Finite Group Databasebeta

← #21429811

Subgroups — #21429811

Order 7552.

1134 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C23×C22⋊D59C211yesmaximal, characteristic, complemented
2C2×#14273711C211yesmaximal, characteristic, complemented
3C2×#14273697C211yesmaximal, characteristic, complemented
4C2×#14273696C211yesmaximal, characteristic, complemented
5#16837318C214yesmaximal, complemented
6#16837311C214yesmaximal, complemented
7#16837263C211yesmaximal, characteristic, complemented
8#16837257C211yesmaximal, characteristic, complemented
9C59×#379C211yesmaximal, characteristic, complemented
10C24×D59C2211yescharacteristic
11C22×D236C2212yescomplemented
12C22×C22⋊D59—24no—
13C22×C22⋊D59—24no—
14C22×C22⋊D59—24no—
15C22×C22⋊D59C2211yescharacteristic
16C22×C22⋊D59C2214yes—
17C22×C22⋊D59C2212yescomplemented
18C23×Dic59C2211yescharacteristic
19#14273711—24no—
20#14273697—24no—
21#14273696—24no—
22C2×C4⋊Dic59C2212yescomplemented
23C2×C22⋊Dic59C2211yescharacteristic
24C2×C22⋊Dic59C2212yescomplemented
25C2×(C2×D59)⋊C4C2212yescomplemented
26C2×(C2×D59)⋊C4C2212yescomplemented
27C2×(C2×D59)⋊C4C2214yescomplemented
28C59⋊#95C2214yescomplemented
29C25×C59C2211yescharacteristic
30C2×C22⋊C4×C59C2211yescharacteristic
31C2×C22⋊C4×C59C2211yescharacteristic
32C2×C22⋊C4×C59C2214yescomplemented
33C24×C59—22no—
34C24×C59—22no—
35C24×C59—44no—
36C24×C59D414yescomplemented
37C24×C59C2312yes—
38C24×C59C2311yescharacteristic
39C23×D59—22no—
40C23×D59—22no—
41C23×D59—44no—
42C23×D59D414yescomplemented
43C23×D59C2311yescharacteristic
44C23×D59C2312yes—
45C22×C4×C59C2312yes—
46C22×C4×C59C2312yes—
47C2×C22⋊D59—232no—
48C2×C22⋊D59—28no—
49C2×C22⋊D59—28no—
50C2×C22⋊D59—22no—
51C2×C22⋊D59—24no—
52C2×C22⋊D59—22no—
53C2×C22⋊D59—22no—
54C2×C22⋊D59—28no—
55C2×C22⋊D59—44no—
56C22×Dic59—22no—
57C22×Dic59—22no—
58C22×Dic59D414yescomplemented
59C22×Dic59C2312yes—
60C22×Dic59C2311yescharacteristic
61C22×Dic59C2312yes—
62C2×D236—24no—
63C2×D236—44no—
64C22⋊C4×C59—22no—
65C22⋊C4×C59—22no—
66C22⋊C4×C59—44no—
67C22⋊Dic59—22no—
68C22⋊Dic59—24no—
69(C2×D59)⋊C4—24no—
70(C2×D59)⋊C4—24no—
71(C2×D59)⋊C4—44no—
72C4⋊Dic59—24no—
73C23×C59—24no—
74C23×C59—28no—
75C23×C59—22no—
76C23×C59—22no—
77C23×C59—24no—
78C23×C59—24no—
79C23×C59—48no—
80C23×C59—48no—
81C23×C59—48no—
82C23×C59C2×D414yes—
83C23×C59C2×D412yes—
84C23×C59C2×D412yes—
85C23×C59C2×D412yes—
86C23×C59C2411yescharacteristic
87C22×D59—28no—
88C22×D59—22no—
89C22×D59—22no—
90C22×D59—24no—
91C22×D59—24no—
92C22×D59—21no—
93C22×D59—21no—
94C22×D59—21no—
95C22×D59—24no—
96C22×D59—48no—
97C22×D59—48no—
98C22×D59—48no—
99C22×D59C2×D414yes—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—24no—
103C2×C4×C59—24no—
104C22⋊D59—232no—
105C22⋊D59—48no—
106C22⋊D59—44no—
107C2×Dic59—28no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—21no—
111C2×Dic59—21no—
112C2×Dic59—21no—
113C2×Dic59—22no—
114C2×Dic59—24no—
115C2×Dic59C2×D414yes—
116D236—48no—
117C22×C59—24no—
118C22×C59—24no—
119C22×C59—24no—
120C22×C59—24no—
121C22×C59—24no—
122C22×C59—24no—
123C22×C59—416no—
124C22×C59—44no—
125C22×C59—44no—
126C22×C59#12014yes—
127C22×C59C22×D411yescharacteristic
128C22×C59C22×D411yescharacteristic
129C22×C59C22×D411yescharacteristic
130C22×C59H(5,𝔽2)14yes—
131C2×D59—24no—
132C2×D59—24no—
133C2×D59—24no—
134C2×D59—416no—
135C2×D59—44no—
136C2×D59—44no—
137C2×D59—41no—
138C2×D59—42no—
139C2×D59—42no—
140C2×D59—42no—
141C4×C59—42no—
142C4×C59—42no—
143Dic59—24no—
144Dic59—41no—
145Dic59—42no—
146#2613—591nomaximal, Sylow
147C2×C59—24no—
148C2×C59—44no—
149C2×C59C2×#12011yescharacteristic
150C2×C59#53412yes—
151C2×C59#53412yes—
152C2×C59#53512yes—
153D59—44no—
154D59—81no—
155C23×D4—591no—
156C2×#123—591no—
157C2×#120—591no—
158C2×#120—591no—
159#394—594no—
160#392—594no—
161#381—591no—
162#379—591no—
163#379D5911yescharacteristic, complemented
164C59#261311yescharacteristic, complemented, Sylow
165C25—591no—
166C25C2×D5911yescharacteristic
167C22×D4—591no—
168C22×D4—594no—
169C22×D4—592no—
170C22×D4—592no—
171C22×D4—1184no—
172C22×D4—1184no—
173C22×D4—1184no—
174C23×C4—591no—
175#123—1184no—
176#120—1184no—
177#120—1184no—
178C2×C4⋊C4—592no—
179C2×C22⋊C4—591no—
180C2×C22⋊C4—592no—
181C2×C22⋊C4—592no—
182C2×C22⋊C4—592no—
183C2×C22⋊C4—594no—
184C2×C22⋊C4C2×D5911yescharacteristic
185C2×C22⋊C4C2×D5911yescharacteristic
186C2×C22⋊C4C2×D5914yescomplemented
187#95—594no—
188C24—22no—
189C24—22no—
190C24—44no—
191C24—592no—
192C24—594no—
193C24—591no—
194C24—1182no—
195C24—1182no—
196C24—2364no—
197C24D23614yescomplemented
198C24C22×D5912yes—
199C24C22×D5911yescharacteristic
200C2×D4—1184no—
201C2×D4—1188no—
202C2×D4—1188no—
203C2×D4—11832no—
204C2×D4—1188no—
205C2×D4—1184no—
206C2×D4—1182no—
207C2×D4—1182no—
208C2×D4—1182no—
209C2×D4—2364no—
210C2×D4—2364no—
211C22×C4—594no—
212C22×C4—592no—
213C22×C4—592no—
214C22×C4—591no—
215C22×C4—1182no—
216C22×C4—1182no—
217C22×C4C22×D5912yes—
218C22×C4C22×D5912yes—
219C4⋊C4—1184no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—44no—
223C22⋊C4—1184no—
224C22⋊C4—1184no—
225C22⋊C4—1184no—
226C22⋊C4—1182no—
227C22⋊C4—2364no—
228C23—24no—
229C23—28no—
230C23—22no—
231C23—22no—
232C23—24no—
233C23—24no—
234C23—48no—
235C23—48no—
236C23—48no—
237C23—594no—
238C23—1188no—
239C23—1182no—
240C23—1182no—
241C23—1184no—
242C23—1184no—
243C23—1181no—
244C23—1181no—
245C23—1181no—
246C23—1184no—
247C23—2368no—
248C23—2368no—
249C23—2368no—
250C23C2×D23614yes—
251C23C2×D23612yes—
252C23D4×D5912yes—
253C23D4×D5912yes—
254C23C23×D5911yescharacteristic
255D4—11832no—
256D4—2368no—
257D4—2368no—
258D4—2364no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—594no—
264C2×C4—1188no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—1181no—
268C2×C4—1181no—
269C2×C4—1181no—
270C2×C4—1182no—
271C2×C4—1184no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—24no—
278C22—416no—
279C22—44no—
280C22—44no—
281C22—1184no—
282C22—1184no—
283C22—1184no—
284C22—23616no—
285C22—2361no—
286C22—2362no—
287C22—2362no—
288C22—2362no—
289C22—2364no—
290C22—2364no—
291C22#1427369714yes—
292C22C2×D4×D5911yescharacteristic
293C22C2×D4×D5911yescharacteristic
294C22C22×D23611yescharacteristic
295C22#1427377214yes—
296C4—42no—
297C4—42no—
298C4—1184no—
299C4—2361no—
300C4—2362no—
301C2—24no—
302C2—44no—
303C2—4721no—
304C2—2364no—
305C2C2×#1427369711yescharacteristic
306C2#1683796212yes—
307C2#1683796412yes—
308C2#1683796712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#379C2
hypercenter#379D59
hypocenterC59#2613
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D59C2semidirect
2. C2×#14273711C2semidirect
3. C2×#14273697C2semidirect
4. C2×#14273696C2semidirect
5. #16837318C2semidirect
6. #16837311C2semidirect
7. #16837263C2semidirect
8. #16837257C2semidirect
9. C59×#379C2semidirect
11. C22×D236C22semidirect
17. C22×C22⋊D59C22semidirect
22. C2×C4⋊Dic59C22semidirect
24. C2×C22⋊Dic59C22semidirect
25. C2×(C2×D59)⋊C4C22semidirect
26. C2×(C2×D59)⋊C4C22semidirect
27. C2×(C2×D59)⋊C4C22semidirect
28. C59⋊#95C22semidirect
32. C2×C22⋊C4×C59C22semidirect
36. C24×C59D4semidirect
42. C23×D59D4semidirect
58. C22×Dic59D4semidirect
163. #379D59semidirect
164. C59#2613semidirect
186. C2×C22⋊C4C2×D59semidirect
197. C24D236semidirect