Finite Group Databasebeta

← #21427456

Subgroups — #21427456

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×#14273705C212yesmaximal, complemented
3C2×#14273689C211yesmaximal, characteristic
4#16837322C211yesmaximal, characteristic, complemented
5#16837315C211yesmaximal, characteristic, complemented
6#16837313C211yesmaximal, characteristic, complemented
7#16837306C212yesmaximal, complemented
8#16837280C212yesmaximal
9#16837273C212yesmaximal, complemented
10#16837261C211yesmaximal, characteristic, complemented
11C59×#395C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×Dic118C2212yes—
14C2×Q8×D59—24no—
15C2×Q8×D59—28no—
16C22×D236C2211yescharacteristic
17C22×C4×D59C2211yescharacteristic
18C22×C4×D59C2212yes—
19C22×C4×D59C2211yescharacteristic
20#14273705—28no—
21#14273689—24no—
22C2×Dic59⋊C4C2212yes—
23C2×Dic59⋊C4C2212yes—
24C2×(C2×D59)⋊C4C2212yes—
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2211yescharacteristic
28C2×(C2×D59)⋊C4C2211yescharacteristic
29C2×C4×Dic59C2211yescharacteristic
30C59⋊#95C2211yescharacteristic
31C59⋊#95C2212yes—
32C59⋊#95C2212yes—
33C59⋊#95C2212yes—
34C22×Q8×C59C2211yescharacteristic
35C2×C4⋊C4×C59C2212yes—
36C2×C4⋊C4×C59C2211yescharacteristic
37#95×C59C2211yescharacteristic
38#95×C59C2212yes—
39C23×D59C2311yescharacteristic
40C23×D59C2311yescharacteristic
41C2×Q8×C59—22no—
42C2×Q8×C59—24no—
43C22×C4×C59C2312yes—
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2311yescharacteristic
46C22×C4×C59C2312yes—
47C22×C4×C59C2311yescharacteristic
48C22×Dic59C2311yescharacteristic
49C22×Dic59C2312yes—
50C22×Dic59C2312yes—
51C22×Dic59C2311yescharacteristic
52Q8×D59—232no—
53C2×D236—22no—
54C2×D236—22no—
55C2×D236Q814yescomplemented
56C2×C4×D59—24no—
57C2×C4×D59—24no—
58C2×C4×D59—24no—
59C2×C4×D59—24no—
60C2×C4×D59—22no—
61C2×C4×D59—24no—
62C2×C4×D59D414yes—
63C2×Dic118—22no—
64C2×Dic118—24no—
65C2×Dic118—24no—
66C2×Dic118—24no—
67C2×Dic118D418yescomplemented
68C4⋊C4×C59—22no—
69C4⋊C4×C59—24no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—24no—
72(C2×D59)⋊C4—22no—
73(C2×D59)⋊C4—24no—
74(C2×D59)⋊C4—24no—
75Dic59⋊C4—24no—
76Dic59⋊C4—24no—
77C4×Dic59—22no—
78C23×C59C2411yescharacteristic
79C22×D59—22no—
80C22×D59—22no—
81C22×D59—22no—
82C22×D59—21no—
83C22×D59—21no—
84C22×D59—22no—
85C22×D59C2×D412yes—
86C22×D59C2×Q812yes—
87C22×D59C2×Q812yescomplemented
88C22×D59#4112yes—
89Q8×C59—28no—
90C2×C4×C59—21no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59C2×D412yes—
101C2×C4×C59C2×D414yes—
102C2×C4×C59C2×Q812yescomplemented
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—21no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59C2×D414yes—
111C2×Dic59C2×D412yes—
112C2×Dic59C2×D414yes—
113D236—28no—
114C4×D59—216no—
115C4×D59—28no—
116C4×D59—44no—
117Dic118—28no—
118Dic118—216no—
119C22×C59C22×D411yescharacteristic
120C22×C59C22×D412yes—
121C22×C59C22×Q811yescharacteristic
122C22×C59C2×#4111yescharacteristic
123C22×C592−1+412yes—
124C2×D59—22no—
125C2×D59—22no—
126C2×D59—24no—
127C2×D59—24no—
128C2×D59—22no—
129C2×D59—24no—
130C2×D59—41no—
131C2×D59—41no—
132C2×D59—42no—
133C2×D59C22⋊Q814yescomplemented
134C4×C59—22no—
135C4×C59—24no—
136C4×C59—22no—
137C4×C59—41no—
138C4×C59—42no—
139Dic59—24no—
140Dic59—41no—
141Dic59—42no—
142Dic59#12014yes—
143#2409—591nomaximal, Sylow
144C2×C59C2×#12011yescharacteristic
145C2×C59C2×C22⋊Q811yescharacteristic
146C2×C59#53611yescharacteristic
147C2×C59#54812yes—
148C2×C59D4×Q812yes—
149D59—24no—
150D59—42no—
151C23×Q8—591no—
152C2×C22⋊Q8—592no—
153C2×C4×D4—591no—
154#395—591no—
155#395D5911yescharacteristic, complemented
156#393—591no—
157#393—591no—
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—592no—
175C2×C4⋊C4—592no—
176C2×C4⋊C4C2×D5911yescharacteristic
177C2×C4⋊C4C2×D5912yes—
178C2×C22⋊C4—591no—
179C2×C22⋊C4—591no—
180C2×C22⋊C4—592no—
181C2×C22⋊C4—592no—
182C2×C22⋊C4—592no—
183C2×C42—591no—
184#95—591no—
185#95—592no—
186#95—592no—
187#95—592no—
188#95C2×D5911yescharacteristic
189#95C2×D5912yes—
190C24—591no—
191C24—591no—
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—592no—
205C22×C4—594no—
206C22×C4—591no—
207C22×C4—592no—
208C22×C4—1184no—
209C22×C4—1184no—
210C22×C4—1182no—
211C22×C4—1184no—
212C22×C4—1184no—
213C22×C4—1184no—
214C22×C4C22×D5912yes—
215C22×C4C22×D5911yescharacteristic
216C22×C4C22×D5911yescharacteristic
217C22×C4C22×D5912yes—
218C22×C4C22×D5911yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—1184no—
222C4⋊C4—1184no—
223C22⋊C4—1184no—
224C22⋊C4—1182no—
225C22⋊C4—1184no—
226C22⋊C4—1182no—
227C22⋊C4—1184no—
228C42—1182no—
229C23—592no—
230C23—592no—
231C23—592no—
232C23—592no—
233C23—1182no—
234C23—1182no—
235C23—1182no—
236C23—1181no—
237C23—1181no—
238C23—1182no—
239C23C23×D5911yescharacteristic
240Q8—28no—
241Q8—11816no—
242Q8—1188no—
243D4—1188no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—592no—
255C2×C4—594no—
256C2×C4—594no—
257C2×C4—1188no—
258C2×C4—11816no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4—1181no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—2364no—
267C2×C4D4×D5912yes—
268C2×C4D4×D5914yes—
269C2×C4Q8×D5912yescomplemented
270C22—594no—
271C22—1182no—
272C22—1184no—
273C22—1184no—
274C22—1182no—
275C22—1182no—
276C22—1184no—
277C22—2361no—
278C22—2362no—
279C22—2361no—
280C22C2×D4×D5911yescharacteristic
281C22C2×D4×D5912yes—
282C22C2×Q8×D5911yescharacteristic
283C22#41×D5911yescharacteristic
284C22#1427377512yes—
285C4—22no—
286C4—22no—
287C4—24no—
288C4—42no—
289C4—41no—
290C4—594no—
291C4—1184no—
292C4—2361no—
293C4—2362no—
294C2—1184no—
295C2—2362no—
296C2#120×D5911yescharacteristic
297C2C22⋊Q8×D5911yescharacteristic
298C2#1683797711yescharacteristic
299C2#1683806212yes—
300C2#1683806712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#395C2
hypercenter#395D59
hypocenterC59#2409
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D59C2semidirect
2. C2×#14273705C2semidirect
4. #16837322C2semidirect
5. #16837315C2semidirect
6. #16837313C2semidirect
7. #16837306C2semidirect
9. #16837273C2semidirect
10. #16837261C2semidirect
11. C59×#395C2semidirect
55. C2×D236Q8semidirect
67. C2×Dic118D4semidirect
87. C22×D59C2×Q8semidirect
102. C2×C4×C59C2×Q8semidirect
133. C2×D59C22⋊Q8semidirect
155. #395D59semidirect
162. C59#2409semidirect
269. C2×C4Q8×D59semidirect