Finite Group Databasebeta

← #21427454

Subgroups — #21427454

Order 7552.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D59C211yesmaximal, characteristic, complemented
2C2×#14273704C212yesmaximal, complemented
3C2×(C22⋊D59)⋊C4C211yesmaximal, characteristic
4#16837323C211yesmaximal, characteristic, complemented
5#16837321C211yesmaximal, characteristic, complemented
6#16837314C211yesmaximal, characteristic, complemented
7#16837306C212yesmaximal, complemented
8#16837278C212yesmaximal
9#16837276C212yesmaximal, complemented
10#16837261C211yesmaximal, characteristic, complemented
11C59×#393C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×Dic118C2212yes—
14C2×Q8×D59—24no—
15C2×Q8×D59—28no—
16C22×C22⋊D59C2211yescharacteristic
17C23×Dic59C2211yescharacteristic
18C22×C4×D59C2212yes—
19C22×C4×D59C2211yescharacteristic
20#14273704—28no—
21(C22⋊D59)⋊C4—24no—
22C2×C4⋊Dic59C2212yes—
23C2×Dic59⋊C4C2212yes—
24C2×Dic59⋊C4C2212yes—
25C2×Dic59⋊C4C2211yescharacteristic
26C2×C22⋊Dic59C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2211yescharacteristic
29C2×C4×Dic59C2211yescharacteristic
30C59⋊#95C2211yescharacteristic
31C59⋊#95C2212yes—
32C59⋊#95C2212yes—
33C59⋊#95C2212yes—
34C22×Q8×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2212yes—
36C2×C22⋊C4×C59C2211yescharacteristic
37#95×C59C2211yescharacteristic
38#95×C59C2212yes—
39C24×C59C2311yescharacteristic
40C23×D59C2311yescharacteristic
41C2×Q8×C59—22no—
42C2×Q8×C59—24no—
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2312yes—
45C22×C4×C59C2311yescharacteristic
46C22×C4×C59C2312yes—
47C2×C22⋊D59—22no—
48C2×C22⋊D59—22no—
49C2×C22⋊D59Q814yescomplemented
50C22×Dic59—22no—
51C22×Dic59—24no—
52C22×Dic59C2312yes—
53C22×Dic59C2311yescharacteristic
54C22×Dic59C2312yes—
55C22×Dic59C2311yescharacteristic
56C22×Dic59C2311yescharacteristic
57Q8×D59—232no—
58C2×C4×D59—24no—
59C2×C4×D59—24no—
60C2×C4×D59—24no—
61C2×C4×D59—24no—
62C2×C4×D59D414yes—
63C2×Dic118—22no—
64C2×Dic118—24no—
65C2×Dic118—24no—
66C2×Dic118—24no—
67C2×Dic118D418yescomplemented
68C22⋊C4×C59—22no—
69C22⋊C4×C59—24no—
70C22⋊Dic59—24no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—24no—
73C4⋊Dic59—24no—
74Dic59⋊C4—22no—
75Dic59⋊C4—24no—
76Dic59⋊C4—24no—
77C4×Dic59—22no—
78C23×C59—22no—
79C23×C59—21no—
80C23×C59—21no—
81C23×C59—22no—
82C23×C59C2×Q812yescomplemented
83C23×C59C2411yescharacteristic
84C22×D59—22no—
85C22×D59—22no—
86C22×D59C2×D412yes—
87C22×D59C2×Q812yes—
88C22×D59#4112yes—
89Q8×C59—28no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59C2×D412yes—
99C2×C4×C59C2×D414yes—
100C22⋊D59—28no—
101C2×Dic59—21no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—21no—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—21no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—44no—
111C2×Dic59C2×D414yes—
112C2×Dic59C2×D412yes—
113C2×Dic59C2×D414yes—
114C2×Dic59C2×Q812yescomplemented
115C4×D59—216no—
116C4×D59—28no—
117Dic118—28no—
118Dic118—216no—
119C22×C59—22no—
120C22×C59—24no—
121C22×C59—41no—
122C22×C59—41no—
123C22×C59—42no—
124C22×C59C22×D411yescharacteristic
125C22×C59C22×D412yes—
126C22×C59C22×Q811yescharacteristic
127C22×C59C2×#4111yescharacteristic
128C22×C592−1+412yes—
129C2×D59—22no—
130C2×D59—22no—
131C2×D59—24no—
132C2×D59—24no—
133C2×D59C22⋊Q814yescomplemented
134C4×C59—22no—
135C4×C59—24no—
136C4×C59—41no—
137C4×C59—42no—
138Dic59—24no—
139Dic59—22no—
140Dic59—41no—
141Dic59—42no—
142Dic59#12014yes—
143#2409—591nomaximal, Sylow
144C2×C59—42no—
145C2×C59C2×#12011yescharacteristic
146C2×C59C2×C22⋊Q811yescharacteristic
147C2×C59#53611yescharacteristic
148C2×C59#54812yes—
149C2×C59D4×Q812yes—
150D59—24no—
151C23×Q8—591no—
152C2×C22⋊Q8—592no—
153C2×C4×D4—591no—
154#395—591no—
155#395—591no—
156#393—591no—
157#393D5911yescharacteristic, complemented
158#391—592no—
159#386—592no—
160#385—592no—
161#380—591no—
162C59#240911yescharacteristic, complemented, Sylow
163C22×Q8—592no—
164C22×Q8—591no—
165C22×Q8—1184no—
166C22×Q8—1188no—
167C22×Q8C2×D5911yescharacteristic
168C22×D4—591no—
169C23×C4—591no—
170C23×C4—591no—
171C23×C4—592no—
172C22⋊Q8—1188no—
173C4×D4—1184no—
174C2×C4⋊C4—591no—
175C2×C4⋊C4—592no—
176C2×C4⋊C4—592no—
177C2×C4⋊C4—592no—
178C2×C22⋊C4—592no—
179C2×C22⋊C4—591no—
180C2×C22⋊C4—592no—
181C2×C22⋊C4C2×D5911yescharacteristic
182C2×C22⋊C4C2×D5912yes—
183C2×C42—591no—
184#95—591no—
185#95—592no—
186#95—592no—
187#95—592no—
188#95C2×D5911yescharacteristic
189#95C2×D5912yes—
190C24—591no—
191C24C22×D5911yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—598no—
195C2×Q8—1184no—
196C2×Q8—1184no—
197C2×Q8—1184no—
198C2×Q8—11832no—
199C2×Q8—1182no—
200C2×D4—594no—
201C2×D4—1182no—
202C2×D4—1182no—
203C22×C4—591no—
204C22×C4—591no—
205C22×C4—592no—
206C22×C4—594no—
207C22×C4—592no—
208C22×C4—591no—
209C22×C4—1184no—
210C22×C4—1184no—
211C22×C4—1184no—
212C22×C4—1182no—
213C22×C4—1184no—
214C22×C4—1184no—
215C22×C4C22×D5911yescharacteristic
216C22×C4C22×D5912yes—
217C22×C4C22×D5911yescharacteristic
218C22×C4C22×D5912yes—
219C4⋊C4—1184no—
220C4⋊C4—1184no—
221C4⋊C4—1182no—
222C4⋊C4—1184no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—1182no—
226C22⋊C4—1184no—
227C22⋊C4—1184no—
228C42—1182no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—592no—
234C23—592no—
235C23—592no—
236C23—1182no—
237C23—1182no—
238C23Q8×D5912yescomplemented
239C23C23×D5911yescharacteristic
240Q8—28no—
241Q8—11816no—
242Q8—1188no—
243D4—1188no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—592no—
253C2×C4—594no—
254C2×C4—594no—
255C2×C4—592no—
256C2×C4—1188no—
257C2×C4—11816no—
258C2×C4—1181no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4—1181no—
262C2×C4—1182no—
263C2×C4—1181no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—2364no—
268C2×C4D4×D5912yes—
269C2×C4D4×D5914yes—
270C22—22no—
271C22—24no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—594no—
276C22—1182no—
277C22—1184no—
278C22—1184no—
279C22—1182no—
280C22C2×D4×D5911yescharacteristic
281C22C2×D4×D5912yes—
282C22C2×Q8×D5911yescharacteristic
283C22#41×D5911yescharacteristic
284C22#1427377412yes—
285C4—22no—
286C4—24no—
287C4—41no—
288C4—42no—
289C4—594no—
290C4—1182no—
291C4—1184no—
292C4—2362no—
293C4—2361no—
294C2—42no—
295C2—1184no—
296C2#120×D5911yescharacteristic
297C2C22⋊Q8×D5911yescharacteristic
298C2#1683797211yescharacteristic
299C2#1683806012yes—
300C2#1683806512yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#393C2
hypercenter#393D59
hypocenterC59#2409
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D59C2semidirect
2. C2×#14273704C2semidirect
4. #16837323C2semidirect
5. #16837321C2semidirect
6. #16837314C2semidirect
7. #16837306C2semidirect
9. #16837276C2semidirect
10. #16837261C2semidirect
11. C59×#393C2semidirect
49. C2×C22⋊D59Q8semidirect
67. C2×Dic118D4semidirect
82. C23×C59C2×Q8semidirect
114. C2×Dic59C2×Q8semidirect
133. C2×D59C22⋊Q8semidirect
157. #393D59semidirect
162. C59#2409semidirect
238. C23Q8×D59semidirect