Finite Group Databasebeta

← #21433571

Subgroups — #21433571

Order 7552.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14273751C212yesmaximal, complemented
2#127×D59C211yesmaximal, characteristic, complemented
3C4×#11745687C211yesmaximal, characteristic, complemented
4#16837603C211yesmaximal, characteristic, complemented
5#16837601C212yesmaximal, complemented
6#16837599C211yesmaximal, characteristic, complemented
7#16837597C211yesmaximal, characteristic, complemented
8#16837595C212yesmaximal, complemented
9#16837424C211yesmaximal, characteristic, complemented
10#16837405C212yesmaximal, complemented
11C59×#459C211yesmaximal, characteristic, complemented
12C2×#11745687C2211yescharacteristic
13C2×D4×D59—21no—
14C2×D4×D59—21no—
15C2×D4×D59C2212yes—
16#14273751—28no—
17C2×#11745660C2212yescomplemented
18C2×#11745670C2212yescomplemented
19C2×#11745669C2212yescomplemented
20C2×#11745659C2212yes—
21#14273724C2211yescharacteristic
22#14273723—21no—
23#14273723—21no—
24#14273723C2212yes—
25C4×Dic118C2211yescharacteristic, complemented
26#14273692C2211yescharacteristic, complemented
27D4×Dic59C2211yescharacteristic, complemented
28C4×C22⋊D59—21no—
29(C22⋊D59)⋊C4—21no—
30#14273685C2211yescharacteristic
31#14273683—21no—
32C2×C4×Dic59—21no—
33C42×D59C2211yescharacteristic
34C4⋊(C59⋊C8)C2211yescharacteristic, complemented
35Dic59⋊C8C2212yescomplemented
36#14273649C2212yescomplemented
37#14273648C2212yescomplemented
38#14273647C2212yescomplemented
39C2×D8×C59C2212yescomplemented
40#127×C59C2211yescharacteristic
41C4×D4×C59C2211yescharacteristic, complemented
42C4⋊C8×C59C2211yescharacteristic, complemented
43#102×C59C2212yescomplemented
44C23×D59—22no—
45C2×D4×C59—21no—
46C2×D4×C59—21no—
47C2×D4×C59C2312yes—
48C2×D4×C59C2311yescharacteristic
49C22×C4×C59—21no—
50C2×C22⋊D59—21no—
51C2×C22⋊D59—22no—
52C2×C22⋊D59—22no—
53C2×C22⋊D59—22no—
54C22×Dic59—21no—
55C22×Dic59—21no—
56#11745687—22no—
57#11745687D414yescomplemented
58D4×D59—24no—
59D4×D59—24no—
60D4×D59—44no—
61D4×D59—44no—
62C2×D236—21no—
63C2×D236—21no—
64C2×D236C2312yes—
65C2×C4×D59D411yescharacteristic
66C2×C4×D59D411yescharacteristic
67C2×C4×D59C2311yescharacteristic
68C2×Dic118C2311yescharacteristic
69D8×C59—24no—
70C2×C8×C59C2312yes—
71C4⋊C4×C59C2311yescharacteristic
72C22⋊C4×C59—21no—
73C42×C59C2311yescharacteristic
74C22⋊Dic59—21no—
75#11745670—24no—
76#11745669—24no—
77(C2×D59)⋊C4—21no—
78C4⋊Dic59C2311yescharacteristic
79Dic59⋊C4—21no—
80Dic59⋊C4—21no—
81C4×Dic59—21no—
82C4×Dic59—21no—
83C4×Dic59—21no—
84C4×Dic59D411yescharacteristic
85C4×Dic59D411yescharacteristic
86C4×Dic59C2311yescharacteristic
87C2×C59⋊C8C2312yes—
88#11745660—24no—
89#11745659—24no—
90C23×C59—21no—
91C23×C59—22no—
92C22×D59—22no—
93C22×D59—44no—
94C22×D59—44no—
95C22×D59—44no—
96C22×D59C2×D411yescharacteristic
97D4×C59—22no—
98D4×C59—21no—
99D4×C59—22no—
100D4×C59—42no—
101D4×C59—42no—
102D4×C59C2×D412yescomplemented
103C2×C4×C59—21no—
104C2×C4×C59—42no—
105C2×C4×C59C2×D411yescharacteristic
106C2×C4×C59C2×D411yescharacteristic
107C2×C4×C59C2411yescharacteristic
108C22⋊D59—24no—
109C22⋊D59—44no—
110C22⋊D59—44no—
111C22⋊D59—44no—
112C2×Dic59—21no—
113C2×Dic59—24no—
114C2×Dic59—21no—
115C2×Dic59—42no—
116C2×Dic59C2×D411yescharacteristic
117C2×Dic59C2×D411yescharacteristic
118C2×Dic59C2×D411yescharacteristic
119D236—22no—
120D236—22no—
121D236—42no—
122D236—42no—
123C4×D59—22no—
124C4×D59—22no—
125C4×D59C2×D412yes—
126C4×D59#4112yes—
127Dic118—21no—
128Dic118C2×D412yescomplemented
129C8×C59—22no—
130C59⋊C8—22no—
131C22×C59—22no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—42no—
137C22×C59C22×D411yescharacteristic
138C2×D59—21no—
139C2×D59—21no—
140C2×D59—42no—
141C2×D59—42no—
142C2×D59—42no—
143C2×D59—88no—
144C2×D59#12112yes—
145C4×C59—21no—
146C4×C59—41no—
147C4×C59AGL(1,ℤ/8)12yescomplemented
148C4×C59C22×D411yescharacteristic
149C4×C59C2×#4111yescharacteristic
150Dic59—22no—
151Dic59—21no—
152Dic59—41no—
153Dic59#12112yes—
154Dic59AGL(1,ℤ/8)12yescomplemented
155#3038—591nomaximal, Sylow
156C2×C59—42no—
157C2×C59—82no—
158C2×C59C2×#12111yescharacteristic
159C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
160C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
161D59—22no—
162D59—82no—
163C2×AGL(1,ℤ/8)—592no—
164C2×#127—591no—
165C4×#41—591no—
166#460—591no—
167#460—591no—
168#460—592no—
169#459—592no—
170#459—591no—
171#459D5911yescharacteristic, complemented
172#423—591no—
173#418—592no—
174C59#303811yescharacteristic, complemented, Sylow
175C2×#41—591no—
176C22×D4—592no—
177C22×D4—1181no—
178C22×D4—1181no—
179AGL(1,ℤ/8)—1188no—
180C2×SD16—592no—
181C2×SD16—592no—
182C2×D8—592no—
183C2×D8C2×D5912yescomplemented
184C2×M16—592no—
185#127—592no—
186#127—591no—
187#127—1181no—
188#127—1181no—
189#127C2×D5911yescharacteristic
190C4×Q8—591no—
191C4×Q8—591no—
192C4×D4—591no—
193C4×D4—1181no—
194C4×D4—1181no—
195C4×D4C2×D5911yescharacteristic, complemented
196#117—591no—
197#117—1181no—
198C2×C42—591no—
199C2×C42—1181no—
200C4⋊C8—592no—
201C4⋊C8—591no—
202C4⋊C8C2×D5911yescharacteristic, complemented
203#102—592no—
204#102—592no—
205#102—592no—
206#102C2×D5912yescomplemented
207C24—1182no—
208#41—594no—
209#41—1182no—
210C2×Q8—591no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—592no—
214C2×D4—1184no—
215C2×D4—1184no—
216C2×D4—1182no—
217C2×D4—1182no—
218C2×D4—1182no—
219C2×D4—1181no—
220C2×D4—1181no—
221C2×D4—1181no—
222C2×D4—2364no—
223C2×D4—2364no—
224C2×D4C22×D5912yes—
225C2×D4C22×D5911yescharacteristic
226C22×C4—21no—
227C22×C4—591no—
228C22×C4—591no—
229C22×C4—591no—
230C22×C4—1181no—
231C22×C4—1181no—
232SD16—1184no—
233SD16—1184no—
234D8—24no—
235D8—1184no—
236M16—1184no—
237C2×C8—592no—
238C2×C8C22×D5912yes—
239C4⋊C4—591no—
240C4⋊C4—1181no—
241C4⋊C4—1181no—
242C4⋊C4C22×D5911yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—1181no—
245C22⋊C4—1181no—
246C42—591no—
247C42—591no—
248C42—591no—
249C42—1181no—
250C42—1181no—
251C42—1181no—
252C42C22×D5911yescharacteristic
253C23—21no—
254C23—22no—
255C23—591no—
256C23—1182no—
257C23—2364no—
258C23—2364no—
259C23—2364no—
260Q8—592no—
261Q8—1181no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—1184no—
268D4—1182no—
269D4—1182no—
270D4—2362no—
271D4—2364no—
272D4—2364no—
273D4—2364no—
274D4—2362no—
275D4D4×D5912yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—591no—
279C2×C4—591no—
280C2×C4—592no—
281C2×C4—592no—
282C2×C4—591no—
283C2×C4—1182no—
284C2×C4—1184no—
285C2×C4—1181no—
286C2×C4—1182no—
287C2×C4—1181no—
288C2×C4—2362no—
289C2×C4D4×D5911yescharacteristic
290C2×C4D4×D5911yescharacteristic
291C2×C4C23×D5911yescharacteristic
292C8—22no—
293C8—1182no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—592no—
301C22—1181no—
302C22—1181no—
303C22—4728no—
304C22—2362no—
305C22—2362no—
306C22—2362no—
307C22C2×D4×D5911yescharacteristic
308C4—21no—
309C4—41no—
310C4—592no—
311C4—592no—
312C4—1182no—
313C4—1181no—
314C4—2361no—
315C4#1427375112yescomplemented
316C4C2×D4×D5911yescharacteristic
317C4#41×D5911yescharacteristic
318C2—42no—
319C2—82no—
320C2—1182no—
321C2—4722no—
322C2#121×D5911yescharacteristic
323C2AGL(1,ℤ/8)×D5911yescharacteristic
324C2C2×#1427375111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#459C2
socleC22×C59C22×D4
hypercenter#459D59
hypocenterC59#3038
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#14273751C2semidirect
2. #127×D59C2semidirect
3. C4×#11745687C2semidirect
4. #16837603C2semidirect
5. #16837601C2semidirect
6. #16837599C2semidirect
7. #16837597C2semidirect
8. #16837595C2semidirect
9. #16837424C2semidirect
10. #16837405C2semidirect
11. C59×#459C2semidirect
17. C2×#11745660C22semidirect
18. C2×#11745670C22semidirect
19. C2×#11745669C22semidirect
25. C4×Dic118C22semidirect
26. #14273692C22semidirect
27. D4×Dic59C22semidirect
34. C4⋊(C59⋊C8)C22semidirect
35. Dic59⋊C8C22semidirect
36. #14273649C22semidirect
37. #14273648C22semidirect
38. #14273647C22semidirect
39. C2×D8×C59C22semidirect
41. C4×D4×C59C22semidirect
42. C4⋊C8×C59C22semidirect
43. #102×C59C22semidirect
57. #11745687D4semidirect
102. D4×C59C2×D4semidirect
128. Dic118C2×D4semidirect
147. C4×C59AGL(1,ℤ/8)semidirect
154. Dic59AGL(1,ℤ/8)semidirect
171. #459D59semidirect
174. C59#3038semidirect
183. C2×D8C2×D59semidirect
195. C4×D4C2×D59semidirect
202. C4⋊C8C2×D59semidirect
206. #102C2×D59semidirect
275. D4D4×D59semidirect
315. C4#14273751semidirect