Finite Group Databasebeta

← #21434849

Subgroups — #21434849

Order 7552.

468 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14273701C211yesmaximal, characteristic
2#16837730C212yesmaximal, complemented
3#16837720C212yesmaximal, complemented
4#16837633C214yesmaximal, complemented
5#16837552C212yesmaximal, complemented
6#16837541C212yesmaximal, complemented
7C2×C22⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C59C211yesmaximal, characteristic, complemented
9C22×D236C2211yescharacteristic
10C22×C22⋊D59—21no—
11C2×#11745671C2214yescomplemented
12C22×C59⋊C8C2212yescomplemented
13#14273701—21no—
14#14273701—21no—
15#14273701C2212yes—
16#14273701C2212yes—
17C2×C4⋊Dic59C2211yescharacteristic
18C2×(C2×D59)⋊C4—21no—
19#14273658C2214yescomplemented
20#14273651C2214yescomplemented
21#14273647C2214yescomplemented
22C22⋊(C59⋊C8)C2212yes—
23C22⋊(C59⋊C8)C2212yescomplemented
24C22×Q8×C59C2211yescharacteristic, complemented
25C23×C4×C59C2211yescharacteristic
26C22⋊Q8×C59—21no—
27C22⋊Q8×C59—21no—
28C22⋊Q8×C59C2212yes—
29C22⋊Q8×C59C2212yescomplemented
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59C2211yescharacteristic, complemented
32C2×C22⋊C4×C59—21no—
33C24×C59D411yescharacteristic
34C23×D59—21no—
35C2×Q8×C59—21no—
36C2×Q8×C59—42no—
37C2×Q8×C59C2312yes—
38C22×C4×C59—21no—
39C22×C4×C59—21no—
40C22×C4×C59—42no—
41C22×C4×C59D411yescharacteristic
42C22×C4×C59D411yescharacteristic
43C22×C4×C59D411yescharacteristic
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2311yescharacteristic
46C22×C4×C59C2311yescharacteristic
47C2×C22⋊D59—22no—
48C2×C22⋊D59—22no—
49C2×C22⋊D59—44no—
50C22×Dic59—21no—
51C2×D236—21no—
52C2×D236—42no—
53C2×D236C2312yes—
54C4⋊C4×C59—22no—
55C4⋊C4×C59—22no—
56C4⋊C4×C59—21no—
57C4⋊C4×C59C2312yes—
58C22⋊C4×C59—22no—
59C22⋊C4×C59—22no—
60#11745671—44no—
61(C2×D59)⋊C4—22no—
62(C2×D59)⋊C4—22no—
63C4⋊Dic59—21no—
64C4⋊Dic59C2312yes—
65C2×C59⋊C8—42no—
66C2×C59⋊C8C2314yes—
67C23×C59—41no—
68C23×C59—41no—
69C23×C59SD1614yescomplemented
70C23×C59C2×D411yescharacteristic
71C23×C59C2×D411yescharacteristic
72C23×C59C2×D411yescharacteristic
73C22×D59—22no—
74C22×D59—42no—
75C22×D59—42no—
76C22×D59—42no—
77Q8×C59—24no—
78Q8×C59—42no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—41no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—42no—
88C2×C4×C59—42no—
89C2×C4×C59C2×D411yescharacteristic
90C2×C4×C59C2×D411yescharacteristic
91C2×C4×C59C2×D411yescharacteristic
92C2×C4×C59C2411yescharacteristic
93C22⋊D59—44no—
94C22⋊D59—44no—
95C2×Dic59—22no—
96C2×Dic59—42no—
97D236—24no—
98D236—42no—
99C59⋊C8—24no—
100C22×C59—21no—
101C22×C59—21no—
102C22×C59—21no—
103C22×C59—21no—
104C22×C59—21no—
105C22×C59—21no—
106C22×C59—44no—
107C22×C59C2×SD1612yes—
108C22×C59C2×SD1612yes—
109C22×C59C2×SD1612yes—
110C22×C59C22×D411yescharacteristic
111C2×D59—42no—
112C2×D59—42no—
113C2×D59—42no—
114C2×D59—84no—
115C4×C59—21no—
116C4×C59—21no—
117C4×C59—21no—
118C4×C59—42no—
119C4×C59—42no—
120C4×C59H(5,𝔽2)12yes—
121Dic59—42no—
122#3187—591nomaximal, Sylow
123C2×C59—22no—
124C2×C59—22no—
125C2×C59—22no—
126C2×C59#53411yescharacteristic
127C2×C59C22×SD1611yescharacteristic
128C2×C59#57511yescharacteristic
129D59—82no—
130C2×C22⋊Q8D5911yescharacteristic, complemented
131C2×#121—591no—
132#483—592no—
133#481—592no—
134#465—594no—
135#450—592no—
136#448—592no—
137C2×C22⋊C8—591no—
138C59#318711yescharacteristic, complemented, Sylow
139C22×Q8C2×D5911yescharacteristic, complemented
140C22×D4—591no—
141C22×D4—1181no—
142C23×C4C2×D5911yescharacteristic
143C2×SD16—594no—
144C22×C8—592no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8C2×D5912yes—
148C22⋊Q8C2×D5912yescomplemented
149#121—592no—
150#121—592no—
151#121—1181no—
152#121—1181no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—591no—
155C2×C4⋊C4C2×D5911yescharacteristic, complemented
156C2×C22⋊C4—21no—
157C2×C22⋊C4—1181no—
158#106—594no—
159#103—594no—
160#102—594no—
161C22⋊C8—592no—
162C22⋊C8—592no—
163C24—1181no—
164C24C22⋊D5911yescharacteristic
165C2×Q8—21no—
166C2×Q8—42no—
167C2×Q8C22×D5912yes—
168C2×D4—592no—
169C2×D4—1182no—
170C2×D4—1182no—
171C2×D4—1181no—
172C2×D4—2362no—
173C2×D4—2364no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4—1181no—
178C22×C4C22⋊D5911yescharacteristic
179C22×C4C22⋊D5911yescharacteristic
180C22×C4C22⋊D5911yescharacteristic
181C22×C4C22×D5911yescharacteristic
182C22×C4C22×D5911yescharacteristic
183C22×C4C22×D5911yescharacteristic
184SD16—2364no—
185C2×C8—594no—
186C2×C8—2362no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—592no—
191C4⋊C4—1181no—
192C4⋊C4C22×D5912yes—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1182no—
196C22⋊C4—1182no—
197C23—41no—
198C23—41no—
199C23—1182no—
200C23—2362no—
201C23—2362no—
202C23—2362no—
203C23#1174567114yescomplemented
204C23C2×C22⋊D5911yescharacteristic
205C23C2×C22⋊D5911yescharacteristic
206C23C2×C22⋊D5911yescharacteristic
207Q8—24no—
208Q8—42no—
209D4—1184no—
210D4—2362no—
211D4—2364no—
212D4—2364no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—1182no—
224C2×C4—2362no—
225C2×C4C2×C22⋊D5911yescharacteristic
226C2×C4C2×C22⋊D5911yescharacteristic
227C2×C4C2×C22⋊D5911yescharacteristic
228C2×C4C23×D5911yescharacteristic
229C8—1184no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—44no—
237C22—4724no—
238C22—2362no—
239C22—2362no—
240C22—2362no—
241C22C2×#1174567112yes—
242C22C2×#1174567112yes—
243C22C2×#1174567112yes—
244C22C22×C22⋊D5911yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—2362no—
251C4#1427377212yes—
252C2—22no—
253C2—22no—
254C2—22no—
255C2—4722no—
256C2#1683796111yescharacteristic
257C2C22×#1174567111yescharacteristic
258C2#1683820011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×C22⋊Q8×C59C2
socleC22×C59C22×D4
hypercenterC2×C22⋊Q8D59
hypocenterC59#3187
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16837730C2semidirect
3. #16837720C2semidirect
4. #16837633C2semidirect
5. #16837552C2semidirect
6. #16837541C2semidirect
7. C2×C22⋊(C59⋊C8)C2semidirect
8. C2×C22⋊Q8×C59C2semidirect
11. C2×#11745671C22semidirect
12. C22×C59⋊C8C22semidirect
19. #14273658C22semidirect
20. #14273651C22semidirect
21. #14273647C22semidirect
23. C22⋊(C59⋊C8)C22semidirect
24. C22×Q8×C59C22semidirect
29. C22⋊Q8×C59C22semidirect
31. C2×C4⋊C4×C59C22semidirect
69. C23×C59SD16semidirect
130. C2×C22⋊Q8D59semidirect
138. C59#3187semidirect
139. C22×Q8C2×D59semidirect
148. C22⋊Q8C2×D59semidirect
155. C2×C4⋊C4C2×D59semidirect
203. C23#11745671semidirect