Finite Group Databasebeta

← #21427756

Subgroups — #21427756

Order 7552.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14273703C212yesmaximal, complemented
2C2×C22⋊C4×D59C212yesmaximal
3#16837338C211yesmaximal, characteristic, complemented
4#16837325C212yesmaximal, complemented
5#16837276C212yesmaximal, complemented
6#16837276C212yesmaximal, complemented
7#16837261C212yesmaximal, complemented
8#16837257C211yesmaximal, characteristic
9C59×#393C211yesmaximal, characteristic, complemented
10C24×D59C2211yescharacteristic
11C22×C4×D59C2212yes—
12C22×C4×D59C2212yes—
13#14273703—28no—
14C2×C4⋊Dic59C2212yescomplemented
15C2×C4⋊Dic59C2211yescharacteristic, complemented
16C2×Dic59⋊C4C2212yescomplemented
17C2×Dic59⋊C4C2212yescomplemented
18C2×Dic59⋊C4C2212yescomplemented
19C2×C22⋊Dic59C2212yes—
20C2×C22⋊Dic59C2211yescharacteristic
21C2×(C2×D59)⋊C4C2212yes—
22C2×(C2×D59)⋊C4C2212yes—
23C2×(C2×D59)⋊C4C2212yes—
24C2×(C2×D59)⋊C4C2212yes—
25C22⋊C4×D59—28no—
26C59⋊#95C2212yescomplemented
27C59⋊#95C2211yescharacteristic, complemented
28C22×Q8×C59C2211yescharacteristic, complemented
29C2×C22⋊C4×C59C2212yes—
30C2×C22⋊C4×C59C2211yescharacteristic
31#95×C59C2211yescharacteristic, complemented
32#95×C59C2212yescomplemented
33C24×C59C2311yescharacteristic
34C23×D59—24no—
35C23×D59—24no—
36C23×D59—41no—
37C23×D59—41no—
38C23×D59Q814yescomplemented
39C23×D59C2312yes—
40C2×Q8×C59—24no—
41C2×Q8×C59—41no—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2312yes—
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2312yes—
46C22×Dic59C2312yes—
47C22×Dic59C2312yes—
48C22×Dic59C2312yes—
49C2×C4×D59—24no—
50C2×C4×D59—24no—
51C2×C4×D59—24no—
52C2×C4×D59—24no—
53C2×C4×D59—42no—
54C2×C4×D59—42no—
55C22⋊C4×C59—24no—
56C22⋊C4×C59—41no—
57C22⋊Dic59—24no—
58C22⋊Dic59—41no—
59(C2×D59)⋊C4—24no—
60(C2×D59)⋊C4—24no—
61(C2×D59)⋊C4—24no—
62(C2×D59)⋊C4D418yes—
63C4⋊Dic59—24no—
64C4⋊Dic59—41no—
65Dic59⋊C4—24no—
66Dic59⋊C4—24no—
67Dic59⋊C4—42no—
68C23×C59—22no—
69C23×C59—22no—
70C23×C59—21no—
71C23×C59—21no—
72C23×C59C2×Q812yes—
73C23×C59C2411yescharacteristic
74C22×D59—22no—
75C22×D59—22no—
76C22×D59—22no—
77C22×D59—22no—
78C22×D59—216no—
79C22×D59—44no—
80C22×D59—44no—
81C22×D59—48no—
82C22×D59—44no—
83C22×D59C2×D414yes—
84C22×D59C2×Q814yes—
85C22×D59#4114yes—
86Q8×C59—44no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59C2×D414yes—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59C2×D414yes—
106C4×D59—48no—
107C4×D59—48no—
108C22×C59—24no—
109C22×C59—22no—
110C22×C59—41no—
111C22×C59—42no—
112C22×C59—41no—
113C22×C59C22×D412yes—
114C22×C59C22×Q811yescharacteristic
115C22×C59C2×#4112yes—
116C22×C59H(5,𝔽2)11yescharacteristic
117C22×C59H(5,𝔽2)11yescharacteristic
118C2×D59—24no—
119C2×D59—24no—
120C2×D59—24no—
121C2×D59—24no—
122C2×D59—24no—
123C2×D59—24no—
124C2×D59—416no—
125C2×D59C22⋊Q818yescomplemented
126C4×C59—24no—
127C4×C59—41no—
128C4×C59—42no—
129C4×C59—41no—
130Dic59—24no—
131Dic59—42no—
132Dic59—42no—
133#2437—591nomaximal, Sylow
134C2×C59—42no—
135C2×C59C2×C22⋊Q812yes—
136C2×C59#54311yescharacteristic
137C2×C59D4211yescharacteristic
138C2×C59#54612yes—
139C2×C59#55111yescharacteristic
140D59—28no—
141C2×C22⋊Q8—592no—
142C22×C22⋊C4—592no—
143#399—591no—
144#396—592no—
145#393D5911yescharacteristic, complemented
146#385—592no—
147#385—592no—
148#380—592no—
149#379—591no—
150C59#243711yescharacteristic, complemented, Sylow
151C25—591no—
152C22×Q8C2×D5911yescharacteristic, complemented
153C23×C4—592no—
154C23×C4—592no—
155C22⋊Q8—1188no—
156C2×C4⋊C4—592no—
157C2×C4⋊C4—592no—
158C2×C4⋊C4—592no—
159C2×C4⋊C4—592no—
160C2×C4⋊C4—591no—
161C2×C22⋊C4—592no—
162C2×C22⋊C4—592no—
163C2×C22⋊C4—592no—
164C2×C22⋊C4—592no—
165C2×C22⋊C4—592no—
166C2×C22⋊C4—591no—
167C2×C22⋊C4—1188no—
168C2×C22⋊C4C2×D5912yes—
169C2×C22⋊C4C2×D5911yescharacteristic
170#95—592no—
171#95—591no—
172#95C2×D5911yescharacteristic, complemented
173#95C2×D5912yescomplemented
174C24—592no—
175C24—594no—
176C24—1184no—
177C24—1184no—
178C24—2361no—
179C24—2361no—
180C24C22×D5911yescharacteristic
181C2×Q8—24no—
182C2×Q8—41no—
183C22×C4—592no—
184C22×C4—592no—
185C22×C4—592no—
186C22×C4—1184no—
187C22×C4—1184no—
188C22×C4—1184no—
189C22×C4—1184no—
190C22×C4—2362no—
191C22×C4—2362no—
192C22×C4C22×D5911yescharacteristic
193C22×C4C22×D5912yes—
194C22×C4C22×D5911yescharacteristic
195C22×C4C22×D5912yes—
196C4⋊C4—1184no—
197C4⋊C4—1184no—
198C4⋊C4—1184no—
199C4⋊C4—2362no—
200C4⋊C4—2361no—
201C22⋊C4—24no—
202C22⋊C4—41no—
203C22⋊C4—598no—
204C22⋊C4—1184no—
205C22⋊C4—1184no—
206C22⋊C4—1184no—
207C22⋊C4—1184no—
208C22⋊C4—2361no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—594no—
214C23—594no—
215C23—594no—
216C23—11816no—
217C23—1182no—
218C23—1182no—
219C23—1182no—
220C23—1182no—
221C23—2364no—
222C23—2364no—
223C23—2368no—
224C23—2364no—
225C23Q8×D5912yes—
226C23C23×D5911yescharacteristic
227Q8—44no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—594no—
238C2×C4—1182no—
239C2×C4—1182no—
240C2×C4—1182no—
241C2×C4—1182no—
242C2×C4—1182no—
243C2×C4—1182no—
244C2×C4—1182no—
245C2×C4—1182no—
246C2×C4—2368no—
247C2×C4—2368no—
248C2×C4D4×D5914yes—
249C22—24no—
250C22—22no—
251C22—41no—
252C22—42no—
253C22—41no—
254C22—598no—
255C22—1184no—
256C22—1184no—
257C22—1184no—
258C22—1184no—
259C22—1184no—
260C22—1184no—
261C22—23616no—
262C22C2×D4×D5912yes—
263C22C2×Q8×D5911yescharacteristic
264C22#41×D5912yes—
265C22#1427377211yescharacteristic
266C22#1427377311yescharacteristic
267C4—24no—
268C4—41no—
269C4—42no—
270C4—41no—
271C4—1184no—
272C4—2362no—
273C4—2362no—
274C2—42no—
275C2—1188no—
276C2C22⋊Q8×D5912yes—
277C2#1683802711yescharacteristic
278C2#1683803411yescharacteristic
279C2#1683804512yes—
280C2#1683808211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#393C2
hypercenter#393D59
hypocenterC59#2437
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14273703C2semidirect
3. #16837338C2semidirect
4. #16837325C2semidirect
5. #16837276C2semidirect
6. #16837276C2semidirect
7. #16837261C2semidirect
9. C59×#393C2semidirect
14. C2×C4⋊Dic59C22semidirect
15. C2×C4⋊Dic59C22semidirect
16. C2×Dic59⋊C4C22semidirect
17. C2×Dic59⋊C4C22semidirect
18. C2×Dic59⋊C4C22semidirect
26. C59⋊#95C22semidirect
27. C59⋊#95C22semidirect
28. C22×Q8×C59C22semidirect
31. #95×C59C22semidirect
32. #95×C59C22semidirect
38. C23×D59Q8semidirect
125. C2×D59C22⋊Q8semidirect
145. #393D59semidirect
150. C59#2437semidirect
152. C22×Q8C2×D59semidirect
172. #95C2×D59semidirect
173. #95C2×D59semidirect