Finite Group Databasebeta

← #21434928

Subgroups — #21434928

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
2C2×#14273706C211yesmaximal, characteristic, complemented
3#16837744C212yesmaximal, complemented
4#16837738C211yesmaximal, characteristic, complemented
5#16837721C211yesmaximal, characteristic, complemented
6#16837660C212yesmaximal, complemented
7#16837648C212yesmaximal, complemented
8#16837566C212yesmaximal, complemented
9#16837560C211yesmaximal, characteristic, complemented
10#16837543C211yesmaximal, characteristic, complemented
11C59×#407C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic
13C22×Dic118C2211yescharacteristic
14C2×Dic236C2212yescomplemented
15C2×#11745660C2212yescomplemented
16#14273710—21no—
17#14273706—21no—
18#14273706—21no—
19#14273706C2212yes—
20#14273706C2212yes—
21#14273706C2211yescharacteristic
22#14273706C2211yescharacteristic
23#14273701C2211yescharacteristic
24#14273701C2211yescharacteristic
25#14273696—21no—
26C4×C22⋊D59—21no—
27C4×C22⋊D59—21no—
28#14273686C2211yescharacteristic
29C2×C4⋊Dic59C2211yescharacteristic
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32#14273661C2212yescomplemented
33#14273659C2212yescomplemented
34#14273652C2212yescomplemented
35#14273652C2212yescomplemented
36#14273652C2212yescomplemented
37#14273649C2212yescomplemented
38C23×C4×C59C2211yescharacteristic
39C2×M16×C59C2212yes—
40C22⋊C8×C59C2211yescharacteristic
41C22⋊C8×C59C2211yescharacteristic
42C22⋊C8×C59C2212yes—
43C24×C59D411yescharacteristic
44C22×C4×C59—42no—
45C22×C4×C59D411yescharacteristic
46C22×C4×C59D411yescharacteristic
47C22×C4×C59D411yescharacteristic
48C22×C4×C59C2311yescharacteristic
49C22×C4×C59C2311yescharacteristic
50C22×C4×C59C2311yescharacteristic
51C2×C22⋊D59—21no—
52C2×C22⋊D59—21no—
53C2×C22⋊D59—21no—
54C22×Dic59—21no—
55C22×Dic59—21no—
56#11745685—42no—
57C2×D236C2311yescharacteristic
58C2×C4×D59—21no—
59C2×Dic118—21no—
60C2×Dic118—42no—
61C2×Dic118C2312yes—
62C2×Dic118C2311yescharacteristic
63M16×C59—42no—
64C2×C8×C59C2312yes—
65C2×C8×C59C2312yes—
66C22⋊Dic59—22no—
67C22⋊Dic59—22no—
68C22⋊Dic59—21no—
69C22⋊Dic59—21no—
70C22⋊Dic59—21no—
71(C2×D59)⋊C4—21no—
72(C2×D59)⋊C4—21no—
73C4⋊Dic59—21no—
74C4⋊Dic59C2312yes—
75C4⋊Dic59C2311yescharacteristic
76C4⋊Dic59C2311yescharacteristic
77Dic59⋊C4—22no—
78Dic59⋊C4—22no—
79Dic59⋊C4—21no—
80Dic59⋊C4—21no—
81C4×Dic59—21no—
82Dic236—42no—
83#11745660—42no—
84C23×C59—22no—
85C23×C59—41no—
86C23×C59—41no—
87C23×C59C2×D411yescharacteristic
88C23×C59C2×D411yescharacteristic
89C23×C59C2×D411yescharacteristic
90C22×D59—21no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59C2×D411yescharacteristic
98C2×C4×C59C2×D411yescharacteristic
99C2×C4×C59C2×D411yescharacteristic
100C2×C4×C59C2411yescharacteristic
101C22⋊D59—42no—
102C22⋊D59—42no—
103C22⋊D59—42no—
104C2×Dic59—21no—
105C2×Dic59—21no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—21no—
109C2×Dic59—42no—
110C2×Dic59—42no—
111D236—22no—
112C4×D59—42no—
113Dic118—22no—
114Dic118—22no—
115Dic118—22no—
116Dic118—42no—
117C8×C59—22no—
118C8×C59—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×C59H(5,𝔽2)12yes—
137Dic59—42no—
138Dic59—41no—
139Dic59—41no—
140Dic59—42no—
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⋊Q8—591no—
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#407D5911yescharacteristic, complemented
161C59#319611yescharacteristic, complemented, Sylow
162C2×#41—591no—
163C22×Q8—591no—
164C23×C4C2×D5911yescharacteristic
165C2×Q16—592no—
166C2×SD16—592no—
167C2×M16C2×D5912yes—
168#123—1181no—
169C22⋊Q8—592no—
170C22⋊Q8—592no—
171C22⋊Q8—591no—
172C22⋊Q8—591no—
173C22⋊Q8—1181no—
174C22⋊Q8—1181no—
175#121—591no—
176#121—591no—
177#120—1181no—
178C4×D4—1181no—
179C4×D4—1181no—
180#117—591no—
181C2×C4⋊C4—591no—
182C2×C4⋊C4—1181no—
183C2×C22⋊C4—1181no—
184#107—592no—
185#106—592no—
186#103—592no—
187#103—592no—
188#103—592no—
189#102—592no—
190C22⋊C8C2×D5911yescharacteristic
191C22⋊C8C2×D5911yescharacteristic
192C22⋊C8C2×D5912yes—
193C24D23611yescharacteristic
194#41—2362no—
195C2×Q8—592no—
196C2×Q8—591no—
197C2×Q8—1181no—
198C2×Q8—2362no—
199C2×D4—591no—
200C2×D4—1181no—
201C2×D4—1181no—
202C2×D4—1181no—
203C22×C4—42no—
204C22×C4—1181no—
205C22×C4—1181no—
206C22×C4—1181no—
207C22×C4D23611yescharacteristic
208C22×C4D23611yescharacteristic
209C22×C4D23611yescharacteristic
210C22×C4C22×D5911yescharacteristic
211C22×C4C22×D5911yescharacteristic
212C22×C4C22×D5911yescharacteristic
213Q16—2362no—
214SD16—2362no—
215M16—42no—
216C2×C8C22×D5912yes—
217C2×C8C22×D5912yes—
218C4⋊C4—592no—
219C4⋊C4—591no—
220C4⋊C4—591no—
221C4⋊C4—1181no—
222C4⋊C4—1182no—
223C4⋊C4—1182no—
224C4⋊C4—1181no—
225C4⋊C4—1181no—
226C22⋊C4—1182no—
227C22⋊C4—1182no—
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×D23611yescharacteristic
239C23C2×D23611yescharacteristic
240C23C2×D23611yescharacteristic
241Q8—1182no—
242Q8—1182no—
243Q8—1182no—
244Q8—2362no—
245D4—1182no—
246D4—2362no—
247D4—2362no—
248D4—2362no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—1181no—
256C2×C4—1181no—
257C2×C4—1182no—
258C2×C4—1181no—
259C2×C4—1182no—
260C2×C4—2362no—
261C2×C4—2362no—
262C2×C4—2362no—
263C2×C4C2×D23611yescharacteristic
264C2×C4C2×D23611yescharacteristic
265C2×C4C2×D23611yescharacteristic
266C2×C4C23×D5911yescharacteristic
267C8—22no—
268C8—22no—
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#1427376012yes—
282C22C22×D23611yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—2361no—
287C4—2362no—
288C4—2361no—
289C4—2361no—
290C4—2362no—
291C4#1427377212yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—4721no—
296C2#1683796411yescharacteristic
297C2C2×#1427376011yescharacteristic
298C2#1683820211yescharacteristic
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#407C2
socleC22×C59C22×D4
hypercenter#407D59
hypocenterC59#3196
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#14273706C2semidirect
3. #16837744C2semidirect
4. #16837738C2semidirect
5. #16837721C2semidirect
6. #16837660C2semidirect
7. #16837648C2semidirect
8. #16837566C2semidirect
9. #16837560C2semidirect
10. #16837543C2semidirect
11. C59×#407C2semidirect
14. C2×Dic236C22semidirect
15. C2×#11745660C22semidirect
32. #14273661C22semidirect
33. #14273659C22semidirect
34. #14273652C22semidirect
35. #14273652C22semidirect
36. #14273652C22semidirect
37. #14273649C22semidirect
160. #407D59semidirect
161. C59#3196semidirect