Finite Group Databasebeta

← #21435094

Subgroups — #21435094

Order 7552.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#16837957C211yesmaximal, characteristic, complemented
2#124×D59C211yesmaximal, characteristic
3#16837794C211yesmaximal, characteristic, complemented
4#16837762C211yesmaximal, characteristic, complemented
5#16837761C211yesmaximal, characteristic, complemented
6#16837626C212yesmaximal, complemented
7#16837622C212yesmaximal, complemented
8#16837556C212yesmaximal, complemented
9#16837551C212yesmaximal
10#16837457C211yesmaximal, characteristic, complemented
11C59×#497C211yesmaximal, characteristic, complemented
12C2×Q8×D59C2211yescharacteristic
13C2×D4×D59C2211yescharacteristic
14C2×#11745672C2212yescomplemented
15C2×#11745671C2212yes—
16#14273727C2211yescharacteristic, complemented
17#14273718—21no—
18#14273716C2211yescharacteristic
19#14273715C2211yescharacteristic
20#14273714C2211yescharacteristic
21#14273713—21no—
22#14273713—21no—
23#14273707—21no—
24#14273703C2212yes—
25Q8×Dic59C2212yescomplemented
26#14273692—21no—
27#14273685—21no—
28C22⋊C4×D59—21no—
29C22⋊C4×D59—21no—
30C42×D59C2211yescharacteristic
31Dic59⋊C8C2212yescomplemented
32#14273652C2212yescomplemented
33#14273651C2212yescomplemented
34#14273649C2212yes—
35#14273648C2212yes—
36(C2×D59)⋊C8C2212yes—
37#14273638C2211yescharacteristic, complemented
38C2×Q16×C59C2212yescomplemented
39C2×SD16×C59C2212yes—
40C4⋊Q8×C59C2211yescharacteristic, complemented
41#124×C59C2211yescharacteristic
42#97×C59C2211yescharacteristic, complemented
43C23×D59—21no—
44C2×Q8×C59C2311yescharacteristic
45C2×Q8×C59C2312yes—
46C2×D4×C59C2311yescharacteristic
47C2×C22⋊D59—21no—
48Q8×D59—42no—
49D4×D59—42no—
50C2×D236C2311yescharacteristic
51C2×C4×D59—21no—
52C2×C4×D59D411yescharacteristic
53C2×C4×D59D411yescharacteristic
54C2×C4×D59C2311yescharacteristic
55C2×Dic118—21no—
56C2×Dic118—21no—
57C2×Dic118C2311yescharacteristic
58Q16×C59—42no—
59SD16×C59—42no—
60C2×C8×C59C2312yes—
61C4⋊C4×C59—21no—
62C4⋊C4×C59—21no—
63C22⋊C4×C59—21no—
64C22⋊C4×C59—21no—
65C42×C59C2311yescharacteristic
66C22⋊Dic59—21no—
67C22⋊Dic59—21no—
68#11745672—42no—
69#11745671—42no—
70(C2×D59)⋊C4—22no—
71(C2×D59)⋊C4—21no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—21no—
75C4⋊Dic59—21no—
76C4⋊Dic59—22no—
77C4⋊Dic59C2312yes—
78Dic59⋊C4—22no—
79C4×Dic59—22no—
80C4×Dic59D411yescharacteristic
81C4×Dic59D411yescharacteristic
82C4×Dic59C2311yescharacteristic
83C2×C59⋊C8C2312yes—
84C23×C59—21no—
85C22×D59—21no—
86C22×D59—42no—
87C22×D59—42no—
88C22×D59—42no—
89C22×D59C2×D411yescharacteristic
90Q8×C59—22no—
91Q8×C59—22no—
92Q8×C59—22no—
93D4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59C2×D411yescharacteristic
97C2×C4×C59C2×D411yescharacteristic
98C2×C4×C59C2411yescharacteristic
99C22⋊D59—42no—
100C2×Dic59—22no—
101C2×Dic59—21no—
102C2×Dic59C2×D411yescharacteristic
103C2×Dic59C2×D411yescharacteristic
104C2×Dic59C2×D411yescharacteristic
105D236—22no—
106C4×D59—22no—
107C4×D59—22no—
108C4×D59—41no—
109C4×D59—42no—
110Dic118—22no—
111Dic118—42no—
112Dic118—42no—
113C8×C59—22no—
114C59⋊C8—22no—
115C22×C59—41no—
116C22×C59—41no—
117C22×C59—41no—
118C22×C59C22×D411yescharacteristic
119C2×D59—21no—
120C2×D59—21no—
121C2×D59—41no—
122C2×D59—41no—
123C2×D59—41no—
124C2×D59—84no—
125C2×D59#13712yes—
126C4×C59—21no—
127C4×C59—21no—
128C4×C59—41no—
129C4×C59—42no—
130C4×C59H(5,𝔽2)12yes—
131Dic59—21no—
132Dic59—21no—
133Dic59—21no—
134Dic59—41no—
135Dic59—42no—
136#3215—591nomaximal, Sylow
137C2×C59—81no—
138C2×C59#53511yescharacteristic
139C2×C59C2×#13711yescharacteristic
140C2×C59#57511yescharacteristic
141D59—22no—
142D59—81no—
143#533—591no—
144C2×#124—591no—
145#497—591no—
146#497D5911yescharacteristic, complemented
147#489—591no—
148#489—591no—
149#464—592no—
150#464—592no—
151#451—592no—
152#450—592no—
153#432—591no—
154C59#321511yescharacteristic, complemented, Sylow
155C22×Q8—591no—
156C22×D4—591no—
157C2×Q16—592no—
158C2×Q16C2×D5912yescomplemented
159C2×SD16—592no—
160C2×SD16C2×D5912yes—
161C4⋊Q8—591no—
162C4⋊Q8C2×D5911yescharacteristic, complemented
163#125—1181no—
164#124—591no—
165#124—591no—
166#124—591no—
167#124—1181no—
168#124—1181no—
169#124C2×D5911yescharacteristic
170C22⋊Q8—592no—
171C22⋊Q8—1181no—
172C4×Q8—592no—
173C4×Q8—1181no—
174#117—1181no—
175C2×C22⋊C4—1181no—
176C2×C22⋊C4—1181no—
177C2×C42—591no—
178C4⋊C8—592no—
179#103—592no—
180#103—592no—
181#102—592no—
182#102—592no—
183C22⋊C8—592no—
184#97—591no—
185#97C2×D5911yescharacteristic, complemented
186C24—1181no—
187C2×Q8—591no—
188C2×Q8—1181no—
189C2×Q8—1181no—
190C2×Q8—2362no—
191C2×Q8C22×D5911yescharacteristic
192C2×Q8C22×D5912yes—
193C2×D4—591no—
194C2×D4—1181no—
195C2×D4—2362no—
196C2×D4C22×D5911yescharacteristic
197C22×C4—591no—
198C22×C4—591no—
199C22×C4—591no—
200C22×C4—1181no—
201Q16—42no—
202Q16—2362no—
203SD16—42no—
204SD16—2362no—
205C2×C8—592no—
206C2×C8C22×D5912yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—592no—
210C4⋊C4—1182no—
211C4⋊C4—1182no—
212C4⋊C4—1181no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—1181no—
216C22⋊C4—1181no—
217C22⋊C4—1182no—
218C22⋊C4—1181no—
219C22⋊C4—1181no—
220C22⋊C4—1181no—
221C22⋊C4—1181no—
222C42—591no—
223C42—591no—
224C42—591no—
225C42—1182no—
226C42C22×D5911yescharacteristic
227C23—21no—
228C23—591no—
229C23—1181no—
230C23—2362no—
231C23—2362no—
232C23—2362no—
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—1182no—
237Q8—2362no—
238Q8—2362no—
239D4—22no—
240D4—1182no—
241D4—2362no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—591no—
245C2×C4—591no—
246C2×C4—591no—
247C2×C4—1182no—
248C2×C4—1182no—
249C2×C4—1182no—
250C2×C4—1181no—
251C2×C4—2361no—
252C2×C4—2362no—
253C2×C4D4×D5911yescharacteristic
254C2×C4D4×D5911yescharacteristic
255C2×C4C23×D5911yescharacteristic
256C8—22no—
257C8—1182no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—592no—
262C22—1181no—
263C22—1181no—
264C22—4724no—
265C22—2361no—
266C22—2361no—
267C22—2361no—
268C22C2×D4×D5911yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—1181no—
274C4—1181no—
275C4—1181no—
276C4—2362no—
277C4—2361no—
278C4#1427377212yes—
279C2—81no—
280C2—1182no—
281C2—4721no—
282C2#1683796511yescharacteristic
283C2#137×D5911yescharacteristic
284C2#1683820111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#497C2
socleC22×C59C22×D4
hypercenter#497D59
hypocenterC59#3215
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #16837957C2semidirect
3. #16837794C2semidirect
4. #16837762C2semidirect
5. #16837761C2semidirect
6. #16837626C2semidirect
7. #16837622C2semidirect
8. #16837556C2semidirect
10. #16837457C2semidirect
11. C59×#497C2semidirect
14. C2×#11745672C22semidirect
16. #14273727C22semidirect
25. Q8×Dic59C22semidirect
31. Dic59⋊C8C22semidirect
32. #14273652C22semidirect
33. #14273651C22semidirect
37. #14273638C22semidirect
38. C2×Q16×C59C22semidirect
40. C4⋊Q8×C59C22semidirect
42. #97×C59C22semidirect
146. #497D59semidirect
154. C59#3215semidirect
158. C2×Q16C2×D59semidirect
162. C4⋊Q8C2×D59semidirect
185. #97C2×D59semidirect