Finite Group Databasebeta

← #21435718

Subgroups — #21435718

Order 7552.

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

normal subgroups only

This group has 556 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×#14273755C212yesmaximal, complemented
2#16838073C211yesmaximal, characteristic, complemented
3D4×D236C211yesmaximal, characteristic, complemented
4#16837779C211yesmaximal, characteristic, complemented
5#16837655C212yesmaximal, complemented
6#16837603C211yesmaximal, characteristic, complemented
7#16837598C211yesmaximal, characteristic, complemented
8#16837549C212yesmaximal, complemented
9#16837538C212yesmaximal, complemented
10C4×#11745660C211yesmaximal, characteristic, complemented
11C59×#436C211yesmaximal, characteristic, complemented
12C2×#11745685C2212yescomplemented
13C22×D236C2212yes—
14C2×D4×D59—21no—
15#14273755—28no—
16C2×#11745660C2211yescharacteristic
17C2×#11745660C2212yescomplemented
18C2×D472C2212yes—
19C2×D472C2212yescomplemented
20C2×D472C2212yescomplemented
21#14273724C2211yescharacteristic
22#14273724C2211yescharacteristic, complemented
23#14273723—21no—
24#14273702—21no—
25#14273701C2212yes—
26#14273701C2212yescomplemented
27#14273700—22no—
28#14273697—22no—
29C4×Dic118C2211yescharacteristic, complemented
30C4×D236C2211yescharacteristic
31#14273689—21no—
32#14273659C2211yescharacteristic, complemented
33#14273652C2211yescharacteristic, complemented
34#14273649C2211yescharacteristic
35#14273649C2212yescomplemented
36#14273649C2212yescomplemented
37C2×M16×C59C2212yescomplemented
38C4×D4×C59C2211yescharacteristic
39C4⋊C8×C59C2211yescharacteristic, complemented
40C22⋊C8×C59C2212yescomplemented
41C4×C8×C59C2211yescharacteristic, complemented
42C23×D59—22no—
43C2×D4×C59D411yescharacteristic
44C22×C4×C59C2312yes—
45C2×C22⋊D59—22no—
46C2×C22⋊D59—22no—
47C2×C22⋊D59—22no—
48D4×D59—44no—
49#11745685—24no—
50#11745685—42no—
51C2×D236—24no—
52C2×D236—21no—
53C2×D236—22no—
54C2×D236—21no—
55C2×D236—21no—
56C2×D236—21no—
57C2×D236—42no—
58C2×D236C2312yes—
59C2×D236C2312yes—
60C2×D236C2311yescharacteristic
61C2×C4×D59—22no—
62C2×C4×D59—21no—
63C2×Dic118C2311yescharacteristic
64M16×C59—24no—
65C2×C8×C59C2311yescharacteristic
66C2×C8×C59C2312yes—
67C2×C8×C59C2311yescharacteristic
68C4⋊C4×C59D411yescharacteristic
69C22⋊C4×C59D412yes—
70C42×C59C2311yescharacteristic
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—22no—
74C4⋊Dic59C2311yescharacteristic
75C4⋊Dic59C2311yescharacteristic
76Dic59⋊C4—21no—
77C4×Dic59—21no—
78D472—24no—
79D472—24no—
80D472—42no—
81#11745660—24no—
82#11745660D414yescomplemented
83C23×C59C2×D412yes—
84C22×D59—21no—
85C22×D59—22no—
86C22×D59—22no—
87C22×D59—44no—
88C22×D59—44no—
89C22×D59—82no—
90D4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59C2×D411yescharacteristic
94C2×C4×C59C2×D412yes—
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59C2411yescharacteristic
97C22⋊D59—44no—
98C22⋊D59—44no—
99C22⋊D59—82no—
100C2×Dic59—21no—
101C2×Dic59—21no—
102C2×Dic59—21no—
103D236—22no—
104D236—22no—
105D236—21no—
106D236—22no—
107D236—22no—
108D236—44no—
109D236—42no—
110D236—42no—
111D236—81no—
112D236—81no—
113D236C2×D412yes—
114C4×D59—44no—
115C4×D59—42no—
116Dic118—21no—
117Dic118C2×D412yescomplemented
118C8×C59—22no—
119C8×C59—21no—
120C8×C59C2×D412yescomplemented
121C22×C59—22no—
122C22×C59—22no—
123C22×C59—22no—
124C22×C59C22×D411yescharacteristic
125C2×D59—22no—
126C2×D59—41no—
127C2×D59—41no—
128C2×D59—42no—
129C2×D59—42no—
130C2×D59—42no—
131C2×D59—42no—
132C2×D59—42no—
133C2×D59—42no—
134C2×D59—88no—
135C4×C59—22no—
136C4×C59—21no—
137C4×C59AGL(1,ℤ/8)12yescomplemented
138C4×C59C22×D411yescharacteristic
139C4×C59H(5,𝔽2)11yescharacteristic
140Dic59—22no—
141Dic59—41no—
142Dic59—41no—
143#3274—591nomaximal, Sylow
144C2×C59—42no—
145C2×C59D4211yescharacteristic
146C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C59#57611yescharacteristic
148D59—42no—
149D59—82no—
150D59—82no—
151C2×AGL(1,ℤ/8)—592no—
152#550—591no—
153D42—591no—
154#493—591no—
155#469—592no—
156#460—591no—
157#459—591no—
158#449—592no—
159#447—592no—
160C4×SD16—591no—
161#436D5911yescharacteristic, complemented
162C59#327411yescharacteristic, complemented, Sylow
163C2×#41—592no—
164C22×D4—592no—
165C22×D4—1181no—
166AGL(1,ℤ/8)—1188no—
167C2×SD16—591no—
168C2×SD16—592no—
169C2×D8—592no—
170C2×D8—592no—
171C2×D8—592no—
172C2×M16C2×D5912yescomplemented
173#127—591no—
174#127—591no—
175#127—1181no—
176#121—592no—
177#121—592no—
178#121—1181no—
179#121—1182no—
180#120—1182no—
181C4×Q8—591no—
182C4×D4—591no—
183C4×D4—1181no—
184C4×D4C2×D5911yescharacteristic
185#106—591no—
186C4⋊C8C2×D5911yescharacteristic, complemented
187#103—591no—
188#102—591no—
189#102—592no—
190#102—592no—
191C22⋊C8C2×D5912yescomplemented
192C4×C8C2×D5911yescharacteristic, complemented
193C24—1182no—
194#41—1184no—
195#41—2362no—
196C2×Q8—591no—
197C2×D4—591no—
198C2×D4—592no—
199C2×D4—592no—
200C2×D4—1184no—
201C2×D4—1182no—
202C2×D4—1182no—
203C2×D4—1182no—
204C2×D4—1181no—
205C2×D4—1181no—
206C2×D4—1181no—
207C2×D4—1182no—
208C2×D4—1181no—
209C2×D4—2362no—
210C2×D4—2364no—
211C2×D4D23611yescharacteristic
212C22×C4—1181no—
213C22×C4—1182no—
214C22×C4C22×D5912yes—
215SD16—594no—
216SD16—1184no—
217D8—1184no—
218D8—1184no—
219D8—2362no—
220M16—24no—
221C2×C8C22×D5911yescharacteristic
222C2×C8C22×D5912yes—
223C2×C8C22×D5911yescharacteristic
224C4⋊C4—591no—
225C4⋊C4—591no—
226C4⋊C4—1181no—
227C4⋊C4D23611yescharacteristic
228C22⋊C4—1181no—
229C22⋊C4—1182no—
230C22⋊C4—1182no—
231C22⋊C4D23612yes—
232C42—1181no—
233C42C22×D5911yescharacteristic
234C23—1181no—
235C23—1182no—
236C23—1182no—
237C23—4722no—
238C23—2364no—
239C23—2364no—
240C23C2×D23612yes—
241Q8—592no—
242Q8—1181no—
243D4—22no—
244D4—592no—
245D4—1182no—
246D4—1182no—
247D4—1181no—
248D4—1182no—
249D4—1182no—
250D4—4721no—
251D4—4721no—
252D4—4722no—
253D4—2364no—
254D4—2364no—
255D4—2364no—
256D4—2362no—
257D4—2362no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—1181no—
261C2×C4—1181no—
262C2×C4—1181no—
263C2×C4—2364no—
264C2×C4—2362no—
265C2×C4C2×D23611yescharacteristic
266C2×C4C2×D23612yes—
267C2×C4C2×D23611yescharacteristic
268C2×C4C23×D5911yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D5912yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—1182no—
276C22—4728no—
277C22—2362no—
278C22—2362no—
279C22—2362no—
280C22—2362no—
281C22—2362no—
282C22—2362no—
283C22—2361no—
284C22—2361no—
285C22C22×D23611yescharacteristic
286C4—21no—
287C4—22no—
288C4—1182no—
289C4—2361no—
290C4—2361no—
291C4#1427375512yescomplemented
292C4C2×D4×D5911yescharacteristic
293C4#1427377311yescharacteristic
294C2—42no—
295C2—4722no—
296C2—4722no—
297C2—2362no—
298C2D4×D23611yescharacteristic
299C2C2×#1427375511yescharacteristic
300C2#1683820711yescharacteristic
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#436C2
socleC22×C59C22×D4
hypercenter#436D59
hypocenterC59#3274
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#14273755C2semidirect
2. #16838073C2semidirect
3. D4×D236C2semidirect
4. #16837779C2semidirect
5. #16837655C2semidirect
6. #16837603C2semidirect
7. #16837598C2semidirect
8. #16837549C2semidirect
9. #16837538C2semidirect
10. C4×#11745660C2semidirect
11. C59×#436C2semidirect
12. C2×#11745685C22semidirect
17. C2×#11745660C22semidirect
19. C2×D472C22semidirect
20. C2×D472C22semidirect
22. #14273724C22semidirect
26. #14273701C22semidirect
29. C4×Dic118C22semidirect
32. #14273659C22semidirect
33. #14273652C22semidirect
35. #14273649C22semidirect
36. #14273649C22semidirect
37. C2×M16×C59C22semidirect
39. C4⋊C8×C59C22semidirect
40. C22⋊C8×C59C22semidirect
41. C4×C8×C59C22semidirect
82. #11745660D4semidirect
117. Dic118C2×D4semidirect
120. C8×C59C2×D4semidirect
137. C4×C59AGL(1,ℤ/8)semidirect
161. #436D59semidirect
162. C59#3274semidirect
172. C2×M16C2×D59semidirect
186. C4⋊C8C2×D59semidirect
191. C22⋊C8C2×D59semidirect
192. C4×C8C2×D59semidirect
271. C8D4×D59semidirect
291. C4#14273755semidirect