Finite Group Databasebeta

← #21434904

Subgroups — #21434904

Order 7552.

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

normal subgroups only

This group has 462 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
1C2×#14273707C211yesmaximal, characteristic, complemented
2#121×D59C211yesmaximal, characteristic
3#16837739C212yesmaximal, complemented
4#16837715C211yesmaximal, characteristic, complemented
5#16837649C212yesmaximal, complemented
6#16837647C212yesmaximal, complemented
7#16837554C212yesmaximal, complemented
8#16837543C211yesmaximal, characteristic, complemented
9#16837540C211yesmaximal, characteristic, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#480C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic, complemented
13C2×D4×D59—21no—
14C2×D4×D59C2211yescharacteristic
15C22×C4×D59C2211yescharacteristic
16C2×#11745660C2212yescomplemented
17C2×#11745670C2212yescomplemented
18C2×#11745659C2212yescomplemented
19#14273707—21no—
20#14273707—21no—
21#14273707C2212yes—
22#14273707C2212yescomplemented
23#14273702C2211yescharacteristic
24#14273701C2211yescharacteristic
25#14273700—21no—
26#14273699—21no—
27#14273698C2211yescharacteristic
28C2×C4⋊Dic59—21no—
29C4⋊C4×D59C2211yescharacteristic
30C2×(C2×D59)⋊C4—21no—
31C22⋊C4×D59—21no—
32#14273660C2212yescomplemented
33#14273652C2212yescomplemented
34#14273650C2212yescomplemented
35#14273647C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic, complemented
37(C2×D59)⋊C8C2212yes—
38#121×C59C2211yescharacteristic
39C2×C4⋊C4×C59C2211yescharacteristic, complemented
40#107×C59C2212yescomplemented
41#102×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic, complemented
43C23×D59—21no—
44C23×D59D411yescharacteristic
45C2×D4×C59—21no—
46C2×D4×C59C2311yescharacteristic
47C22×C4×C59—21no—
48C22×C4×C59C2311yescharacteristic
49C2×C22⋊D59—21no—
50C2×C22⋊D59—21no—
51C2×C22⋊D59—21no—
52C22×Dic59—21no—
53C22×Dic59D411yescharacteristic
54D4×D59—42no—
55D4×D59—44no—
56C2×D236—21no—
57C2×D236C2311yescharacteristic
58C2×C4×D59—21no—
59C2×C4×D59—42no—
60C2×C4×D59D411yescharacteristic
61C2×C4×D59D411yescharacteristic
62C2×C4×D59C2311yescharacteristic
63C2×C4×D59C2311yescharacteristic
64C2×Dic118—21no—
65C2×Dic118—42no—
66C2×Dic118C2312yes—
67C2×C8×C59C2312yes—
68C4⋊C4×C59—21no—
69C4⋊C4×C59C2312yes—
70C4⋊C4×C59C2311yescharacteristic
71C22⋊C4×C59—21no—
72C22⋊Dic59—21no—
73#11745670—42no—
74(C2×D59)⋊C4—22no—
75(C2×D59)⋊C4—22no—
76(C2×D59)⋊C4—21no—
77(C2×D59)⋊C4—21no—
78C4⋊Dic59—22no—
79C4⋊Dic59—22no—
80C4⋊Dic59C2311yescharacteristic
81Dic59⋊C4—21no—
82C2×C59⋊C8C2312yes—
83#11745660—42no—
84#11745659—42no—
85C23×C59—21no—
86C23×C59C2×D411yescharacteristic
87C22×D59—22no—
88C22×D59—22no—
89C22×D59—21no—
90C22×D59—41no—
91C22×D59—42no—
92C22×D59—42no—
93C22×D59—42no—
94C22×D59C2×D411yescharacteristic
95C22×D59C2×D411yescharacteristic
96D4×C59—22no—
97D4×C59—42no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—42no—
102C2×C4×C59C2×D411yescharacteristic
103C2×C4×C59C2411yescharacteristic
104C22⋊D59—42no—
105C22⋊D59—42no—
106C22⋊D59—42no—
107C2×Dic59—21no—
108C2×Dic59—22no—
109C2×Dic59—41no—
110C2×Dic59—42no—
111C2×Dic59C2×D411yescharacteristic
112C2×Dic59C2×D411yescharacteristic
113D236—22no—
114D236—42no—
115C4×D59—22no—
116C4×D59—22no—
117C4×D59—41no—
118C4×D59—41no—
119C4×D59—42no—
120Dic118—22no—
121Dic118—22no—
122Dic118—42no—
123C8×C59—22no—
124C59⋊C8—22no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59—41no—
130C22×C59AGL(1,ℤ/8)12yes—
131C22×C59C22×D411yescharacteristic
132C2×D59—21no—
133C2×D59—21no—
134C2×D59—21no—
135C2×D59—21no—
136C2×D59—21no—
137C2×D59—44no—
138C2×D59—41no—
139C2×D59—41no—
140C2×D59—41no—
141C2×D59—84no—
142C2×D59#13712yes—
143C4×C59—21no—
144C4×C59—41no—
145C4×C59—42no—
146C4×C59H(5,𝔽2)12yes—
147Dic59—21no—
148Dic59—21no—
149Dic59—42no—
150Dic59—41no—
151#3193—591nomaximal, Sylow
152C2×C59—22no—
153C2×C59—81no—
154C2×C59#53411yescharacteristic
155C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C59C2×#13711yescharacteristic
157D59—22no—
158D59—41no—
159D59—81no—
160C2×C22⋊Q8—591no—
161C2×#121—591no—
162#484—592no—
163#480—591no—
164#480D5911yescharacteristic, complemented
165#468—592no—
166#468—592no—
167#450—592no—
168#448—591no—
169#448—591no—
170#407—591no—
171C59#319311yescharacteristic, complemented, Sylow
172C22×Q8—591no—
173C22×D4—591no—
174C22×D4—1181no—
175C23×C4—591no—
176C2×SD16—592no—
177C2×SD16—592no—
178C2×M16—592no—
179C22⋊Q8—592no—
180C22⋊Q8—592no—
181C22⋊Q8—1181no—
182C22⋊Q8—1181no—
183#121—591no—
184#121—591no—
185#121—591no—
186#121—1181no—
187#121—1181no—
188#121C2×D5911yescharacteristic
189C2×C4⋊C4—591no—
190C2×C4⋊C4—1181no—
191C2×C4⋊C4C2×D5911yescharacteristic, complemented
192C2×C22⋊C4—1181no—
193C2×C22⋊C4—1181no—
194#107—592no—
195#107C2×D5912yescomplemented
196#103—592no—
197#103—592no—
198#102—592no—
199#102C2×D5912yescomplemented
200C22⋊C8—591no—
201C22⋊C8—592no—
202C22⋊C8C2×D5911yescharacteristic, complemented
203C24—591no—
204C24—1181no—
205C2×Q8—592no—
206C2×Q8—1181no—
207C2×Q8—2362no—
208C2×D4—21no—
209C2×D4—591no—
210C2×D4—1181no—
211C2×D4—1181no—
212C2×D4—1181no—
213C2×D4—1181no—
214C2×D4—2362no—
215C2×D4—2364no—
216C2×D4C22×D5911yescharacteristic
217C22×C4—21no—
218C22×C4—591no—
219C22×C4—591no—
220C22×C4—591no—
221C22×C4—591no—
222C22×C4—591no—
223C22×C4—1181no—
224C22×C4—1181no—
225C22×C4—2362no—
226C22×C4C22×D5911yescharacteristic
227SD16—2362no—
228SD16—2362no—
229M16—2362no—
230C2×C8—592no—
231C2×C8C22×D5912yes—
232C4⋊C4—21no—
233C4⋊C4—591no—
234C4⋊C4—1181no—
235C4⋊C4—1182no—
236C4⋊C4—1182no—
237C4⋊C4C22×D5912yes—
238C4⋊C4C22×D5911yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—1181no—
241C22⋊C4—1182no—
242C22⋊C4—1182no—
243C22⋊C4—1181no—
244C22⋊C4—1181no—
245C23—21no—
246C23—591no—
247C23—591no—
248C23—1182no—
249C23—1182no—
250C23—1181no—
251C23—2361no—
252C23—2362no—
253C23—2362no—
254C23—2362no—
255C23D4×D5911yescharacteristic
256Q8—1182no—
257Q8—1182no—
258Q8—2362no—
259D4—22no—
260D4—42no—
261D4—1182no—
262D4—2362no—
263D4—2362no—
264D4—2362no—
265D4—2362no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—591no—
271C2×C4—591no—
272C2×C4—1182no—
273C2×C4—1182no—
274C2×C4—1181no—
275C2×C4—1182no—
276C2×C4—2361no—
277C2×C4—2361no—
278C2×C4—2362no—
279C2×C4—2361no—
280C2×C4—2362no—
281C2×C4D4×D5911yescharacteristic
282C2×C4C23×D5911yescharacteristic
283C8—22no—
284C8—1182no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—592no—
291C22—1181no—
292C22—1181no—
293C22—1181no—
294C22—1181no—
295C22—1181no—
296C22—4724no—
297C22—2364no—
298C22—2361no—
299C22—2361no—
300C22—2361no—
301C22#1427375112yes—
302C22C2×D4×D5911yescharacteristic
303C4—21no—
304C4—42no—
305C4—41no—
306C4—1181no—
307C4—1181no—
308C4—2361no—
309C4—2362no—
310C4#1427377312yes—
311C2—22no—
312C2—81no—
313C2—1182no—
314C2—4721no—
315C2—2361no—
316C2#1683796311yescharacteristic
317C2C2×#1427375111yescharacteristic
318C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#480C2
socleC22×C59C22×D4
hypercenter#480D59
hypocenterC59#3193
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
1. C2×#14273707C2semidirect
3. #16837739C2semidirect
4. #16837715C2semidirect
5. #16837649C2semidirect
6. #16837647C2semidirect
7. #16837554C2semidirect
8. #16837543C2semidirect
9. #16837540C2semidirect
10. #16837364C2semidirect
11. C59×#480C2semidirect
12. C22×Dic118C22semidirect
16. C2×#11745660C22semidirect
17. C2×#11745670C22semidirect
18. C2×#11745659C22semidirect
22. #14273707C22semidirect
32. #14273660C22semidirect
33. #14273652C22semidirect
34. #14273650C22semidirect
35. #14273647C22semidirect
36. C22⋊(C59⋊C8)C22semidirect
39. C2×C4⋊C4×C59C22semidirect
40. #107×C59C22semidirect
41. #102×C59C22semidirect
42. C22⋊C8×C59C22semidirect
164. #480D59semidirect
171. C59#3193semidirect
191. C2×C4⋊C4C2×D59semidirect
195. #107C2×D59semidirect
199. #102C2×D59semidirect
202. C22⋊C8C2×D59semidirect