, complemented
121#450—594no—
122#447—594no—
123#418—592no—
124#407—591no—
125C59#300311yescharacteristic, complemented, Sylow
126C25—1181no—
127C2×#41C2×D5911yescharacteristic, complemented
128C22×D4—592no—
129C22×D4—594no—
130C22×D4—1188no—
131C22×D4—1182no—
132C22×D4—1181no—
133C23×C4C2×D5911yescharacteristic
134AGL(1,ℤ/8)—1188no—
135C2×SD16—594no—
136C2×D8—594no—
137C2×M16—592no—
138#123—21no—
139C22⋊Q8C2×D5912yescomplemented
140#121C2×D5912yescomplemented
141#120—21no—
142C4×D4—22no—
143#117C2×D5911yescharacteristic, complemented
144#102—598no—
145C22⋊C8—594no—
146C24—1188no—
147C24—1184no—
148C24—1182no—
149C24—2368no—
150C24C22⋊D5911yescharacteristic
151#41—24no—
152C2×Q8C22×D5911yescharacteristic, complemented
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—598no—
156C2×D4—594no—
157C2×D4—11816no—
158C2×D4—1188no—
159C2×D4—11816no—
160C2×D4—1184no—
161C2×D4—1184no—
162C2×D4—1182no—
163C2×D4C22×D5911yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4C22⋊D5912yes—
167C22×C4C22⋊D5911yescharacteristic
168C22×C4C22×D5912yes—
169C22×C4C22×D5911yescharacteristic
170SD16—1188no—
171D8—1188no—
172M16—1184no—
173C2×C8—594no—
174C4⋊C4—22no—
175C4⋊C4C22×D5912yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—22no—
181C23—24no—
182C23—21no—
183C23—11816no—
184C23—1188no—
185C23—1184no—
186C23—23632no—
187C23—2368no—
188C23—2368no—
189C23—2368no—
190C23C2×C22⋊D5911yescharacteristic
191C23C2×C22⋊D5912yes—
192Q8—22no—
193D4—22no—
194D4—42no—
195D4—44no—
196D4—598no—
197D4—11816no—
198D4—1188no—
199D4—1184no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4D4×D5914yes—
207C2×C4C2×C22⋊D5912yes—
208C2×C4C2×C22⋊D5911yescharacteristic
209C2×C4C23×D5911yescharacteristic
210C8—1184no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22—41no—
217C22—1188no—
218C22—23616no—
219C22—23632no—
220C22—2368no—
221C22#1427369512yes—
222C22#1427375214yescomplemented
223C22C22×C22⋊D5911yescharacteristic
224C4—22no—
225C4—42no—
226C4—41no—
227C4#1427369512yes—
228C4C2×D4×D5912yes—
229C2—22no—
230C2—24no—
231C2—81no—
232C2—2368no—
233C2C2×#1427369511yescharacteristic
234C2C2×#1427375212yes—
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#525C2
socleC22×C59C22×D4
hypercenter#525D59
hypocenterC59#3003
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×D236C2semidirect
2. C2×#14273752C2semidirect
3. #16837552C2semidirect
4. #16837535C2semidirect
5. #16837406C2semidirect
6. #16837366C2semidirect
7. C59×#525C2semidirect
15. C2×#11745671C22semidirect
16. C2×#11745669C22semidirect
17. C2×#11745664C22semidirect
18. #14273647C22semidirect
19. C22⋊(C59⋊C8)C22semidirect
20. C2×#41×C59C22semidirect
23. C22⋊Q8×C59C22semidirect
24. #121×C59C22semidirect
27. #117×C59C22semidirect
34. C2×Q8×C59C23semidirect
37. C2×D4×C59C23semidirect
50. C2×D236D4semidirect
53. C4⋊C4×C59C23semidirect
61. C2×C59⋊C8C23semidirect
100. C22×C59AGL(1,ℤ/8)semidirect
120. #525D59semidirect
125. C59#3003semidirect
127. C2×#41C2×D59semidirect
139. C22⋊Q8C2×D59semidirect
140. #121C2×D59semidirect
143. #117C2×D59semidirect
152. C2×Q8C22×D59semidirect
163. C2×D4C22×D59semidirect
175. C4⋊C4C22×D59semidirect
222. C22#14273752semidirect