Finite Group Databasebeta

← #21435128

Subgroups — #21435128

Order 7552.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#16837950C211yesmaximal, characteristic, complemented
2#16837931C211yesmaximal, characteristic, complemented
3#16837797C211yesmaximal, characteristic, complemented
4#16837771C211yesmaximal, characteristic, complemented
5#16837766C211yesmaximal, characteristic, complemented
6#16837622C212yesmaximal, complemented
7#16837615C212yesmaximal, complemented
8#16837561C212yesmaximal, complemented
9#16837544C212yesmaximal, complemented
10#16837457C211yesmaximal, characteristic, complemented
11C59×#496C211yesmaximal, characteristic, complemented
12C2×#11745687C2212yes—
13C2×#11745672C2212yes—
14C2×#11745670C2212yes—
15#14273727C2211yescharacteristic
16#14273720—21no—
17#14273719C2211yescharacteristic
18#14273714C2212yes—
19#14273709—21no—
20#14273703C2211yescharacteristic
21#14273698—21no—
22#14273698—21no—
23#14273698C2211yescharacteristic
24Q8×Dic59C2211yescharacteristic
25D4×Dic59—21no—
26D4×Dic59—21no—
27D4×Dic59C2211yescharacteristic
28(C22⋊D59)⋊C4—21no—
29#14273683—21no—
30C42×D59C2211yescharacteristic
31Dic59⋊C8C2212yes—
32#14273652C2212yes—
33#14273652C2212yes—
34#14273648C2212yes—
35#14273648C2212yes—
36(C2×D59)⋊C8C2212yes—
37#14273638C2211yescharacteristic
38C2×SD16×C59C2212yes—
39C2×D8×C59C2212yes—
40#127×C59C2211yescharacteristic
41#124×C59C2211yescharacteristic
42#97×C59C2211yescharacteristic
43C2×Q8×C59C2311yescharacteristic
44C2×D4×C59—21no—
45C2×D4×C59—21no—
46C2×D4×C59C2311yescharacteristic
47C2×D4×C59C2312yes—
48C2×C22⋊D59—21no—
49C2×C22⋊D59—22no—
50C22×Dic59—21no—
51C22×Dic59—22no—
52#11745687—42no—
53#11745687—42no—
54C2×C4×D59D411yescharacteristic
55C2×C4×D59D411yescharacteristic
56C2×C4×D59C2311yescharacteristic
57C2×Dic118C2312yes—
58SD16×C59—42no—
59D8×C59—42no—
60C2×C8×C59C2312yes—
61C22⋊C4×C59—21no—
62C22⋊C4×C59—21no—
63C42×C59C2311yescharacteristic
64C22⋊Dic59—21no—
65C22⋊Dic59—22no—
66C22⋊Dic59—22no—
67C22⋊Dic59—21no—
68#11745672—42no—
69#11745670—42no—
70(C2×D59)⋊C4—21no—
71C4⋊Dic59—21no—
72C4⋊Dic59—21no—
73C4⋊Dic59—21no—
74C4⋊Dic59C2311yescharacteristic
75C4⋊Dic59C2311yescharacteristic
76Dic59⋊C4—21no—
77Dic59⋊C4—21no—
78Dic59⋊C4—21no—
79C4×Dic59—21no—
80C4×Dic59D411yescharacteristic
81C4×Dic59D411yescharacteristic
82C4×Dic59C2311yescharacteristic
83C2×C59⋊C8C2312yes—
84C23×C59—21no—
85C23×C59—22no—
86C22×D59C2×D411yescharacteristic
87Q8×C59—22no—
88D4×C59—22no—
89D4×C59—22no—
90D4×C59—22no—
91D4×C59—42no—
92D4×C59—42no—
93C2×C4×C59—21no—
94C2×C4×C59C2×D411yescharacteristic
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59C2411yescharacteristic
97C22⋊D59—44no—
98C22⋊D59—42no—
99C2×Dic59—21no—
100C2×Dic59—21no—
101C2×Dic59—22no—
102C2×Dic59—44no—
103C2×Dic59—42no—
104C2×Dic59C2×D411yescharacteristic
105C2×Dic59C2×D411yescharacteristic
106C2×Dic59C2×D411yescharacteristic
107C4×D59—22no—
108C4×D59—41no—
109C4×D59—41no—
110Dic118—22no—
111Dic118—22no—
112C8×C59—22no—
113C59⋊C8—22no—
114C22×C59—42no—
115C22×C59—41no—
116C22×C59—41no—
117C22×C59—41no—
118C22×C59—42no—
119C22×C59—42no—
120C22×C59C22×D411yescharacteristic
121C2×D59—21no—
122C2×D59—21no—
123C2×D59—21no—
124C4×C59—21no—
125C4×C59—21no—
126C4×C59—41no—
127C4×C59H(5,𝔽2)12yes—
128Dic59—21no—
129Dic59—21no—
130Dic59—21no—
131Dic59—42no—
132Dic59—41no—
133Dic59—41no—
134#3219—591nomaximal, Sylow
135C2×C59—82no—
136C2×C59—81no—
137C2×C59#53511yescharacteristic
138C2×C59#57511yescharacteristic
139C2×C59#57511yescharacteristic
140D59—41no—
141#532—591no—
142#529—591no—
143#497—591no—
144#496D5911yescharacteristic, complemented
145#491—591no—
146#490—591no—
147#464—592no—
148#463—592no—
149#452—592no—
150#449—592no—
151#432—591no—
152C59#321911yescharacteristic, complemented, Sylow
153C2×#41—592no—
154C2×Q16—592no—
155C2×SD16—592no—
156C2×SD16C2×D5912yes—
157C2×D8C2×D5912yes—
158C4⋊Q8—591no—
159#127C2×D5911yescharacteristic
160#126—1181no—
161#125—591no—
162#124—592no—
163#124C2×D5911yescharacteristic
164#123—1181no—
165C22⋊Q8—591no—
166#121—591no—
167#121—1181no—
168#121—1181no—
169C4×Q8—591no—
170C4×D4—591no—
171C4×D4—1181no—
172C4×D4—1181no—
173C4×D4—1181no—
174#117—1181no—
175C2×C42—591no—
176C4⋊C8—592no—
177#103—592no—
178#103—592no—
179#102—592no—
180#102—592no—
181C22⋊C8—592no—
182#97—591no—
183#97C2×D5911yescharacteristic
184#41—2362no—
185#41—2362no—
186C2×Q8—592no—
187C2×Q8C22×D5911yescharacteristic
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—1181no—
191C2×D4—1182no—
192C2×D4C22×D5912yes—
193C2×D4C22×D5911yescharacteristic
194C22×C4—591no—
195C22×C4—591no—
196C22×C4—591no—
197C22×C4—1181no—
198C22×C4—1182no—
199Q16—2362no—
200SD16—42no—
201SD16—2362no—
202D8—42no—
203C2×C8—592no—
204C2×C8C22×D5912yes—
205C4⋊C4—591no—
206C4⋊C4—591no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4—1181no—
210C4⋊C4—1181no—
211C4⋊C4—1181no—
212C4⋊C4—1181no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—1181no—
216C22⋊C4—1181no—
217C22⋊C4—1181no—
218C22⋊C4—1182no—
219C22⋊C4—1182no—
220C42—591no—
221C42—591no—
222C42—591no—
223C42—1181no—
224C42C22×D5911yescharacteristic
225C23—21no—
226C23—22no—
227C23—591no—
228Q8—22no—
229Q8—1182no—
230Q8—1182no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—42no—
235D4—42no—
236D4—2362no—
237D4—2364no—
238C2×C4—21no—
239C2×C4—591no—
240C2×C4—591no—
241C2×C4—591no—
242C2×C4—1182no—
243C2×C4—1181no—
244C2×C4—1181no—
245C2×C4—1182no—
246C2×C4—2361no—
247C2×C4—2361no—
248C2×C4—2362no—
249C2×C4—2364no—
250C2×C4D4×D5911yescharacteristic
251C2×C4D4×D5911yescharacteristic
252C2×C4C23×D5911yescharacteristic
253C8—22no—
254C8—1182no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—1181no—
262C22—1181no—
263C22—1181no—
264C22C2×D4×D5911yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—1181no—
269C4—1181no—
270C4—1181no—
271C4—2361no—
272C4—2361no—
273C4—2362no—
274C4#1427377212yes—
275C2—81no—
276C2—82no—
277C2—2361no—
278C2#1683796511yescharacteristic
279C2#1683819811yescharacteristic
280C2#1683819811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#496C2
socleC22×C59C22×D4
hypercenter#496D59
hypocenterC59#3219
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #16837950C2semidirect
2. #16837931C2semidirect
3. #16837797C2semidirect
4. #16837771C2semidirect
5. #16837766C2semidirect
6. #16837622C2semidirect
7. #16837615C2semidirect
8. #16837561C2semidirect
9. #16837544C2semidirect
10. #16837457C2semidirect
11. C59×#496C2semidirect
144. #496D59semidirect
152. C59#3219semidirect