Finite Group Databasebeta

← #21434923

Subgroups — #21434923

Order 7552.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 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#16837912C211yesmaximal, characteristic
2C2×#14273703C211yesmaximal, characteristic, complemented
3#16837742C212yesmaximal, complemented
4#16837738C211yesmaximal, characteristic, complemented
5#16837718C211yesmaximal, characteristic, complemented
6#16837659C212yesmaximal, complemented
7#16837645C212yesmaximal, complemented
8#16837561C212yesmaximal, complemented
9#16837558C211yesmaximal, characteristic, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#448C211yesmaximal, characteristic, complemented
12C2×#11745687C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C2×#11745672C2212yescomplemented
15C2×#11745659C2212yes—
16#14273709—21no—
17#14273707C2211yescharacteristic
18#14273706C2211yescharacteristic
19#14273703—21no—
20#14273703—21no—
21#14273703C2212yes—
22#14273703C2212yes—
23#14273698C2211yescharacteristic
24#14273695—21no—
25D4×Dic59—21no—
26(C22⋊D59)⋊C4—21no—
27#14273685C2211yescharacteristic
28C2×C4⋊Dic59C2211yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×(C2×D59)⋊C4—21no—
31#14273661C2212yescomplemented
32#14273658C2212yescomplemented
33#14273652C2212yescomplemented
34#14273651C2212yescomplemented
35#14273648C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37(C2×D59)⋊C8C2212yes—
38C22×Q8×C59C2211yescharacteristic
39C2×SD16×C59C2212yescomplemented
40#121×C59C2211yescharacteristic
41#103×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C23×D59D411yescharacteristic
44C2×Q8×C59—21no—
45C2×Q8×C59—42no—
46C2×Q8×C59C2312yes—
47C2×D4×C59—21no—
48C2×D4×C59C2311yescharacteristic
49C22×C4×C59—21no—
50C22×C4×C59C2311yescharacteristic
51C2×C22⋊D59—21no—
52C2×C22⋊D59—21no—
53C22×Dic59—21no—
54C22×Dic59—21no—
55C22×Dic59D411yescharacteristic
56#11745687—42no—
57C2×C4×D59—42no—
58C2×C4×D59D411yescharacteristic
59C2×C4×D59D411yescharacteristic
60C2×C4×D59C2311yescharacteristic
61C2×C4×D59C2311yescharacteristic
62C2×Dic118C2311yescharacteristic
63SD16×C59—42no—
64C2×C8×C59C2312yes—
65C4⋊C4×C59C2311yescharacteristic
66C22⋊C4×C59—21no—
67C22⋊Dic59—21no—
68C22⋊Dic59—21no—
69#11745672—42no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74C4⋊Dic59—21no—
75C4⋊Dic59—21no—
76C4⋊Dic59C2312yes—
77C4⋊Dic59C2311yescharacteristic
78Dic59⋊C4—21no—
79Dic59⋊C4—22no—
80Dic59⋊C4—22no—
81C4×Dic59—21no—
82C2×C59⋊C8C2312yes—
83#11745659—42no—
84C23×C59—21no—
85C23×C59C2×D411yescharacteristic
86C22×D59—22no—
87C22×D59—41no—
88C22×D59—41no—
89C22×D59C2×D411yescharacteristic
90C22×D59C2×D411yescharacteristic
91Q8×C59—22no—
92Q8×C59—22no—
93Q8×C59—42no—
94D4×C59—22no—
95D4×C59—42no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—42no—
100C2×C4×C59C2×D411yescharacteristic
101C2×C4×C59C2411yescharacteristic
102C22⋊D59—42no—
103C22⋊D59—42no—
104C2×Dic59—21no—
105C2×Dic59—22no—
106C2×Dic59—21no—
107C2×Dic59—41no—
108C2×Dic59—42no—
109C2×Dic59—42no—
110C2×Dic59C2×D411yescharacteristic
111C2×Dic59C2×D411yescharacteristic
112C4×D59—22no—
113C4×D59—22no—
114C4×D59—41no—
115C4×D59—41no—
116Dic118—22no—
117C8×C59—22no—
118C59⋊C8—22no—
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—41no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59#13712yes—
125C22×C59C22×D411yescharacteristic
126C2×D59—21no—
127C2×D59—21no—
128C2×D59—21no—
129C2×D59—21no—
130C2×D59—21no—
131C2×D59—21no—
132C2×D59—44no—
133C4×C59—21no—
134C4×C59—42no—
135C4×C59—41no—
136C4×C59H(5,𝔽2)12yes—
137Dic59—21no—
138Dic59—21no—
139Dic59—41no—
140Dic59—42no—
141Dic59—41no—
142#3196—591nomaximal, Sylow
143C2×C59—22no—
144C2×C59—81no—
145C2×C59#53411yescharacteristic
146C2×C59C2×#13711yescharacteristic
147C2×C59#57511yescharacteristic
148D59—41no—
149D59—41no—
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#448D5911yescharacteristic, complemented
160#407—591no—
161C59#319611yescharacteristic, complemented, Sylow
162C2×#41—591no—
163C22×Q8C2×D5911yescharacteristic
164C23×C4—591no—
165C2×Q16—592no—
166C2×SD16C2×D5912yescomplemented
167C2×M16—592no—
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#121C2×D5911yescharacteristic
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#103C2×D5912yescomplemented
189#102—592no—
190C22⋊C8—591no—
191C22⋊C8—592no—
192C22⋊C8C2×D5911yescharacteristic
193C24—591no—
194#41—2362no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—591no—
198C2×Q8C22×D5912yes—
199C2×D4—21no—
200C2×D4—1181no—
201C2×D4—1181no—
202C2×D4C22×D5911yescharacteristic
203C22×C4—21no—
204C22×C4—591no—
205C22×C4—591no—
206C22×C4—591no—
207C22×C4—591no—
208C22×C4—591no—
209C22×C4—1181no—
210C22×C4—1181no—
211C22×C4—2362no—
212C22×C4C22×D5911yescharacteristic
213Q16—2362no—
214SD16—42no—
215M16—2362no—
216C2×C8—592no—
217C2×C8C22×D5912yes—
218C4⋊C4—592no—
219C4⋊C4—591no—
220C4⋊C4—1182no—
221C4⋊C4—1182no—
222C4⋊C4—1181no—
223C4⋊C4—1181no—
224C4⋊C4—1181no—
225C4⋊C4C22×D5911yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1182no—
228C22⋊C4—1182no—
229C22⋊C4—1181no—
230C22⋊C4—1181no—
231C22⋊C4—1181no—
232C22⋊C4—1181no—
233C42—1181no—
234C23—21no—
235C23—591no—
236C23—591no—
237C23—1182no—
238C23—2361no—
239C23—2361no—
240C23D4×D5911yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—1182no—
245D4—22no—
246D4—42no—
247D4—2362no—
248D4—2362no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—591no—
254C2×C4—591no—
255C2×C4—1182no—
256C2×C4—1182no—
257C2×C4—1181no—
258C2×C4—1182no—
259C2×C4—1181no—
260C2×C4—2361no—
261C2×C4—2361no—
262C2×C4—2361no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4D4×D5911yescharacteristic
266C2×C4C23×D5911yescharacteristic
267C8—22no—
268C8—1182no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—1181no—
275C22—1181no—
276C22—1181no—
277C22—1181no—
278C22—1181no—
279C22—1181no—
280C22—2364no—
281C22#1427375612yes—
282C22C2×D4×D5911yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—1181no—
287C4—1181no—
288C4—2362no—
289C4—2361no—
290C4—2361no—
291C4#1427377212yes—
292C2—22no—
293C2—81no—
294C2—2361no—
295C2—2361no—
296C2#1683796211yescharacteristic
297C2C2×#1427375611yescharacteristic
298C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#448C2
socleC22×C59C22×D4
hypercenter#448D59
hypocenterC59#3196
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C2×#14273703C2semidirect
3. #16837742C2semidirect
4. #16837738C2semidirect
5. #16837718C2semidirect
6. #16837659C2semidirect
7. #16837645C2semidirect
8. #16837561C2semidirect
9. #16837558C2semidirect
10. #16837364C2semidirect
11. C59×#448C2semidirect
14. C2×#11745672C22semidirect
31. #14273661C22semidirect
32. #14273658C22semidirect
33. #14273652C22semidirect
34. #14273651C22semidirect
35. #14273648C22semidirect
39. C2×SD16×C59C22semidirect
41. #103×C59C22semidirect
159. #448D59semidirect
161. C59#3196semidirect
166. C2×SD16C2×D59semidirect
188. #103C2×D59semidirect