Finite Group Databasebeta

← #21428371

Subgroups — #21428371

Order 7552.

672 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 139 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×#14273685C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D59C211yesmaximal, characteristic
3#16837337C211yesmaximal, characteristic, complemented
4#16837329C211yesmaximal, characteristic, complemented
5#16837327C211yesmaximal, characteristic, complemented
6#16837296C212yesmaximal, complemented
7#16837292C212yesmaximal, complemented
8#16837273C212yesmaximal, complemented
9#16837264C211yesmaximal, characteristic, complemented
10#16837257C212yesmaximal
11C59×#399C211yesmaximal, characteristic, complemented
12C24×D59C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C22×C4×D59C2211yescharacteristic
15#14273685—24no—
16C2×C4⋊Dic59C2211yescharacteristic, complemented
17C2×Dic59⋊C4C2212yescomplemented
18C2×C22⋊Dic59C2212yes—
19C2×C22⋊Dic59C2211yescharacteristic
20C2×(C2×D59)⋊C4C2212yes—
21C2×(C2×D59)⋊C4C2212yes—
22C2×(C2×D59)⋊C4C2212yes—
23C2×(C2×D59)⋊C4C2211yescharacteristic
24C2×(C2×D59)⋊C4C2211yescharacteristic
25C2×(C2×D59)⋊C4C2212yes—
26C22⋊C4×D59—24no—
27C2×C4×Dic59C2212yescomplemented
28C59⋊#95C2212yescomplemented
29C59⋊#95C2211yescharacteristic, complemented
30C59⋊#95C2212yescomplemented
31C59⋊#95C2212yescomplemented
32C2×C4⋊C4×C59C2211yescharacteristic, complemented
33C2×C4⋊C4×C59C2212yescomplemented
34C2×C22⋊C4×C59C2211yescharacteristic
35C2×C22⋊C4×C59C2212yes—
36#95×C59C2211yescharacteristic, complemented
37C24×C59C2311yescharacteristic
38C23×D59—22no—
39C23×D59—22no—
40C23×D59—22no—
41C23×D59—24no—
42C23×D59—42no—
43C23×D59C2311yescharacteristic
44C23×D59C2311yescharacteristic
45C22×C4×C59C2312yes—
46C22×C4×C59C2311yescharacteristic
47C22×C4×C59C2312yes—
48C22×C4×C59C2311yescharacteristic
49C22×Dic59C2312yes—
50C22×Dic59C2312yes—
51C22×Dic59C2311yescharacteristic
52C22×Dic59C2311yescharacteristic
53C2×C4×D59—24no—
54C2×C4×D59—22no—
55C2×C4×D59—42no—
56C2×C4×D59D414yes—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—42no—
59C22⋊C4×C59—22no—
60C22⋊C4×C59—42no—
61C22⋊Dic59—24no—
62C22⋊Dic59—22no—
63(C2×D59)⋊C4—24no—
64(C2×D59)⋊C4—24no—
65(C2×D59)⋊C4—24no—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—42no—
69C4⋊Dic59—22no—
70Dic59⋊C4—42no—
71C4×Dic59—24no—
72C23×C59—21no—
73C23×C59—21no—
74C23×C59—22no—
75C23×C59—22no—
76C23×C59#4112yes—
77C23×C59C2411yescharacteristic
78C22×D59—21no—
79C22×D59—21no—
80C22×D59—22no—
81C22×D59—28no—
82C22×D59—21no—
83C22×D59—22no—
84C22×D59—21no—
85C22×D59—22no—
86C22×D59—48no—
87C22×D59—48no—
88C22×D59—44no—
89C22×D59—44no—
90C22×D59C2×D412yes—
91C22×D59#4112yes—
92C22×D59#4114yes—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59C2×D412yes—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—21no—
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—22no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59C2×D412yes—
113C4×D59—28no—
114C4×D59—44no—
115C22×C59—22no—
116C22×C59—24no—
117C22×C59—41no—
118C22×C59—41no—
119C22×C59—42no—
120C22×C59C22×D411yescharacteristic
121C22×C59C2×#4112yes—
122C22×C59C2×#4111yescharacteristic
123C22×C59C2×#4111yescharacteristic
124C22×C59H(5,𝔽2)12yes—
125C2×D59—22no—
126C2×D59—22no—
127C2×D59—24no—
128C2×D59—24no—
129C2×D59—22no—
130C2×D59—24no—
131C2×D59—416no—
132C2×D59—41no—
133C2×D59—41no—
134C2×D59—42no—
135C2×D59#12314yes—
136C4×C59—22no—
137C4×C59—41no—
138C4×C59—42no—
139C4×C59—42no—
140Dic59—22no—
141Dic59—42no—
142Dic59—41no—
143Dic59—42no—
144#2482—591nomaximal, Sylow
145C2×C59—42no—
146C2×C59C2×#12311yescharacteristic
147C2×C59#52511yescharacteristic
148C2×C59#53511yescharacteristic
149C2×C59#55112yes—
150C2×C59#55112yes—
151D59—24no—
152D59—42no—
153C2×#117—591no—
154C22×C22⋊C4—591no—
155#399—591no—
156#399D5911yescharacteristic, complemented
157#397—591no—
158#397—591no—
159#388—592no—
160#388—592no—
161#385—592no—
162#381—591no—
163#379—592no—
164C59#248211yescharacteristic, complemented, Sylow
165C25—591no—
166C23×C4—591no—
167C23×C4—591no—
168#117—1184no—
169C2×C4⋊C4—591no—
170C2×C4⋊C4—592no—
171C2×C4⋊C4C2×D5911yescharacteristic, complemented
172C2×C4⋊C4C2×D5912yescomplemented
173C2×C22⋊C4—591no—
174C2×C22⋊C4—592no—
175C2×C22⋊C4—591no—
176C2×C22⋊C4—591no—
177C2×C22⋊C4—592no—
178C2×C22⋊C4—592no—
179C2×C22⋊C4—592no—
180C2×C22⋊C4—592no—
181C2×C22⋊C4—1184no—
182C2×C22⋊C4C2×D5911yescharacteristic
183C2×C22⋊C4C2×D5912yes—
184C2×C42—592no—
185#95—592no—
186#95—591no—
187#95—592no—
188#95—592no—
189#95C2×D5911yescharacteristic, complemented
190C24—591no—
191C24—591no—
192C24—1182no—
193C24—1182no—
194C24—1184no—
195C24—1182no—
196C24—2362no—
197C24C22×D5911yescharacteristic
198C22×C4—591no—
199C22×C4—594no—
200C22×C4—592no—
201C22×C4—592no—
202C22×C4—591no—
203C22×C4—1184no—
204C22×C4—1182no—
205C22×C4—2362no—
206C22×C4C22×D5912yes—
207C22×C4C22×D5911yescharacteristic
208C22×C4C22×D5912yes—
209C22×C4C22×D5911yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—42no—
212C4⋊C4—1182no—
213C4⋊C4—2362no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—1182no—
217C22⋊C4—1184no—
218C22⋊C4—1184no—
219C22⋊C4—1184no—
220C22⋊C4—1182no—
221C22⋊C4—1184no—
222C22⋊C4—1182no—
223C22⋊C4—2362no—
224C42—1184no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—592no—
230C23—594no—
231C23—592no—
232C23—1188no—
233C23—1181no—
234C23—1181no—
235C23—1182no—
236C23—1181no—
237C23—1182no—
238C23—1181no—
239C23—1182no—
240C23—2368no—
241C23—2364no—
242C23—2368no—
243C23—2364no—
244C23#1174568912yes—
245C23C23×D5911yescharacteristic
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—592no—
256C2×C4—1188no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—1181no—
260C2×C4—1182no—
261C2×C4—1182no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—2364no—
267C2×C4D4×D5912yes—
268C22—22no—
269C22—24no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—594no—
274C22—1182no—
275C22—1182no—
276C22—1184no—
277C22—1184no—
278C22—1182no—
279C22—1184no—
280C22—23616no—
281C22—2361no—
282C22—2361no—
283C22—2362no—
284C22C2×D4×D5911yescharacteristic
285C22#41×D5912yes—
286C22#41×D5911yescharacteristic
287C22C2×#1174568911yescharacteristic
288C22#1427377212yes—
289C4—22no—
290C4—41no—
291C4—42no—
292C4—42no—
293C4—1182no—
294C4—2362no—
295C4—2361no—
296C4—2362no—
297C2—42no—
298C2—1184no—
299C2—2362no—
300C2#123×D5911yescharacteristic
301C2#1683791511yescharacteristic
302C2#1683796511yescharacteristic
303C2#1683807912yes—
304C2#1683808012yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#399C2
hypercenter#399D59
hypocenterC59#2482
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14273685C2semidirect
3. #16837337C2semidirect
4. #16837329C2semidirect
5. #16837327C2semidirect
6. #16837296C2semidirect
7. #16837292C2semidirect
8. #16837273C2semidirect
9. #16837264C2semidirect
11. C59×#399C2semidirect
16. C2×C4⋊Dic59C22semidirect
17. C2×Dic59⋊C4C22semidirect
27. C2×C4×Dic59C22semidirect
28. C59⋊#95C22semidirect
29. C59⋊#95C22semidirect
30. C59⋊#95C22semidirect
31. C59⋊#95C22semidirect
32. C2×C4⋊C4×C59C22semidirect
33. C2×C4⋊C4×C59C22semidirect
36. #95×C59C22semidirect
156. #399D59semidirect
164. C59#2482semidirect
171. C2×C4⋊C4C2×D59semidirect
172. C2×C4⋊C4C2×D59semidirect
189. #95C2×D59semidirect