Finite Group Databasebeta

← #21434876

Subgroups — #21434876

Order 7552.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14273707C211yesmaximal, characteristic, complemented
3#16837745C212yesmaximal, complemented
4#16837732C211yesmaximal, characteristic, complemented
5#16837644C212yesmaximal, complemented
6#16837630C212yesmaximal, complemented
7#16837565C212yesmaximal, complemented
8#16837560C211yesmaximal, characteristic, complemented
9#16837540C211yesmaximal, characteristic, complemented
10C22⋊C8×D59C211yesmaximal, characteristic, complemented
11C59×#480C211yesmaximal, characteristic, complemented
12C2×#11745687C2211yescharacteristic
13C22×Dic118C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15C2×Dic236C2212yescomplemented
16C2×#11745670C2212yescomplemented
17C2×C8×D59C2212yes—
18#14273709—21no—
19#14273707—21no—
20#14273707—21no—
21#14273707C2211yescharacteristic
22#14273707C2212yes—
23#14273707C2212yes—
24#14273706C2211yescharacteristic
25#14273698C2211yescharacteristic
26#14273695—21no—
27D4×Dic59—21no—
28(C22⋊D59)⋊C4—21no—
29#14273685C2211yescharacteristic
30C2×C4⋊Dic59—21no—
31C2×(C2×D59)⋊C4—21no—
32#14273658C2212yescomplemented
33#14273652C2212yescomplemented
34#14273650C2212yescomplemented
35#14273650C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic
37(C2×D59)⋊C8C2212yes—
38#121×C59C2211yescharacteristic
39C2×C4⋊C4×C59C2211yescharacteristic
40#107×C59C2212yescomplemented
41#102×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C23×D59D411yescharacteristic
44C2×D4×C59—21no—
45C2×D4×C59C2311yescharacteristic
46C22×C4×C59—21no—
47C22×C4×C59C2311yescharacteristic
48C2×C22⋊D59—21no—
49C2×C22⋊D59—21no—
50C22×Dic59—21no—
51C22×Dic59—21no—
52C22×Dic59D411yescharacteristic
53#11745687—42no—
54C2×C4×D59—42no—
55C2×C4×D59D411yescharacteristic
56C2×C4×D59D411yescharacteristic
57C2×C4×D59C2311yescharacteristic
58C2×C4×D59C2311yescharacteristic
59C2×Dic118—21no—
60C2×Dic118—42no—
61C2×Dic118C2311yescharacteristic
62C2×Dic118C2312yes—
63C2×C8×C59C2312yes—
64C4⋊C4×C59—21no—
65C4⋊C4×C59C2312yes—
66C4⋊C4×C59C2311yescharacteristic
67C22⋊C4×C59—21no—
68C22⋊Dic59—21no—
69C22⋊Dic59—21no—
70#11745670—42no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—22no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—21no—
75C4⋊Dic59—22no—
76C4⋊Dic59—22no—
77C4⋊Dic59—21no—
78C4⋊Dic59C2311yescharacteristic
79Dic59⋊C4—21no—
80C4×Dic59—21no—
81C2×C59⋊C8C2312yes—
82Dic236—42no—
83C8×D59—42no—
84C23×C59—21no—
85C23×C59C2×D411yescharacteristic
86C22×D59—22no—
87C22×D59—22no—
88C22×D59—41no—
89C22×D59C2×D411yescharacteristic
90C22×D59C2×D411yescharacteristic
91D4×C59—22no—
92D4×C59—42no—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—42no—
97C2×C4×C59C2×D411yescharacteristic
98C2×C4×C59C2411yescharacteristic
99C22⋊D59—42no—
100C22⋊D59—42no—
101C2×Dic59—21no—
102C2×Dic59—21no—
103C2×Dic59—22no—
104C2×Dic59—41no—
105C2×Dic59—42no—
106C2×Dic59—42no—
107C2×Dic59C2×D411yescharacteristic
108C2×Dic59C2×D411yescharacteristic
109C4×D59—22no—
110C4×D59—22no—
111C4×D59—41no—
112C4×D59—41no—
113Dic118—22no—
114Dic118—22no—
115Dic118—22no—
116Dic118—42no—
117C8×C59—22no—
118C59⋊C8—22no—
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—41no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59#13512yes—
125C22×C59C22×D411yescharacteristic
126C2×D59—21no—
127C2×D59—21no—
128C2×D59—21no—
129C2×D59—21no—
130C2×D59—21no—
131C2×D59—44no—
132C2×D59#13712yes—
133C4×C59—21no—
134C4×C59—42no—
135C4×C59—41no—
136C4×C59H(5,𝔽2)12yes—
137Dic59—21no—
138Dic59—21no—
139Dic59—42no—
140Dic59—41no—
141Dic59—41no—
142#3190—591nomaximal, Sylow
143C2×C59—22no—
144C2×C59—81no—
145C2×C59#53411yescharacteristic
146C2×C59C2×#13511yescharacteristic
147C2×C59C2×#13711yescharacteristic
148D59—22no—
149D59—41no—
150#525—591no—
151C2×C22⋊Q8—591no—
152#485—592no—
153#483—591no—
154#480D5911yescharacteristic, complemented
155#467—592no—
156#465—592no—
157#452—592no—
158#451—591no—
159#448—591no—
160C2×C22⋊C8—591no—
161C59#319011yescharacteristic, complemented, Sylow
162C2×#41—591no—
163C22×Q8—591no—
164C23×C4—591no—
165C2×Q16—592no—
166C2×SD16—592no—
167C22×C8—592no—
168#123—1181no—
169C22⋊Q8—591no—
170C22⋊Q8—592no—
171C22⋊Q8—592no—
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—1181no—
182C2×C4⋊C4C2×D5911yescharacteristic
183C2×C22⋊C4—1181no—
184#107C2×D5912yescomplemented
185#106—592no—
186#103—592no—
187#103—592no—
188#103—592no—
189#102C2×D5912yescomplemented
190C22⋊C8—591no—
191C22⋊C8—592no—
192C22⋊C8C2×D5911yescharacteristic
193C24—591no—
194#41—2362no—
195C2×Q8—592no—
196C2×Q8—591no—
197C2×Q8—1181no—
198C2×Q8—2362no—
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—2362no—
215C2×C8—592no—
216C2×C8—2362no—
217C2×C8C22×D5912yes—
218C4⋊C4—21no—
219C4⋊C4—591no—
220C4⋊C4—1182no—
221C4⋊C4—1182no—
222C4⋊C4—1181no—
223C4⋊C4—1181no—
224C4⋊C4C22×D5912yes—
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—1182no—
239C23—2361no—
240C23D4×D5911yescharacteristic
241Q8—1182no—
242Q8—1182no—
243Q8—1182no—
244Q8—2362no—
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—1181no—
259C2×C4—1182no—
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—592no—
275C22—1181no—
276C22—1181no—
277C22—1181no—
278C22—1181no—
279C22—1181no—
280C22—2364no—
281C22#1427374612yes—
282C22C2×D4×D5911yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—1181no—
287C4—1181no—
288C4—2361no—
289C4—2361no—
290C4—2362no—
291C4#1427377312yes—
292C2—22no—
293C2—81no—
294C2—1182no—
295C2—2361no—
296C2#1683796311yescharacteristic
297C2C2×#1427374611yescharacteristic
298C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#480C2
socleC22×C59C22×D4
hypercenter#480D59
hypocenterC59#3190
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#14273707C2semidirect
3. #16837745C2semidirect
4. #16837732C2semidirect
5. #16837644C2semidirect
6. #16837630C2semidirect
7. #16837565C2semidirect
8. #16837560C2semidirect
9. #16837540C2semidirect
10. C22⋊C8×D59C2semidirect
11. C59×#480C2semidirect
15. C2×Dic236C22semidirect
16. C2×#11745670C22semidirect
32. #14273658C22semidirect
33. #14273652C22semidirect
34. #14273650C22semidirect
35. #14273650C22semidirect
40. #107×C59C22semidirect
41. #102×C59C22semidirect
154. #480D59semidirect
161. C59#3190semidirect
184. #107C2×D59semidirect
189. #102C2×D59semidirect