Finite Group Databasebeta

← #21435699

Subgroups — #21435699

Order 7552.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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#16838055C211yesmaximal, characteristic, complemented
3D4×D236C211yesmaximal, characteristic, complemented
4#16837776C211yesmaximal, characteristic, complemented
5#16837651C212yesmaximal, complemented
6#16837598C211yesmaximal, characteristic, complemented
7#16837551C212yesmaximal, complemented
8#16837550C212yesmaximal, complemented
9C4×#11745669C211yesmaximal, characteristic, complemented
10#16837479C211yesmaximal, characteristic, complemented
11C59×#461C211yesmaximal, characteristic, complemented
12C2×#11745689C2212yescomplemented
13C22×D236—21no—
14C2×D4×D59C2212yes—
15#14273754—28no—
16C2×#11745671C2212yescomplemented
17C2×#11745671C2212yescomplemented
18C2×D472C2212yescomplemented
19C2×#11745669C2211yescharacteristic
20C2×#11745659C2212yescomplemented
21#14273724C2211yescharacteristic
22#14273712—22no—
23#14273703C2212yescomplemented
24#14273702—21no—
25#14273702C2212yes—
26#14273701—21no—
27#14273697—22no—
28C4×D236C2211yescharacteristic
29C4×D236C2211yescharacteristic, complemented
30C4⋊C4×D59—21no—
31#14273660C2211yescharacteristic, complemented
32#14273649C2212yescomplemented
33#14273648C2211yescharacteristic
34#14273647C2211yescharacteristic, complemented
35(C2×D59)⋊C8C2212yescomplemented
36C4×C59⋊C8C2211yescharacteristic, complemented
37C2×SD16×C59C2212yes—
38C4⋊Q8×C59C2211yescharacteristic, complemented
39C4×D4×C59C2211yescharacteristic
40C4⋊C8×C59C2211yescharacteristic, complemented
41#103×C59C2212yescomplemented
42C23×D59—22no—
43C2×Q8×C59C2312yes—
44C2×D4×C59C2311yescharacteristic
45C22×C4×C59—21no—
46C2×C22⋊D59—22no—
47#11745689—24no—
48#11745689—42no—
49D4×D59—24no—
50D4×D59—42no—
51C2×D236—22no—
52C2×D236—22no—
53C2×D236—21no—
54C2×D236—21no—
55C2×D236—44no—
56C2×D236D411yescharacteristic
57C2×D236C2312yes—
58C2×D236C2311yescharacteristic
59C2×C4×D59—21no—
60C2×C4×D59—22no—
61C2×C4×D59C2312yes—
62SD16×C59—24no—
63C2×C8×C59C2312yes—
64C4⋊C4×C59—21no—
65C4⋊C4×C59—21no—
66C4⋊C4×C59C2311yescharacteristic
67C22⋊C4×C59—21no—
68C42×C59C2311yescharacteristic
69#11745671—24no—
70#11745671—42no—
71#11745669D414yescomplemented
72(C2×D59)⋊C4—22no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—22no—
75(C2×D59)⋊C4D412yes—
76C4⋊Dic59D411yescharacteristic
77C4⋊Dic59C2311yescharacteristic
78Dic59⋊C4—22no—
79C2×C59⋊C8C2311yescharacteristic
80C2×C59⋊C8C2311yescharacteristic
81D472—24no—
82#11745659—24no—
83C23×C59—21no—
84C22×D59—21no—
85C22×D59—22no—
86C22×D59—44no—
87C22×D59—44no—
88C22×D59—82no—
89C22×D59C2×D412yes—
90Q8×C59—22no—
91Q8×C59—22no—
92D4×C59—21no—
93D4×C59C2×D412yes—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—42no—
97C2×C4×C59C2×D411yescharacteristic
98C2×C4×C59C2×D411yescharacteristic
99C2×C4×C59C2411yescharacteristic
100C22⋊D59—44no—
101C2×Dic59—21no—
102C2×Dic59C2×D412yes—
103D236—22no—
104D236—22no—
105D236—21no—
106D236—22no—
107D236—44no—
108D236—44no—
109D236—42no—
110D236—81no—
111D236C2×D412yescomplemented
112C4×D59—22no—
113C4×D59—22no—
114C4×D59—44no—
115C4×D59—42no—
116C8×C59—22no—
117C59⋊C8—21no—
118C59⋊C8C2×D412yescomplemented
119C22×C59—22no—
120C22×C59—41no—
121C22×C59—41no—
122C22×C59C22×D411yescharacteristic
123C2×D59—22no—
124C2×D59—22no—
125C2×D59—22no—
126C2×D59—22no—
127C2×D59—41no—
128C2×D59—41no—
129C2×D59—42no—
130C2×D59—42no—
131C2×D59—42no—
132C2×D59—88no—
133C4×C59—21no—
134C4×C59—42no—
135C4×C59—41no—
136C4×C59AGL(1,ℤ/8)12yescomplemented
137C4×C59C22×D411yescharacteristic
138C4×C59H(5,𝔽2)11yescharacteristic
139Dic59—22no—
140Dic59—41no—
141#3272—591nomaximal, Sylow
142C2×C59—42no—
143C2×C59D4211yescharacteristic
144C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C59#57711yescharacteristic
146D59—42no—
147D59—42no—
148D59—82no—
149C2×AGL(1,ℤ/8)—592no—
150#547—591no—
151D42—591no—
152#492—591no—
153#468—592no—
154#461D5911yescharacteristic, complemented
155#459—591no—
156#450—592no—
157#449—592no—
158C4×D8—591no—
159#436—591no—
160C59#327211yescharacteristic, complemented, Sylow
161C2×#41—592no—
162C22×D4—592no—
163C22×D4—1181no—
164AGL(1,ℤ/8)—1188no—
165C2×SD16—592no—
166C2×SD16—592no—
167C2×SD16C2×D5912yes—
168C2×D8—591no—
169C2×D8—592no—
170C2×M16—592no—
171C4⋊Q8C2×D5911yescharacteristic, complemented
172#127—591no—
173#123—1182no—
174C22⋊Q8—592no—
175#121—592no—
176#121—1181no—
177#121—1181no—
178#120—1182no—
179C4×D4—591no—
180C4×D4—591no—
181C4×D4C2×D5911yescharacteristic
182C2×C4⋊C4—1181no—
183#107—591no—
184C4⋊C8C2×D5911yescharacteristic, complemented
185#103C2×D5912yescomplemented
186#102—591no—
187#102—592no—
188#102—591no—
189C22⋊C8—592no—
190C4×C8—591no—
191C24—1182no—
192#41—1184no—
193#41—2362no—
194C2×Q8C22×D5912yes—
195C2×D4—591no—
196C2×D4—591no—
197C2×D4—592no—
198C2×D4—1182no—
199C2×D4—1182no—
200C2×D4—1182no—
201C2×D4—1184no—
202C2×D4—1181no—
203C2×D4—1181no—
204C2×D4—2364no—
205C2×D4—2362no—
206C2×D4C22×D5911yescharacteristic
207C22×C4—21no—
208C22×C4—592no—
209C22×C4—1182no—
210C22×C4—1181no—
211SD16—24no—
212SD16—1184no—
213SD16—2362no—
214D8—594no—
215D8—1184no—
216M16—1184no—
217C2×C8—591no—
218C2×C8—591no—
219C2×C8C22×D5912yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—591no—
223C4⋊C4—591no—
224C4⋊C4—1182no—
225C4⋊C4C22×D5911yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—592no—
228C22⋊C4—1182no—
229C22⋊C4—1182no—
230C22⋊C4—1181no—
231C42C22×D5911yescharacteristic
232C23—21no—
233C23—592no—
234C23—1181no—
235C23—1182no—
236C23—4722no—
237C23—2364no—
238C23—2364no—
239Q8—22no—
240Q8—22no—
241D4—21no—
242D4—592no—
243D4—1182no—
244D4—1182no—
245D4—1181no—
246D4—1182no—
247D4—4721no—
248D4—2364no—
249D4—2364no—
250D4—2364no—
251D4—2362no—
252D4D4×D5912yes—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—592no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—1181no—
260C2×C4—2364no—
261C2×C4—2362no—
262C2×C4D4×D5911yescharacteristic
263C2×C4D4×D5911yescharacteristic
264C2×C4C23×D5911yescharacteristic
265C8—22no—
266C8—592no—
267C8—1181no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—1182no—
272C22—1182no—
273C22—1182no—
274C22—1182no—
275C22—4728no—
276C22—2362no—
277C22—2362no—
278C22—2362no—
279C22—2361no—
280C22—2361no—
281C22C2×D4×D5911yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—1182no—
286C4—2361no—
287C4#1427375412yescomplemented
288C4C2×D4×D5911yescharacteristic
289C4#1427377311yescharacteristic
290C2—42no—
291C2—4722no—
292C2—2362no—
293C2—2362no—
294C2#1683803611yescharacteristic
295C2C2×#1427375411yescharacteristic
296C2#1683821111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#461C2
socleC22×C59C22×D4
hypercenter#461D59
hypocenterC59#3272
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#14273754C2semidirect
2. #16838055C2semidirect
3. D4×D236C2semidirect
4. #16837776C2semidirect
5. #16837651C2semidirect
6. #16837598C2semidirect
7. #16837551C2semidirect
8. #16837550C2semidirect
9. C4×#11745669C2semidirect
10. #16837479C2semidirect
11. C59×#461C2semidirect
12. C2×#11745689C22semidirect
16. C2×#11745671C22semidirect
17. C2×#11745671C22semidirect
18. C2×D472C22semidirect
20. C2×#11745659C22semidirect
23. #14273703C22semidirect
29. C4×D236C22semidirect
31. #14273660C22semidirect
32. #14273649C22semidirect
34. #14273647C22semidirect
35. (C2×D59)⋊C8C22semidirect
36. C4×C59⋊C8C22semidirect
38. C4⋊Q8×C59C22semidirect
40. C4⋊C8×C59C22semidirect
41. #103×C59C22semidirect
71. #11745669D4semidirect
111. D236C2×D4semidirect
118. C59⋊C8C2×D4semidirect
136. C4×C59AGL(1,ℤ/8)semidirect
154. #461D59semidirect
160. C59#3272semidirect
171. C4⋊Q8C2×D59semidirect
184. C4⋊C8C2×D59semidirect
185. #103C2×D59semidirect
287. C4#14273754semidirect