Finite Group Databasebeta

← #21435098

Subgroups — #21435098

Order 7552.

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

normal subgroups only

This group has 400 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#16837954C211yesmaximal, characteristic, complemented
2C2×#14273716C211yesmaximal, characteristic
3#16837795C212yesmaximal, complemented
4#16837763C212yesmaximal, complemented
5#16837627C212yesmaximal, complemented
6#16837626C212yesmaximal, complemented
7#16837558C211yesmaximal, characteristic, complemented
8#16837555C211yesmaximal, characteristic
9#16837552C211yesmaximal, characteristic
10#16837456C211yesmaximal, characteristic, complemented
11C59×#451C211yesmaximal, characteristic, complemented
12C22×D236C2211yescharacteristic
13C2×#11745672C2212yescomplemented
14C2×#11745660C2212yes—
15C2×#11745671C2212yes—
16#14273726C2211yescharacteristic, complemented
17#14273725C2211yescharacteristic, complemented
18#14273717—21no—
19#14273716—21no—
20#14273716—21no—
21#14273716C2212yes—
22#14273716C2212yes—
23#14273706C2211yescharacteristic
24#14273704—21no—
25Q8×Dic59C2211yescharacteristic, complemented
26#14273692—21no—
27#14273692C2211yescharacteristic, complemented
28#14273683—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
37#14273637C2212yescomplemented
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
50C22×Dic59D411yescharacteristic
51C22×Dic59D411yescharacteristic
52C2×D236—21no—
53C2×D236—42no—
54C2×D236C2312yes—
55C2×Dic118—21no—
56C2×Dic118C2311yescharacteristic
57Q16×C59—42no—
58C2×C8×C59C2312yes—
59C4⋊C4×C59—21no—
60C4⋊C4×C59C2311yescharacteristic
61C22⋊C4×C59—21no—
62C22⋊Dic59—21no—
63#11745672—42no—
64#11745671—42no—
65(C2×D59)⋊C4—22no—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—22no—
69C4⋊Dic59—21no—
70C4⋊Dic59C2311yescharacteristic
71Dic59⋊C4—21no—
72Dic59⋊C4—21no—
73Dic59⋊C4—21no—
74Dic59⋊C4—21no—
75Dic59⋊C4—21no—
76C4×Dic59—21no—
77C4×Dic59—21no—
78C4×Dic59D411yescharacteristic
79C4×Dic59D411yescharacteristic
80C4×Dic59C2311yescharacteristic
81C4×Dic59C2311yescharacteristic
82C2×C59⋊C8C2312yes—
83#11745660—42no—
84C23×C59C2×D411yescharacteristic
85C22×D59—22no—
86C22×D59—42no—
87C22×D59—42no—
88C22×D59—42no—
89Q8×C59—22no—
90Q8×C59—22no—
91Q8×C59—22no—
92Q8×C59—42no—
93C2×C4×C59—22no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—42no—
98C2×C4×C59C2×D411yescharacteristic
99C2×C4×C59C2411yescharacteristic
100C2×Dic59—22no—
101C2×Dic59—21no—
102C2×Dic59—21no—
103C2×Dic59—41no—
104C2×Dic59C2×D411yescharacteristic
105C2×Dic59C2×D411yescharacteristic
106C2×Dic59C2×D411yescharacteristic
107C2×Dic59C2×D411yescharacteristic
108D236—22no—
109D236—22no—
110D236—42no—
111Dic118—22no—
112Dic118—42no—
113C8×C59—22no—
114C59⋊C8—22no—
115C22×C59—21no—
116C22×C59—21no—
117C22×C59#13712yes—
118C22×C59C22×D411yescharacteristic
119C2×D59—42no—
120C2×D59—42no—
121C2×D59—42no—
122C2×D59—84no—
123C4×C59—21no—
124C4×C59—42no—
125C4×C59—41no—
126C4×C59—41no—
127C4×C59H(5,𝔽2)12yes—
128Dic59—21no—
129Dic59—21no—
130Dic59—21no—
131Dic59—21no—
132Dic59—41no—
133Dic59—41no—
134#3215—591nomaximal, Sylow
135C2×C59—22no—
136C2×C59#53511yescharacteristic
137C2×C59C2×#13711yescharacteristic
138C2×C59#57511yescharacteristic
139D59—82no—
140#533—591no—
141C2×#124—591no—
142#497—592no—
143#489—592no—
144#464—592no—
145#464—592no—
146#451—591no—
147#451D5911yescharacteristic, complemented
148#450—591no—
149#450—591no—
150#432—591no—
151C59#321511yescharacteristic, complemented, Sylow
152C22×Q8C2×D5911yescharacteristic
153C22×D4—591no—
154C2×Q16—592no—
155C2×Q16C2×D5912yescomplemented
156C2×SD16—592no—
157C2×SD16—592no—
158C4⋊Q8—591no—
159C4⋊Q8—591no—
160#125—1181no—
161#124—592no—
162#124—592no—
163#124—1181no—
164#124—1181no—
165C22⋊Q8—591no—
166C22⋊Q8—1181no—
167C22⋊Q8C2×D5911yescharacteristic
168C4×Q8—591no—
169C4×Q8—591no—
170C4×Q8—1181no—
171#117—1181no—
172C2×C22⋊C4—1181no—
173C2×C22⋊C4—1181no—
174C2×C42—591no—
175C4⋊C8—592no—
176#103—592no—
177#103C2×D5912yescomplemented
178#102—592no—
179#102—592no—
180C22⋊C8—591no—
181C22⋊C8C2×D5911yescharacteristic
182#97—592no—
183C24—1181no—
184C2×Q8—21no—
185C2×Q8—42no—
186C2×Q8—591no—
187C2×Q8—1181no—
188C2×Q8C22×D5912yes—
189C2×Q8C22×D5911yescharacteristic
190C2×D4—592no—
191C2×D4—1181no—
192C2×D4—2362no—
193C22×C4—21no—
194C22×C4—591no—
195C22×C4—591no—
196C22×C4C22×D5911yescharacteristic
197Q16—42no—
198Q16—2362no—
199SD16—2362no—
200SD16—2362no—
201C2×C8—592no—
202C2×C8C22×D5912yes—
203C4⋊C4—21no—
204C4⋊C4—591no—
205C4⋊C4—1181no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4—1181no—
210C4⋊C4—1181no—
211C4⋊C4C22×D5911yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—1182no—
214C22⋊C4—1182no—
215C22⋊C4—1182no—
216C22⋊C4—1182no—
217C22⋊C4—1181no—
218C42—591no—
219C42—591no—
220C42—591no—
221C42—591no—
222C42—1181no—
223C42—1181no—
224C23—1182no—
225C23—2362no—
226C23—2362no—
227C23—2362no—
228C23D4×D5911yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—22no—
232Q8—42no—
233Q8—1182no—
234Q8—2362no—
235D4—1182no—
236D4—1182no—
237D4—2362no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—591no—
244C2×C4—591no—
245C2×C4—591no—
246C2×C4—591no—
247C2×C4—1182no—
248C2×C4—1181no—
249C2×C4—1181no—
250C2×C4—2361no—
251C2×C4D4×D5911yescharacteristic
252C2×C4C23×D5911yescharacteristic
253C8—22no—
254C8—1182no—
255C22—21no—
256C22—21no—
257C22—4724no—
258C22—2362no—
259C22—2362no—
260C22—2362no—
261C22#1427375912yes—
262C22C2×D4×D5911yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—1181no—
268C4—1181no—
269C4—1181no—
270C4—1181no—
271C4—2361no—
272C4—2361no—
273C4#1427377212yes—
274C2—22no—
275C2—4722no—
276C2#1683796711yescharacteristic
277C2C2×#1427375911yescharacteristic
278C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#451C2
socleC22×C59C22×D4
hypercenter#451D59
hypocenterC59#3215
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. #16837954C2semidirect
3. #16837795C2semidirect
4. #16837763C2semidirect
5. #16837627C2semidirect
6. #16837626C2semidirect
7. #16837558C2semidirect
10. #16837456C2semidirect
11. C59×#451C2semidirect
13. C2×#11745672C22semidirect
16. #14273726C22semidirect
17. #14273725C22semidirect
25. Q8×Dic59C22semidirect
27. #14273692C22semidirect
32. Dic59⋊C8C22semidirect
33. #14273651C22semidirect
37. #14273637C22semidirect
39. C2×Q16×C59C22semidirect
41. #103×C59C22semidirect
147. #451D59semidirect
151. C59#3215semidirect
155. C2×Q16C2×D59semidirect
177. #103C2×D59semidirect