Finite Group Databasebeta

← #21433573

Subgroups — #21433573

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×#14273754C212yesmaximal, complemented
2#127×D59C211yesmaximal, characteristic, complemented
3C4×#11745689C211yesmaximal, characteristic, complemented
4#16837602C211yesmaximal, characteristic, complemented
5#16837600C212yesmaximal, complemented
6#16837598C211yesmaximal, characteristic, complemented
7#16837596C212yesmaximal, complemented
8#16837594C211yesmaximal, characteristic, complemented
9#16837424C211yesmaximal, characteristic, complemented
10#16837405C212yesmaximal, complemented
11C59×#460C211yesmaximal, characteristic, complemented
12C2×#11745689C2211yescharacteristic
13C2×D4×D59—21no—
14C2×D4×D59—21no—
15C2×D4×D59C2212yes—
16#14273754—28no—
17C2×#11745671C2212yescomplemented
18C2×D472C2212yescomplemented
19C2×#11745669C2212yescomplemented
20C2×#11745659C2212yes—
21#14273724C2211yescharacteristic
22#14273723—21no—
23#14273723—21no—
24#14273723C2212yes—
25Q8×Dic59C2211yescharacteristic, complemented
26C4×D236—21no—
27C4×D236C2211yescharacteristic, complemented
28#14273689—21no—
29#14273689C2211yescharacteristic, complemented
30#14273685—21no—
31#14273685C2211yescharacteristic
32C42×D59—21no—
33C42×D59C2211yescharacteristic
34C4⋊(C59⋊C8)C2211yescharacteristic, complemented
35Dic59⋊C8C2212yescomplemented
36#14273649C2212yescomplemented
37#14273648C2212yescomplemented
38#14273647C2212yescomplemented
39C2×SD16×C59C2212yescomplemented
40#127×C59C2211yescharacteristic
41C4×Q8×C59C2211yescharacteristic, complemented
42C4⋊C8×C59C2211yescharacteristic, complemented
43#102×C59C2212yescomplemented
44C23×D59—22no—
45C2×Q8×C59C2311yescharacteristic
46C2×D4×C59—21no—
47C2×D4×C59—21no—
48C2×D4×C59C2312yes—
49C2×C22⋊D59—22no—
50C2×C22⋊D59—22no—
51C2×C22⋊D59—22no—
52#11745689—22no—
53#11745689D414yescomplemented
54D4×D59—24no—
55D4×D59—24no—
56D4×D59—44no—
57D4×D59—44no—
58C2×D236—21no—
59C2×D236—21no—
60C2×D236—21no—
61C2×D236C2312yes—
62C2×D236C2311yescharacteristic
63C2×C4×D59—21no—
64C2×C4×D59—21no—
65C2×C4×D59—21no—
66C2×C4×D59D411yescharacteristic
67C2×C4×D59D411yescharacteristic
68C2×C4×D59C2311yescharacteristic
69SD16×C59—24no—
70C2×C8×C59C2312yes—
71C4⋊C4×C59—21no—
72C4⋊C4×C59C2311yescharacteristic
73C42×C59—21no—
74C42×C59C2311yescharacteristic
75#11745671—24no—
76#11745669—24no—
77(C2×D59)⋊C4—21no—
78(C2×D59)⋊C4—21no—
79(C2×D59)⋊C4—21no—
80C4⋊Dic59—21no—
81C4⋊Dic59C2311yescharacteristic
82C4×Dic59—21no—
83C4×Dic59—21no—
84C4×Dic59D411yescharacteristic
85C4×Dic59D411yescharacteristic
86C4×Dic59C2311yescharacteristic
87C2×C59⋊C8C2312yes—
88D472—24no—
89#11745659—24no—
90C23×C59—22no—
91C22×D59—21no—
92C22×D59—22no—
93C22×D59—44no—
94C22×D59—44no—
95C22×D59—44no—
96C22×D59C2×D411yescharacteristic
97Q8×C59—21no—
98Q8×C59C2×D412yescomplemented
99D4×C59—22no—
100D4×C59—22no—
101D4×C59—42no—
102D4×C59—42no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59C2×D411yescharacteristic
106C2×C4×C59C2×D411yescharacteristic
107C2×C4×C59C2411yescharacteristic
108C22⋊D59—44no—
109C22⋊D59—44no—
110C22⋊D59—44no—
111C2×Dic59—21no—
112C2×Dic59C2×D411yescharacteristic
113C2×Dic59C2×D411yescharacteristic
114C2×Dic59C2×D411yescharacteristic
115D236—24no—
116D236—22no—
117D236—21no—
118D236—22no—
119D236—42no—
120D236—42no—
121D236C2×D412yescomplemented
122C4×D59—22no—
123C4×D59—24no—
124C4×D59—22no—
125C4×D59—42no—
126C4×D59—42no—
127C4×D59C2×D412yes—
128C4×D59#4112yes—
129C8×C59—22no—
130C59⋊C8—22no—
131C22×C59—42no—
132C22×C59—42no—
133C22×C59—42no—
134C22×C59C22×D411yescharacteristic
135C2×D59—21no—
136C2×D59—21no—
137C2×D59—22no—
138C2×D59—41no—
139C2×D59—41no—
140C2×D59—42no—
141C2×D59—42no—
142C2×D59—42no—
143C2×D59—88no—
144C2×D59#12112yes—
145C4×C59—22no—
146C4×C59—21no—
147C4×C59—41no—
148C4×C59AGL(1,ℤ/8)12yescomplemented
149C4×C59C22×D411yescharacteristic
150C4×C59C2×#4111yescharacteristic
151Dic59—21no—
152Dic59—41no—
153Dic59#12112yes—
154Dic59AGL(1,ℤ/8)12yescomplemented
155#3038—591nomaximal, Sylow
156C2×C59—82no—
157C2×C59C2×#12111yescharacteristic
158C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
160D59—22no—
161D59—42no—
162D59—82no—
163C2×AGL(1,ℤ/8)—592no—
164C2×#127—591no—
165C4×#41—591no—
166#460—592no—
167#460—591no—
168#460D5911yescharacteristic, complemented
169#459—591no—
170#459—591no—
171#459—592no—
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×SD16C2×D5912yescomplemented
182C2×D8—592no—
183C2×D8—592no—
184C2×M16—592no—
185#127—592no—
186#127—591no—
187#127—1181no—
188#127—1181no—
189#127C2×D5911yescharacteristic
190C4×Q8—591no—
191C4×Q8C2×D5911yescharacteristic, complemented
192C4×D4—591no—
193C4×D4—591no—
194C4×D4—1181no—
195C4×D4—1181no—
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×Q8C22×D5911yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—591no—
214C2×D4—592no—
215C2×D4—1181no—
216C2×D4—1184no—
217C2×D4—1184no—
218C2×D4—1182no—
219C2×D4—1182no—
220C2×D4—1182no—
221C2×D4—1181no—
222C2×D4—1181no—
223C2×D4—2364no—
224C2×D4—2364no—
225C2×D4C22×D5912yes—
226C22×C4—591no—
227C22×C4—591no—
228C22×C4—591no—
229C22×C4—1181no—
230C22×C4—1181no—
231C22×C4—1181no—
232SD16—24no—
233SD16—1184no—
234D8—1184no—
235D8—1184no—
236M16—1184no—
237C2×C8—592no—
238C2×C8C22×D5912yes—
239C4⋊C4—21no—
240C4⋊C4—591no—
241C4⋊C4—1181no—
242C4⋊C4C22×D5911yescharacteristic
243C22⋊C4—1181no—
244C22⋊C4—1181no—
245C22⋊C4—1181no—
246C42—21no—
247C42—591no—
248C42—591no—
249C42—591no—
250C42—1181no—
251C42—1181no—
252C42C22×D5911yescharacteristic
253C23—22no—
254C23—591no—
255C23—1181no—
256C23—1182no—
257C23—2364no—
258C23—2364no—
259C23—2364no—
260Q8—21no—
261Q8D4×D5912yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—592no—
267D4—1184no—
268D4—1182no—
269D4—1181no—
270D4—1182no—
271D4—2362no—
272D4—2364no—
273D4—2364no—
274D4—2364no—
275D4—2362no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—591no—
279C2×C4—591no—
280C2×C4—592no—
281C2×C4—592no—
282C2×C4—591no—
283C2×C4—1182no—
284C2×C4—1184no—
285C2×C4—1182no—
286C2×C4—1181no—
287C2×C4—2362no—
288C2×C4—2362no—
289C2×C4D4×D5911yescharacteristic
290C2×C4D4×D5911yescharacteristic
291C2×C4C23×D5911yescharacteristic
292C8—22no—
293C8—1182no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—592no—
298C22—1181no—
299C22—1181no—
300C22—1182no—
301C22—4728no—
302C22—2362no—
303C22—2362no—
304C22—2362no—
305C22—2361no—
306C22—2361no—
307C22C2×D4×D5911yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—592no—
312C4—592no—
313C4—1181no—
314C4—2361no—
315C4#1427375412yescomplemented
316C4C2×D4×D5911yescharacteristic
317C4#41×D5911yescharacteristic
318C2—82no—
319C2—1182no—
320C2—4722no—
321C2—2362no—
322C2#121×D5911yescharacteristic
323C2AGL(1,ℤ/8)×D5911yescharacteristic
324C2C2×#1427375411yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#460C2
socleC22×C59C22×D4
hypercenter#460D59
hypocenterC59#3038
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#14273754C2semidirect
2. #127×D59C2semidirect
3. C4×#11745689C2semidirect
4. #16837602C2semidirect
5. #16837600C2semidirect
6. #16837598C2semidirect
7. #16837596C2semidirect
8. #16837594C2semidirect
9. #16837424C2semidirect
10. #16837405C2semidirect
11. C59×#460C2semidirect
17. C2×#11745671C22semidirect
18. C2×D472C22semidirect
19. C2×#11745669C22semidirect
25. Q8×Dic59C22semidirect
27. C4×D236C22semidirect
29. #14273689C22semidirect
34. C4⋊(C59⋊C8)C22semidirect
35. Dic59⋊C8C22semidirect
36. #14273649C22semidirect
37. #14273648C22semidirect
38. #14273647C22semidirect
39. C2×SD16×C59C22semidirect
41. C4×Q8×C59C22semidirect
42. C4⋊C8×C59C22semidirect
43. #102×C59C22semidirect
53. #11745689D4semidirect
98. Q8×C59C2×D4semidirect
121. D236C2×D4semidirect
148. C4×C59AGL(1,ℤ/8)semidirect
154. Dic59AGL(1,ℤ/8)semidirect
168. #460D59semidirect
174. C59#3038semidirect
181. C2×SD16C2×D59semidirect
191. C4×Q8C2×D59semidirect
202. C4⋊C8C2×D59semidirect
206. #102C2×D59semidirect
261. Q8D4×D59semidirect
315. C4#14273754semidirect