Finite Group Databasebeta

← #21434026

Subgroups — #21434026

Order 7552.

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

normal subgroups only

This group has 414 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
1#16837909C211yesmaximal, characteristic
2C2×#14273685C211yesmaximal, characteristic, complemented
3#16837742C212yesmaximal, complemented
4#16837741C212yesmaximal, complemented
5#16837725C212yesmaximal, complemented
6#16837724C212yesmaximal, complemented
7#16837444C211yesmaximal, characteristic, complemented
8#16837442C211yesmaximal, characteristic, complemented
9#16837403C211yesmaximal, characteristic, complemented
10#16837365C211yesmaximal, characteristic, complemented
11C59×#417C211yesmaximal, characteristic, complemented
12C22×C4×D59C2211yescharacteristic
13C2×#11745664C2211yescharacteristic
14#14273708—21no—
15#14273703C2211yescharacteristic
16#14273703C2211yescharacteristic
17#14273698C2211yescharacteristic
18#14273698C2211yescharacteristic
19#14273695—21no—
20C4×C22⋊D59—21no—
21C4×C22⋊D59—21no—
22#14273686C2211yescharacteristic
23#14273685—21no—
24#14273685—21no—
25#14273685C2212yes—
26#14273685C2212yes—
27C2×C4⋊Dic59C2211yescharacteristic
28C2×(C2×D59)⋊C4—21no—
29C2×C4×Dic59—21no—
30#14273661C2212yescomplemented
31#14273660C2212yescomplemented
32#14273659C2212yescomplemented
33#14273658C2212yescomplemented
34#14273651C2212yescomplemented
35#14273648C2212yescomplemented
36(C2×D59)⋊C8C2212yes—
37(C2×D59)⋊C8C2212yes—
38C2×#41×C59C2211yescharacteristic
39C2×M16×C59C2211yescharacteristic
40C2×C4⋊C4×C59C2211yescharacteristic
41#103×C59C2212yescomplemented
42#102×C59C2212yescomplemented
43C23×D59D411yescharacteristic
44#41×C59—22no—
45#41×C59—22no—
46C2×Q8×C59C2311yescharacteristic
47C2×D4×C59—21no—
48C2×D4×C59C2311yescharacteristic
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59C2311yescharacteristic
52C2×C22⋊D59—21no—
53C2×C22⋊D59—21no—
54C22×Dic59—21no—
55C22×Dic59D411yescharacteristic
56C2×C4×D59—22no—
57C2×C4×D59—22no—
58C2×C4×D59D411yescharacteristic
59C2×C4×D59D411yescharacteristic
60C2×C4×D59C2311yescharacteristic
61C2×C4×D59C2311yescharacteristic
62M16×C59—22no—
63C2×C8×C59C2312yes—
64C4⋊C4×C59—21no—
65C4⋊C4×C59C2312yes—
66C22⋊Dic59—21no—
67C22⋊Dic59—21no—
68C22⋊Dic59—21no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—21no—
72(C2×D59)⋊C4—21no—
73C4⋊Dic59—21no—
74C4⋊Dic59C2312yes—
75C4⋊Dic59C2311yescharacteristic
76C4⋊Dic59C2311yescharacteristic
77Dic59⋊C4—21no—
78Dic59⋊C4—21no—
79C4×Dic59—21no—
80C4×Dic59—22no—
81C4×Dic59—22no—
82#11745664—22no—
83C2×C59⋊C8C2312yes—
84C23×C59—21no—
85C23×C59C2×D411yescharacteristic
86C22×D59—22no—
87C22×D59—41no—
88C22×D59—41no—
89C22×D59C2×D411yescharacteristic
90C22×D59C2×D411yescharacteristic
91Q8×C59—21no—
92Q8×C59—21no—
93D4×C59—21no—
94D4×C59—21no—
95D4×C59—42no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59C2×D411yescharacteristic
101C2×C4×C59#4112yes—
102C2×C4×C59#4112yes—
103C2×C4×C59C2411yescharacteristic
104C22⋊D59—42no—
105C22⋊D59—42no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—21no—
109C2×Dic59—21no—
110C2×Dic59—42no—
111C2×Dic59C2×D411yescharacteristic
112C2×Dic59C2×D411yescharacteristic
113C4×D59—22no—
114C4×D59—22no—
115C4×D59—21no—
116C4×D59—21no—
117C4×D59—21no—
118C4×D59—21no—
119C8×C59—22no—
120C59⋊C8—22no—
121C22×C59—21no—
122C22×C59—21no—
123C22×C59—41no—
124C22×C59—41no—
125C22×C59—41no—
126C22×C59#12312yes—
127C22×C59C22×D411yescharacteristic
128C2×D59—21no—
129C2×D59—21no—
130C2×D59—21no—
131C2×D59—21no—
132C2×D59—21no—
133C2×D59—21no—
134C2×D59—44no—
135C4×C59—41no—
136C4×C59—42no—
137C4×C59#12312yes—
138C4×C59C2×#4111yescharacteristic
139C4×C59C2×#4111yescharacteristic
140Dic59—21no—
141Dic59—21no—
142Dic59—42no—
143Dic59—41no—
144Dic59—41no—
145#3093—591nomaximal, Sylow
146C2×C59—22no—
147C2×C59—81no—
148C2×C59C2×#12311yescharacteristic
149C2×C59#57511yescharacteristic
150C2×C59#57511yescharacteristic
151D59—41no—
152D59—41no—
153#525—591no—
154C2×#117—591no—
155#485—592no—
156#485—592no—
157#482—592no—
158#482—592no—
159#428—591no—
160#428—591no—
161#417—591no—
162#417D5911yescharacteristic, complemented
163#407—591no—
164C59#309311yescharacteristic, complemented, Sylow
165C2×#41C2×D5911yescharacteristic
166C23×C4—591no—
167C2×M16—591no—
168C2×M16C2×D5911yescharacteristic
169#123—1181no—
170C22⋊Q8—591no—
171C22⋊Q8—591no—
172#121—591no—
173#121—591no—
174#120—1181no—
175C4×D4—1181no—
176C4×D4—1181no—
177#117—591no—
178#117—592no—
179#117—592no—
180#117—1181no—
181#117—1181no—
182C2×C4⋊C4—591no—
183C2×C4⋊C4C2×D5911yescharacteristic
184C2×C22⋊C4—1181no—
185C2×C42—1181no—
186#107—592no—
187#107—592no—
188#106—592no—
189#106—592no—
190#103—592no—
191#103C2×D5912yescomplemented
192#102—592no—
193#102C2×D5912yescomplemented
194C22⋊C8—592no—
195C22⋊C8—592no—
196C24—591no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D5911yescharacteristic
200C2×D4—21no—
201C2×D4—1181no—
202C2×D4—1181no—
203C2×D4C22×D5911yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—591no—
207C22×C4—591no—
208C22×C4—591no—
209C22×C4—591no—
210C22×C4—591no—
211C22×C4—1182no—
212C22×C4—1182no—
213C22×C4—1181no—
214C22×C4C22×D5911yescharacteristic
215M16—22no—
216M16—1182no—
217C2×C8—592no—
218C2×C8C22×D5912yes—
219C4⋊C4—21no—
220C4⋊C4—592no—
221C4⋊C4—591no—
222C4⋊C4—591no—
223C4⋊C4—1181no—
224C4⋊C4—1181no—
225C4⋊C4—1181no—
226C4⋊C4C22×D5912yes—
227C22⋊C4—1182no—
228C22⋊C4—1182no—
229C22⋊C4—1181no—
230C22⋊C4—1181no—
231C22⋊C4—1181no—
232C22⋊C4—1181no—
233C22⋊C4—1181no—
234C42—1182no—
235C42—1182no—
236C42—1181no—
237C23—21no—
238C23—591no—
239C23—591no—
240C23—1182no—
241C23—2361no—
242C23—2361no—
243C23D4×D5911yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—2362no—
250D4—2362no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—591no—
256C2×C4—591no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—1182no—
260C2×C4—1181no—
261C2×C4—1181no—
262C2×C4—1181no—
263C2×C4—1181no—
264C2×C4—1182no—
265C2×C4—1181no—
266C2×C4—1181no—
267C2×C4—2362no—
268C2×C4#1174568512yes—
269C2×C4D4×D5911yescharacteristic
270C2×C4#1174568712yes—
271C2×C4C23×D5911yescharacteristic
272C8—22no—
273C8—1182no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—1181no—
280C22—1181no—
281C22—1181no—
282C22—1181no—
283C22—1181no—
284C22—1181no—
285C22—2364no—
286C22#1427370912yes—
287C22C2×D4×D5911yescharacteristic
288C4—41no—
289C4—42no—
290C4—1181no—
291C4—1181no—
292C4—2361no—
293C4—2362no—
294C4—2361no—
295C4#1427370912yes—
296C4C2×#1174568711yescharacteristic
297C4C2×#1174568511yescharacteristic
298C2—22no—
299C2—81no—
300C2—2361no—
301C2—2361no—
302C2C2×#1427370911yescharacteristic
303C2#1683819811yescharacteristic
304C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#417C2
socleC22×C59C22×D4
hypercenter#417D59
hypocenterC59#3093
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#14273685C2semidirect
3. #16837742C2semidirect
4. #16837741C2semidirect
5. #16837725C2semidirect
6. #16837724C2semidirect
7. #16837444C2semidirect
8. #16837442C2semidirect
9. #16837403C2semidirect
10. #16837365C2semidirect
11. C59×#417C2semidirect
30. #14273661C22semidirect
31. #14273660C22semidirect
32. #14273659C22semidirect
33. #14273658C22semidirect
34. #14273651C22semidirect
35. #14273648C22semidirect
41. #103×C59C22semidirect
42. #102×C59C22semidirect
162. #417D59semidirect
164. C59#3093semidirect
191. #103C2×D59semidirect
193. #102C2×D59semidirect