Finite Group Databasebeta

← #21434031

Subgroups — #21434031

Order 7552.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#121×D59C211yesmaximal, characteristic
2C2×#14273685C211yesmaximal, characteristic, complemented
3#16837729C212yesmaximal, complemented
4#16837725C212yesmaximal, complemented
5#16837721C211yesmaximal, characteristic, complemented
6#16837718C211yesmaximal, characteristic, complemented
7#16837715C211yesmaximal, characteristic, complemented
8#16837443C212yesmaximal, complemented
9#16837405C212yesmaximal, complemented
10#16837364C211yesmaximal, characteristic, complemented
11C59×#480C211yesmaximal, characteristic, complemented
12C2×D4×D59—21no—
13C2×D4×D59C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15C2×#11745659C2212yes—
16#14273702C2211yescharacteristic
17#14273701C2211yescharacteristic
18#14273700—21no—
19#14273699—21no—
20#14273698C2211yescharacteristic
21#14273685—21no—
22#14273685—21no—
23#14273685C2212yes—
24#14273685C2212yescomplemented
25C2×C4⋊Dic59C2211yescharacteristic, complemented
26C4⋊C4×D59C2211yescharacteristic
27C2×(C2×D59)⋊C4—21no—
28C22⋊C4×D59—21no—
29C2×C4×Dic59—21no—
30#14273660C2212yescomplemented
31#14273659C2212yescomplemented
32#14273658C2212yescomplemented
33#14273649C2212yescomplemented
34#14273648C2212yescomplemented
35#14273647C2212yescomplemented
36C22⋊(C59⋊C8)C2211yescharacteristic, complemented
37(C2×D59)⋊C8C2212yescomplemented
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—24no—
55D4×D59—44no—
56C2×D236—21no—
57C2×D236C2311yescharacteristic
58C2×C4×D59—24no—
59C2×C4×D59—21no—
60C2×C4×D59D411yescharacteristic
61C2×C4×D59D411yescharacteristic
62C2×C4×D59C2311yescharacteristic
63C2×C4×D59C2311yescharacteristic
64C2×C8×C59C2312yes—
65C4⋊C4×C59—21no—
66C4⋊C4×C59C2312yes—
67C4⋊C4×C59C2311yescharacteristic
68C22⋊C4×C59—21no—
69C22⋊Dic59—21no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74C4⋊Dic59—21no—
75C4⋊Dic59C2312yes—
76C4⋊Dic59C2311yescharacteristic
77Dic59⋊C4—21no—
78C4×Dic59—22no—
79C4×Dic59—22no—
80C2×C59⋊C8C2312yes—
81#11745659—24no—
82C23×C59—21no—
83C23×C59C2×D411yescharacteristic
84C22×D59—22no—
85C22×D59—22no—
86C22×D59—21no—
87C22×D59—41no—
88C22×D59—42no—
89C22×D59—42no—
90C22×D59—42no—
91C22×D59C2×D411yescharacteristic
92C22×D59C2×D411yescharacteristic
93D4×C59—22no—
94D4×C59—42no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—42no—
99C2×C4×C59C2×D411yescharacteristic
100C2×C4×C59C2411yescharacteristic
101C22⋊D59—42no—
102C22⋊D59—42no—
103C22⋊D59—42no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59—21no—
107C2×Dic59—42no—
108C2×Dic59C2×D411yescharacteristic
109C2×Dic59C2×D411yescharacteristic
110D236—22no—
111D236—42no—
112C4×D59—22no—
113C4×D59—22no—
114C4×D59—22no—
115C4×D59—42no—
116C4×D59#4114yes—
117C8×C59—22no—
118C59⋊C8—22no—
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—41no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59AGL(1,ℤ/8)12yes—
125C22×C59C22×D411yescharacteristic
126C2×D59—21no—
127C2×D59—21no—
128C2×D59—21no—
129C2×D59—21no—
130C2×D59—21no—
131C2×D59—44no—
132C2×D59—41no—
133C2×D59—41no—
134C2×D59—41no—
135C2×D59—84no—
136C2×D59#12312yes—
137C4×C59—21no—
138C4×C59—41no—
139C4×C59—42no—
140C4×C59C2×#4112yes—
141Dic59—21no—
142Dic59—41no—
143Dic59—42no—
144Dic59#12312yes—
145#3094—591nomaximal, Sylow
146C2×C59—22no—
147C2×C59—81no—
148C2×C59C2×#12311yescharacteristic
149C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C59#57511yescharacteristic
151D59—22no—
152D59—41no—
153D59—81no—
154C2×#121—591no—
155C2×#117—591no—
156#482—592no—
157#482—592no—
158#481—591no—
159#481—591no—
160#480—591no—
161#480D5911yescharacteristic, complemented
162#428—592no—
163#418—592no—
164#407—591no—
165C59#309411yescharacteristic, complemented, Sylow
166C22×D4—591no—
167C22×D4—1181no—
168C23×C4—591no—
169C2×M16—592no—
170#121—591no—
171#121—591no—
172#121—591no—
173#121—1181no—
174#121—1181no—
175#121C2×D5911yescharacteristic
176#117—592no—
177#117—592no—
178#117—1181no—
179#117—1181no—
180C2×C4⋊C4—591no—
181C2×C4⋊C4—591no—
182C2×C4⋊C4C2×D5911yescharacteristic, complemented
183C2×C22⋊C4—1181no—
184C2×C22⋊C4—1181no—
185C2×C42—1181no—
186#107—592no—
187#107C2×D5912yescomplemented
188#106—592no—
189#106—592no—
190#102—592no—
191#102—592no—
192#102—592no—
193#102C2×D5912yescomplemented
194C22⋊C8—591no—
195C22⋊C8—592no—
196C22⋊C8C2×D5911yescharacteristic, complemented
197C24—591no—
198C24—1181no—
199C2×D4—21no—
200C2×D4—591no—
201C2×D4—1181no—
202C2×D4—1184no—
203C2×D4—1181no—
204C2×D4—1181no—
205C2×D4—1181no—
206C2×D4—2364no—
207C2×D4C22×D5911yescharacteristic
208C22×C4—21no—
209C22×C4—591no—
210C22×C4—591no—
211C22×C4—591no—
212C22×C4—591no—
213C22×C4—591no—
214C22×C4—1184no—
215C22×C4—1181no—
216C22×C4—1181no—
217C22×C4C22×D5911yescharacteristic
218M16—1184no—
219C2×C8—592no—
220C2×C8C22×D5912yes—
221C4⋊C4—21no—
222C4⋊C4—592no—
223C4⋊C4—591no—
224C4⋊C4—1181no—
225C4⋊C4—1181no—
226C4⋊C4C22×D5912yes—
227C4⋊C4C22×D5911yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—1181no—
230C22⋊C4—1182no—
231C22⋊C4—1182no—
232C22⋊C4—1181no—
233C22⋊C4—1181no—
234C42—1182no—
235C42—1182no—
236C23—21no—
237C23—591no—
238C23—591no—
239C23—1182no—
240C23—1182no—
241C23—1181no—
242C23—2361no—
243C23—2362no—
244C23—2362no—
245C23—2362no—
246C23D4×D5911yescharacteristic
247D4—22no—
248D4—42no—
249D4—1182no—
250D4—2362no—
251D4—2362no—
252D4—2362no—
253D4—2362no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—591no—
259C2×C4—594no—
260C2×C4—591no—
261C2×C4—1182no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1181no—
267C2×C4—2362no—
268C2×C4—2362no—
269C2×C4D4×D5911yescharacteristic
270C2×C4C23×D5911yescharacteristic
271C8—22no—
272C8—1182no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—592no—
279C22—1181no—
280C22—1181no—
281C22—1181no—
282C22—1181no—
283C22—1181no—
284C22—4724no—
285C22—2364no—
286C22—2361no—
287C22—2361no—
288C22—2361no—
289C22#1427375112yes—
290C22C2×D4×D5911yescharacteristic
291C4—21no—
292C4—42no—
293C4—41no—
294C4—592no—
295C4—1181no—
296C4—2361no—
297C4—2362no—
298C4#41×D5912yes—
299C2—22no—
300C2—81no—
301C2—1182no—
302C2—4721no—
303C2—2361no—
304C2#123×D5911yescharacteristic
305C2C2×#1427375111yescharacteristic
306C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#480C2
socleC22×C59C22×D4
hypercenter#480D59
hypocenterC59#3094
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
2. C2×#14273685C2semidirect
3. #16837729C2semidirect
4. #16837725C2semidirect
5. #16837721C2semidirect
6. #16837718C2semidirect
7. #16837715C2semidirect
8. #16837443C2semidirect
9. #16837405C2semidirect
10. #16837364C2semidirect
11. C59×#480C2semidirect
24. #14273685C22semidirect
25. C2×C4⋊Dic59C22semidirect
30. #14273660C22semidirect
31. #14273659C22semidirect
32. #14273658C22semidirect
33. #14273649C22semidirect
34. #14273648C22semidirect
35. #14273647C22semidirect
36. C22⋊(C59⋊C8)C22semidirect
37. (C2×D59)⋊C8C22semidirect
39. C2×C4⋊C4×C59C22semidirect
40. #107×C59C22semidirect
41. #102×C59C22semidirect
42. C22⋊C8×C59C22semidirect
161. #480D59semidirect
165. C59#3094semidirect
182. C2×C4⋊C4C2×D59semidirect
187. #107C2×D59semidirect
193. #102C2×D59semidirect
196. C22⋊C8C2×D59semidirect