Finite Group Databasebeta

← #21435066

Subgroups — #21435066

Order 7552.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#16837938C211yesmaximal, characteristic, complemented
2C2×#14273716C211yesmaximal, characteristic, complemented
3#16837787C212yesmaximal, complemented
4#16837758C212yesmaximal, complemented
5#16837626C212yesmaximal, complemented
6#16837620C212yesmaximal, complemented
7#16837552C211yesmaximal, characteristic, complemented
8#16837541C211yesmaximal, characteristic, complemented
9#16837538C211yesmaximal, characteristic
10#16837452C211yesmaximal, characteristic, complemented
11C59×#451C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C2×#11745671C2212yes—
14C2×#11745671C2212yescomplemented
15C2×D472C2212yes—
16#14273722—21no—
17#14273718C2211yescharacteristic, complemented
18#14273716—21no—
19#14273716—21no—
20#14273716C2212yes—
21#14273716C2212yes—
22#14273716C2211yescharacteristic, complemented
23#14273711—21no—
24#14273701C2211yescharacteristic
25Q8×Dic59C2211yescharacteristic, complemented
26#14273689C2211yescharacteristic, complemented
27(C22⋊D59)⋊C4—21no—
28#14273685—21no—
29C2×(C2×D59)⋊C4—21no—
30C2×(C2×D59)⋊C4—21no—
31C2×C4×Dic59C2211yescharacteristic
32Dic59⋊C8C2212yescomplemented
33#14273651C2212yescomplemented
34#14273649C2212yes—
35#14273647C2212yes—
36C22⋊(C59⋊C8)C2211yescharacteristic
37C8×Dic59C2212yescomplemented
38C22×Q8×C59C2211yescharacteristic
39C2×Q16×C59C2212yescomplemented
40C22⋊Q8×C59C2211yescharacteristic
41#103×C59C2212yescomplemented
42C22⋊C8×C59C2211yescharacteristic
43C23×D59—21no—
44C2×Q8×C59—21no—
45C2×Q8×C59—42no—
46C2×Q8×C59C2312yes—
47C2×Q8×C59C2311yescharacteristic
48C22×C4×C59—21no—
49C22×C4×C59C2311yescharacteristic
50C2×C22⋊D59—21no—
51C22×Dic59D411yescharacteristic
52C22×Dic59D411yescharacteristic
53C2×D236—21no—
54C2×D236—42no—
55C2×D236C2312yes—
56C2×D236C2311yescharacteristic
57C2×C4×D59—21no—
58Q16×C59—42no—
59C2×C8×C59C2312yes—
60C4⋊C4×C59—21no—
61C4⋊C4×C59C2311yescharacteristic
62C22⋊C4×C59—21no—
63#11745671—42no—
64#11745671—42no—
65(C2×D59)⋊C4—21no—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—22no—
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—21no—
71(C2×D59)⋊C4—21no—
72(C2×D59)⋊C4—21no—
73C4⋊Dic59—21no—
74C4⋊Dic59—21no—
75C4⋊Dic59C2311yescharacteristic
76Dic59⋊C4—21no—
77C4×Dic59—21no—
78C4×Dic59D411yescharacteristic
79C4×Dic59D411yescharacteristic
80C4×Dic59C2311yescharacteristic
81C4×Dic59C2311yescharacteristic
82C2×C59⋊C8C2312yes—
83D472—42no—
84C23×C59C2×D411yescharacteristic
85C22×D59—21no—
86C22×D59—22no—
87C22×D59—42no—
88C22×D59—42no—
89C22×D59—42no—
90Q8×C59—22no—
91Q8×C59—22no—
92Q8×C59—22no—
93Q8×C59—42no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—22no—
98C2×C4×C59—42no—
99C2×C4×C59C2×D411yescharacteristic
100C2×C4×C59C2411yescharacteristic
101C22⋊D59—42no—
102C2×Dic59—22no—
103C2×Dic59—21no—
104C2×Dic59—41no—
105C2×Dic59C2×D411yescharacteristic
106C2×Dic59C2×D411yescharacteristic
107C2×Dic59C2×D411yescharacteristic
108C2×Dic59C2×D411yescharacteristic
109D236—22no—
110D236—22no—
111D236—22no—
112D236—42no—
113C4×D59—42no—
114C8×C59—22no—
115C59⋊C8—22no—
116C22×C59—21no—
117C22×C59—21no—
118C22×C59#13512yes—
119C22×C59C22×D411yescharacteristic
120C2×D59—42no—
121C2×D59—41no—
122C2×D59—41no—
123C2×D59—42no—
124C2×D59—41no—
125C2×D59—42no—
126C2×D59—84no—
127C4×C59—21no—
128C4×C59—42no—
129C4×C59—41no—
130C4×C59—41no—
131C4×C59H(5,𝔽2)12yes—
132Dic59—21no—
133Dic59—21no—
134Dic59—21no—
135Dic59—21no—
136Dic59—41no—
137#3211—591nomaximal, Sylow
138C2×C59—22no—
139C2×C59#53511yescharacteristic
140C2×C59C2×#13511yescharacteristic
141C2×C59#57511yescharacteristic
142D59—81no—
143D59—82no—
144#529—591no—
145C2×#124—591no—
146#495—592no—
147#488—592no—
148#464—592no—
149#463—592no—
150#451D5911yescharacteristic, complemented
151#450—591no—
152#448—591no—
153#447—591no—
154#431—591no—
155C59#321111yescharacteristic, complemented, Sylow
156C22×Q8C2×D5911yescharacteristic
157C22×D4—591no—
158C2×Q16C2×D5912yescomplemented
159C2×SD16—592no—
160C2×SD16—592no—
161C2×D8—592no—
162#126—1181no—
163#125—591no—
164#124—592no—
165#124—592no—
166#124—591no—
167#124—1181no—
168#124—1181no—
169#123—1181no—
170C22⋊Q8C2×D5911yescharacteristic
171#121—591no—
172C4×Q8—591no—
173C4×D4—591no—
174C4×D4—1181no—
175#117—1181no—
176C2×C22⋊C4—1181no—
177C2×C22⋊C4—1181no—
178C2×C42—591no—
179C4⋊C8—592no—
180#103—592no—
181#103C2×D5912yescomplemented
182#102—592no—
183#102—592no—
184C22⋊C8—591no—
185C22⋊C8C2×D5911yescharacteristic
186C4×C8—592no—
187C24—1181no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8C22×D5912yes—
191C2×Q8C22×D5911yescharacteristic
192C2×D4—592no—
193C2×D4—591no—
194C2×D4—1181no—
195C2×D4—1181no—
196C2×D4—2362no—
197C22×C4—21no—
198C22×C4—591no—
199C22×C4—591no—
200C22×C4—1181no—
201C22×C4C22×D5911yescharacteristic
202Q16—42no—
203SD16—2362no—
204SD16—2362no—
205D8—2362no—
206C2×C8—592no—
207C2×C8C22×D5912yes—
208C4⋊C4—21no—
209C4⋊C4—591no—
210C4⋊C4—1181no—
211C4⋊C4—1181no—
212C4⋊C4—1181no—
213C4⋊C4C22×D5911yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1182no—
216C22⋊C4—1182no—
217C22⋊C4—1182no—
218C22⋊C4—1182no—
219C22⋊C4—1181no—
220C22⋊C4—1181no—
221C22⋊C4—1181no—
222C22⋊C4—1181no—
223C42—591no—
224C42—591no—
225C42—591no—
226C42—591no—
227C42—1181no—
228C23—1181no—
229C23—1182no—
230C23—2362no—
231C23—2362no—
232C23—2362no—
233C23D4×D5911yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—22no—
237Q8—42no—
238D4—1182no—
239D4—1182no—
240D4—1182no—
241D4—2362no—
242D4—2362no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—591no—
249C2×C4—591no—
250C2×C4—591no—
251C2×C4—591no—
252C2×C4—1182no—
253C2×C4—1181no—
254C2×C4—2362no—
255C2×C4—2361no—
256C2×C4D4×D5911yescharacteristic
257C2×C4C23×D5911yescharacteristic
258C8—22no—
259C8—1182no—
260C22—21no—
261C22—21no—
262C22—4724no—
263C22—2361no—
264C22—2362no—
265C22—2361no—
266C22—2362no—
267C22—2362no—
268C22—2361no—
269C22#1427375012yes—
270C22C2×D4×D5911yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—1181no—
276C4—1181no—
277C4—1181no—
278C4—1181no—
279C4—2361no—
280C4#1427377212yes—
281C2—22no—
282C2—4722no—
283C2—4721no—
284C2#1683796711yescharacteristic
285C2C2×#1427375011yescharacteristic
286C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#451C2
socleC22×C59C22×D4
hypercenter#451D59
hypocenterC59#3211
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #16837938C2semidirect
2. C2×#14273716C2semidirect
3. #16837787C2semidirect
4. #16837758C2semidirect
5. #16837626C2semidirect
6. #16837620C2semidirect
7. #16837552C2semidirect
8. #16837541C2semidirect
10. #16837452C2semidirect
11. C59×#451C2semidirect
14. C2×#11745671C22semidirect
17. #14273718C22semidirect
22. #14273716C22semidirect
25. Q8×Dic59C22semidirect
26. #14273689C22semidirect
32. Dic59⋊C8C22semidirect
33. #14273651C22semidirect
37. C8×Dic59C22semidirect
39. C2×Q16×C59C22semidirect
41. #103×C59C22semidirect
150. #451D59semidirect
155. C59#3211semidirect
158. C2×Q16C2×D59semidirect
181. #103C2×D59semidirect