Finite Group Databasebeta

← #21434926

Subgroups — #21434926

Order 7552.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 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#16837913C211yesmaximal, characteristic
2#16837740C212yesmaximal, complemented
3#16837737C211yesmaximal, characteristic, complemented
4#16837720C211yesmaximal, characteristic, complemented
5#16837658C212yesmaximal, complemented
6#16837650C212yesmaximal, complemented
7#16837562C212yesmaximal, complemented
8#16837558C211yesmaximal, characteristic, complemented
9#16837541C211yesmaximal, characteristic, complemented
10#16837366C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C59C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic
13C2×#11745672C2212yescomplemented
14C2×#11745671C2212yescomplemented
15C2×#11745664C2212yes—
16#14273710—21no—
17#14273706C2211yescharacteristic
18#14273706C2211yescharacteristic
19#14273701C2211yescharacteristic
20#14273701C2211yescharacteristic
21#14273696—21no—
22C4×C22⋊D59—21no—
23C4×C22⋊D59—21no—
24#14273686C2211yescharacteristic
25#14273660C2212yescomplemented
26#14273658C2212yescomplemented
27#14273651C2212yescomplemented
28#14273651C2212yescomplemented
29#14273650C2212yescomplemented
30#14273647C2212yescomplemented
31C22⋊(C59⋊C8)C2211yescharacteristic
32C22⋊(C59⋊C8)C2211yescharacteristic
33C22⋊(C59⋊C8)C2212yes—
34C22×Q8×C59C2211yescharacteristic
35C23×C4×C59C2211yescharacteristic
36C22⋊Q8×C59—21no—
37C22⋊Q8×C59—21no—
38C22⋊Q8×C59C2212yes—
39C22⋊Q8×C59C2212yes—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59C2211yescharacteristic
42C2×C22⋊C4×C59—21no—
43C24×C59D411yescharacteristic
44C2×Q8×C59—21no—
45C2×Q8×C59—42no—
46C2×Q8×C59C2312yes—
47C22×C4×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59—42no—
50C22×C4×C59D411yescharacteristic
51C22×C4×C59D411yescharacteristic
52C22×C4×C59D411yescharacteristic
53C22×C4×C59C2311yescharacteristic
54C22×C4×C59C2311yescharacteristic
55C22×C4×C59C2311yescharacteristic
56C2×C22⋊D59—21no—
57C2×C22⋊D59—21no—
58C2×C22⋊D59—21no—
59#11745685—42no—
60C2×D236C2311yescharacteristic
61C2×C4×D59—21no—
62C2×Dic118C2311yescharacteristic
63C4⋊C4×C59—22no—
64C4⋊C4×C59—22no—
65C4⋊C4×C59—21no—
66C4⋊C4×C59C2312yes—
67C22⋊C4×C59—22no—
68C22⋊C4×C59—22no—
69C22⋊Dic59—21no—
70C22⋊Dic59—21no—
71C22⋊Dic59—21no—
72#11745672—42no—
73#11745671—42no—
74(C2×D59)⋊C4—21no—
75(C2×D59)⋊C4—21no—
76C4⋊Dic59C2311yescharacteristic
77C4⋊Dic59C2311yescharacteristic
78Dic59⋊C4—21no—
79Dic59⋊C4—21no—
80C4×Dic59—21no—
81#11745664—42no—
82C2×C59⋊C8C2312yes—
83C2×C59⋊C8C2312yes—
84C23×C59—22no—
85C23×C59—41no—
86C23×C59—41no—
87C23×C59C2×D411yescharacteristic
88C23×C59C2×D411yescharacteristic
89C23×C59C2×D411yescharacteristic
90C22×D59—21no—
91Q8×C59—22no—
92Q8×C59—22no—
93Q8×C59—42no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59C2×D411yescharacteristic
105C2×C4×C59C2×D411yescharacteristic
106C2×C4×C59C2×D411yescharacteristic
107C2×C4×C59C2411yescharacteristic
108C22⋊D59—42no—
109C22⋊D59—42no—
110C22⋊D59—42no—
111C2×Dic59—21no—
112C2×Dic59—21no—
113C2×Dic59—21no—
114D236—22no—
115C4×D59—42no—
116Dic118—22no—
117C59⋊C8—22no—
118C59⋊C8—22no—
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—21no—
122C22×C59—21no—
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—44no—
128C22×C59#13712yes—
129C22×C59C22×D411yescharacteristic
130C2×D59—41no—
131C2×D59—41no—
132C2×D59—41no—
133C4×C59—21no—
134C4×C59—21no—
135C4×C59—21no—
136C4×C59—42no—
137C4×C59—42no—
138C4×C59H(5,𝔽2)12yes—
139Dic59—41no—
140Dic59—41no—
141Dic59—41no—
142#3196—591nomaximal, Sylow
143C2×C59—22no—
144C2×C59—41no—
145C2×C59—41no—
146C2×C59#53411yescharacteristic
147C2×C59C2×#13711yescharacteristic
148C2×C59#57511yescharacteristic
149D59—81no—
150#525—591no—
151C2×C22⋊Q8D5911yescharacteristic, complemented
152#485—592no—
153#484—591no—
154#481—591no—
155#470—592no—
156#468—592no—
157#452—592no—
158#451—591no—
159#448—591no—
160#407—591no—
161C59#319611yescharacteristic, complemented, Sylow
162C2×#41—591no—
163C22×Q8C2×D5911yescharacteristic
164C23×C4C2×D5911yescharacteristic
165C2×Q16—592no—
166C2×SD16—592no—
167C2×M16—592no—
168#123—1181no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—591no—
172C22⋊Q8—591no—
173C22⋊Q8C2×D5912yes—
174C22⋊Q8C2×D5912yes—
175#121—591no—
176#121—591no—
177#120—1181no—
178C4×D4—1181no—
179C4×D4—1181no—
180#117—591no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×D5911yescharacteristic
183C2×C22⋊C4—21no—
184#107—592no—
185#106—592no—
186#103—592no—
187#103—592no—
188#103—592no—
189#102—592no—
190C22⋊C8—591no—
191C22⋊C8—591no—
192C22⋊C8—592no—
193C24C22⋊D5911yescharacteristic
194#41—2362no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—591no—
198C2×Q8C22×D5912yes—
199C2×D4—591no—
200C2×D4—1181no—
201C2×D4—1181no—
202C2×D4—1181no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—1181no—
207C22×C4C22⋊D5911yescharacteristic
208C22×C4C22⋊D5911yescharacteristic
209C22×C4C22⋊D5911yescharacteristic
210C22×C4C22×D5911yescharacteristic
211C22×C4C22×D5911yescharacteristic
212C22×C4C22×D5911yescharacteristic
213Q16—2362no—
214SD16—2362no—
215M16—2362no—
216C2×C8—592no—
217C2×C8—592no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—591no—
222C4⋊C4—591no—
223C4⋊C4—1181no—
224C4⋊C4—1181no—
225C4⋊C4C22×D5912yes—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—1181no—
229C22⋊C4—1181no—
230C22⋊C4—1181no—
231C22⋊C4—1181no—
232C22⋊C4—1181no—
233C42—1181no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—1181no—
238C23C2×C22⋊D5911yescharacteristic
239C23C2×C22⋊D5911yescharacteristic
240C23C2×C22⋊D5911yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—1182no—
245D4—1182no—
246D4—2362no—
247D4—2362no—
248D4—2362no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1181no—
260C2×C4—1181no—
261C2×C4—1181no—
262C2×C4—2362no—
263C2×C4C2×C22⋊D5911yescharacteristic
264C2×C4C2×C22⋊D5911yescharacteristic
265C2×C4C2×C22⋊D5911yescharacteristic
266C2×C4C23×D5911yescharacteristic
267C8—1182no—
268C8—1182no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—2361no—
279C22—2361no—
280C22—2361no—
281C22#1427375812yes—
282C22C22×C22⋊D5911yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—2361no—
289C4—2361no—
290C4—2361no—
291C4#1427377212yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—4721no—
296C2#1683796111yescharacteristic
297C2C2×#1427375811yescharacteristic
298C2#1683820011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×C22⋊Q8×C59C2
socleC22×C59C22×D4
hypercenterC2×C22⋊Q8D59
hypocenterC59#3196
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. #16837740C2semidirect
3. #16837737C2semidirect
4. #16837720C2semidirect
5. #16837658C2semidirect
6. #16837650C2semidirect
7. #16837562C2semidirect
8. #16837558C2semidirect
9. #16837541C2semidirect
10. #16837366C2semidirect
11. C2×C22⋊Q8×C59C2semidirect
13. C2×#11745672C22semidirect
14. C2×#11745671C22semidirect
25. #14273660C22semidirect
26. #14273658C22semidirect
27. #14273651C22semidirect
28. #14273651C22semidirect
29. #14273650C22semidirect
30. #14273647C22semidirect
151. C2×C22⋊Q8D59semidirect
161. C59#3196semidirect