Finite Group Databasebeta

← #21430472

Subgroups — #21430472

Order 7552.

688 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#14273695C211yesmaximal, characteristic
2#16837330C212yesmaximal, complemented
3#16837317C212yesmaximal, complemented
4#16837313C212yesmaximal, complemented
5#16837293C212yesmaximal, complemented
6#16837282C211yesmaximal, characteristic, complemented
7#16837279C212yesmaximal, complemented
8#16837257C212yesmaximal, complemented
9C2×#124×C59C211yesmaximal, characteristic, complemented
10C24×D59C2211yescharacteristic
11C22×C22⋊D59C2212yes—
12C23×Dic59C2211yescharacteristic, complemented
13#14273695—24no—
14C2×Dic59⋊C4C2212yescomplemented
15C2×C22⋊Dic59C2211yescharacteristic
16C2×C22⋊Dic59C2212yes—
17C2×C22⋊Dic59C2211yescharacteristic, complemented
18C2×(C2×D59)⋊C4C2212yes—
19C2×(C2×D59)⋊C4C2212yes—
20C2×(C2×D59)⋊C4C2212yes—
21C2×(C2×D59)⋊C4C2212yes—
22C2×(C2×D59)⋊C4C2212yes—
23C59⋊#95C2212yescomplemented
24C59⋊#95C2212yescomplemented
25C59⋊#95C2212yescomplemented
26C59⋊#95C2212yescomplemented
27C22×Q8×C59C2211yescharacteristic, complemented
28C22×D4×C59C2211yescharacteristic
29#124×C59—24no—
30C2×C22⋊C4×C59C2212yes—
31C2×C22⋊C4×C59C2212yescomplemented
32C2×C42×C59C2211yescharacteristic, complemented
33C24×C59C2311yescharacteristic
34C24×C59C2311yescharacteristic
35C23×D59—22no—
36C23×D59—24no—
37C23×D59—41no—
38C23×D59—41no—
39C23×D59—42no—
40C23×D59C2312yes—
41C2×Q8×C59—22no—
42C2×Q8×C59—41no—
43C2×Q8×C59—41no—
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59D414yes—
47C22×C4×C59C2312yes—
48C22×C4×C59C2312yes—
49C22×C4×C59C2311yescharacteristic
50C2×C22⋊D59—24no—
51C2×C22⋊D59—24no—
52C2×C22⋊D59—42no—
53C22×Dic59—41no—
54C22×Dic59—42no—
55C22×Dic59C2312yes—
56C22×Dic59C2312yes—
57C22×Dic59C2312yes—
58C22⋊C4×C59—24no—
59C22⋊C4×C59—24no—
60C42×C59—22no—
61C22⋊Dic59—22no—
62C22⋊Dic59—42no—
63C22⋊Dic59—41no—
64(C2×D59)⋊C4—24no—
65(C2×D59)⋊C4—24no—
66(C2×D59)⋊C4—42no—
67(C2×D59)⋊C4—42no—
68(C2×D59)⋊C4—42no—
69Dic59⋊C4—42no—
70C23×C59—22no—
71C23×C59—22no—
72C23×C59—21no—
73C23×C59—22no—
74C23×C59—22no—
75C23×C59—21no—
76C23×C59—21no—
77C23×C59—21no—
78C23×C59C2×D412yes—
79C23×C59C2×D412yes—
80C23×C59C2411yescharacteristic
81C22×D59—22no—
82C22×D59—22no—
83C22×D59—22no—
84C22×D59—22no—
85C22×D59—22no—
86C22×D59—22no—
87C22×D59—44no—
88C22×D59—48no—
89C22×D59—48no—
90C22×D59—44no—
91C22×D59—44no—
92C22×D59#4114yes—
93Q8×C59—44no—
94D4×C59—44no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59C2×D412yes—
104C22⋊D59—48no—
105C2×Dic59—22no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59—44no—
115C22×C59—22no—
116C22×C59—22no—
117C22×C59—24no—
118C22×C59—24no—
119C22×C59—42no—
120C22×C59—42no—
121C22×C59—41no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59—41no—
125C22×C59C22×D411yescharacteristic
126C22×C59C2×#4112yes—
127C22×C59H(5,𝔽2)11yescharacteristic
128C22×C59H(5,𝔽2)11yescharacteristic
129C22×C59H(5,𝔽2)12yes—
130C2×D59—24no—
131C2×D59—24no—
132C2×D59—24no—
133C2×D59—416no—
134C2×D59—42no—
135C2×D59—42no—
136C2×D59—42no—
137C2×D59—42no—
138C4×C59—22no—
139C4×C59—42no—
140C4×C59—42no—
141Dic59—42no—
142Dic59—42no—
143Dic59—42no—
144#2684—591nomaximal, Sylow
145C2×C59—42no—
146C2×C59—42no—
147C2×C59#53411yescharacteristic
148C2×C59#53812yes—
149C2×C59#54612yes—
150C2×C59#55111yescharacteristic
151C2×C59H(3,𝔽4)11yescharacteristic
152D59—44no—
153C2×#124D5911yescharacteristic, complemented
154C2×#120—591no—
155#397—592no—
156#394—592no—
157#393—592no—
158#388—592no—
159#386—591no—
160#386—592no—
161#379—592no—
162C59#268411yescharacteristic, complemented, Sylow
163C25—591no—
164C22×Q8C2×D5911yescharacteristic, complemented
165C22×D4—592no—
166C22×D4C2×D5911yescharacteristic
167C23×C4—591no—
168#124—24no—
169#120—1184no—
170C2×C4⋊C4—592no—
171C2×C22⋊C4—591no—
172C2×C22⋊C4—592no—
173C2×C22⋊C4—592no—
174C2×C22⋊C4—592no—
175C2×C22⋊C4—591no—
176C2×C22⋊C4—592no—
177C2×C22⋊C4—592no—
178C2×C22⋊C4—592no—
179C2×C22⋊C4C2×D5912yes—
180C2×C22⋊C4C2×D5912yescomplemented
181C2×C42C2×D5911yescharacteristic, complemented
182#95—592no—
183#95—592no—
184#95—592no—
185#95—592no—
186C24—592no—
187C24—1184no—
188C24—1182no—
189C24—2361no—
190C24—2361no—
191C24—2362no—
192C24C22×D5911yescharacteristic
193C24C22×D5911yescharacteristic
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—1184no—
200C2×D4—1184no—
201C2×D4—2362no—
202C2×D4C22⋊D5914yes—
203C22×C4—592no—
204C22×C4—592no—
205C22×C4—592no—
206C22×C4—2361no—
207C22×C4—2362no—
208C22×C4C22×D5912yes—
209C22×C4C22×D5912yes—
210C22×C4C22×D5911yescharacteristic
211C4⋊C4—2362no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—1184no—
215C22⋊C4—1182no—
216C22⋊C4—1184no—
217C22⋊C4—2362no—
218C22⋊C4—2362no—
219C22⋊C4—2362no—
220C22⋊C4—2361no—
221C22⋊C4—2362no—
222C42—22no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—594no—
232C23—1182no—
233C23—1182no—
234C23—1182no—
235C23—1182no—
236C23—1182no—
237C23—1182no—
238C23—2368no—
239C23—2364no—
240C23—2364no—
241C23—2368no—
242C23—2364no—
243C23C2×C22⋊D5912yes—
244C23C2×C22⋊D5912yes—
245C23C23×D5911yescharacteristic
246Q8—44no—
247D4—44no—
248D4—2368no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4—1182no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—2364no—
267C2×C4C2×C22⋊D5912yes—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—1184no—
279C22—1184no—
280C22—1184no—
281C22—23616no—
282C22—2362no—
283C22—2362no—
284C22—2362no—
285C22—2362no—
286C22C22×C22⋊D5911yescharacteristic
287C22#41×D5912yes—
288C22#1427377211yescharacteristic
289C22#1427377211yescharacteristic
290C22#1427377312yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—2362no—
295C4—2362no—
296C4—2362no—
297C2—42no—
298C2—42no—
299C2—2364no—
300C2#1683796111yescharacteristic
301C2#1683798612yes—
302C2#1683804012yes—
303C2#1683808211yescharacteristic
304C2#1683814411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#124×C59C2
hypercenterC2×#124D59
hypocenterC59#2684
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #16837330C2semidirect
3. #16837317C2semidirect
4. #16837313C2semidirect
5. #16837293C2semidirect
6. #16837282C2semidirect
7. #16837279C2semidirect
8. #16837257C2semidirect
9. C2×#124×C59C2semidirect
12. C23×Dic59C22semidirect
14. C2×Dic59⋊C4C22semidirect
17. C2×C22⋊Dic59C22semidirect
23. C59⋊#95C22semidirect
24. C59⋊#95C22semidirect
25. C59⋊#95C22semidirect
26. C59⋊#95C22semidirect
27. C22×Q8×C59C22semidirect
31. C2×C22⋊C4×C59C22semidirect
32. C2×C42×C59C22semidirect
153. C2×#124D59semidirect
162. C59#2684semidirect
164. C22×Q8C2×D59semidirect
180. C2×C22⋊C4C2×D59semidirect
181. C2×C42C2×D59semidirect