Finite Group Databasebeta

← #21435119

Subgroups — #21435119

Order 7552.

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

normal subgroups only

This group has 386 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
1C4⋊Q8×D59C211yesmaximal, characteristic, complemented
2#16837931C211yesmaximal, characteristic
3#16837797C211yesmaximal, characteristic, complemented
4#16837771C211yesmaximal, characteristic, complemented
5#16837770C211yesmaximal, characteristic, complemented
6#16837610C212yesmaximal, complemented
7#16837605C212yesmaximal, complemented
8#16837556C212yesmaximal, complemented
9#16837539C212yesmaximal
10#16837457C211yesmaximal, characteristic, complemented
11C59×#497C211yesmaximal, characteristic, complemented
12C2×Q8×D59C2212yes—
13C2×#11745672C2212yes—
14C2×#11745670C2212yes—
15#14273727C2211yescharacteristic
16#14273726—21no—
17#14273726—21no—
18#14273725C2212yes—
19#14273720—21no—
20#14273719C2211yescharacteristic
21#14273709—21no—
22#14273703C2211yescharacteristic
23#14273698C2211yescharacteristic
24Q8×Dic59C2211yescharacteristic
25D4×Dic59C2211yescharacteristic
26(C22⋊D59)⋊C4—21no—
27#14273683—21no—
28C4⋊C4×D59—21no—
29C4⋊C4×D59—21no—
30C42×D59C2211yescharacteristic
31Dic59⋊C8C2212yes—
32#14273652C2212yes—
33#14273652C2212yes—
34#14273651C2212yes—
35#14273651C2212yes—
36(C2×D59)⋊C8C2212yes—
37#14273638C2211yescharacteristic
38C2×Q16×C59C2212yes—
39C2×SD16×C59C2212yes—
40C4⋊Q8×C59C2211yescharacteristic
41#124×C59C2211yescharacteristic
42#97×C59C2211yescharacteristic
43C2×Q8×C59C2311yescharacteristic
44C2×Q8×C59C2312yes—
45C2×D4×C59C2311yescharacteristic
46C2×C22⋊D59—21no—
47C22×Dic59—21no—
48Q8×D59—42no—
49Q8×D59—42no—
50C2×C4×D59—22no—
51C2×C4×D59D411yescharacteristic
52C2×C4×D59D411yescharacteristic
53C2×C4×D59C2311yescharacteristic
54C2×Dic118—22no—
55C2×Dic118C2312yes—
56Q16×C59—42no—
57SD16×C59—42no—
58C2×C8×C59C2312yes—
59C4⋊C4×C59—21no—
60C4⋊C4×C59—21no—
61C22⋊C4×C59—21no—
62C22⋊C4×C59—21no—
63C42×C59C2311yescharacteristic
64C22⋊Dic59—21no—
65C22⋊Dic59—21no—
66#11745672—42no—
67#11745670—42no—
68(C2×D59)⋊C4—21no—
69C4⋊Dic59—21no—
70C4⋊Dic59—21no—
71C4⋊Dic59—21no—
72C4⋊Dic59C2311yescharacteristic
73C4⋊Dic59C2311yescharacteristic
74Dic59⋊C4—21no—
75Dic59⋊C4—21no—
76Dic59⋊C4—22no—
77Dic59⋊C4—22no—
78Dic59⋊C4—21no—
79C4×Dic59—21no—
80C4×Dic59D411yescharacteristic
81C4×Dic59D411yescharacteristic
82C4×Dic59C2311yescharacteristic
83C2×C59⋊C8C2312yes—
84C23×C59—21no—
85C22×D59C2×D411yescharacteristic
86Q8×C59—22no—
87Q8×C59—22no—
88Q8×C59—22no—
89D4×C59—22no—
90C2×C4×C59—21no—
91C2×C4×C59—22no—
92C2×C4×C59C2×D411yescharacteristic
93C2×C4×C59C2×D411yescharacteristic
94C2×C4×C59C2411yescharacteristic
95C22⋊D59—42no—
96C2×Dic59—21no—
97C2×Dic59—21no—
98C2×Dic59—22no—
99C2×Dic59—42no—
100C2×Dic59C2×D411yescharacteristic
101C2×Dic59C2×D411yescharacteristic
102C2×Dic59C2×D411yescharacteristic
103C4×D59—22no—
104C4×D59—22no—
105C4×D59—22no—
106C4×D59—44no—
107Dic118—22no—
108Dic118—22no—
109Dic118—44no—
110C8×C59—22no—
111C59⋊C8—22no—
112C22×C59—41no—
113C22×C59—41no—
114C22×C59—41no—
115C22×C59C22×D411yescharacteristic
116C2×D59—21no—
117C2×D59—21no—
118C2×D59#13712yes—
119C4×C59—21no—
120C4×C59—21no—
121C4×C59—42no—
122C4×C59—41no—
123C4×C59H(5,𝔽2)12yes—
124Dic59—21no—
125Dic59—21no—
126Dic59—42no—
127Dic59—41no—
128Dic59—41no—
129Dic59#13712yes—
130#3218—591nomaximal, Sylow
131C2×C59—81no—
132C2×C59#53511yescharacteristic
133C2×C59C2×#13711yescharacteristic
134C2×C59C2×#13711yescharacteristic
135D59—22no—
136C2×C4⋊Q8—591no—
137#529—591no—
138#497—591no—
139#497D5911yescharacteristic, complemented
140#491—591no—
141#491—591no—
142#462—592no—
143#461—592no—
144#451—592no—
145#448—592no—
146#432—591no—
147C59#321811yescharacteristic, complemented, Sylow
148C22×Q8—592no—
149C2×Q16—592no—
150C2×Q16C2×D5912yes—
151C2×SD16—592no—
152C2×SD16C2×D5912yes—
153C4⋊Q8—592no—
154C4⋊Q8—591no—
155C4⋊Q8—1181no—
156C4⋊Q8—1181no—
157C4⋊Q8C2×D5911yescharacteristic
158#126—1181no—
159#125—591no—
160#124C2×D5911yescharacteristic
161#123—1181no—
162C22⋊Q8—591no—
163#121—591no—
164C4×Q8—591no—
165C4×D4—591no—
166C4×D4—1181no—
167#117—1181no—
168C2×C4⋊C4—1181no—
169C2×C4⋊C4—1181no—
170C2×C42—591no—
171C4⋊C8—592no—
172#103—592no—
173#103—592no—
174#103—592no—
175#103—592no—
176C22⋊C8—592no—
177#97—591no—
178#97C2×D5911yescharacteristic
179C2×Q8—592no—
180C2×Q8—1182no—
181C2×Q8—2362no—
182C2×Q8—2362no—
183C2×Q8C22×D5912yes—
184C2×Q8C22×D5911yescharacteristic
185C2×D4—1181no—
186C2×D4C22×D5911yescharacteristic
187C22×C4—591no—
188C22×C4—591no—
189C22×C4—591no—
190C22×C4—1182no—
191C22×C4—1181no—
192Q16—42no—
193Q16—2362no—
194SD16—42no—
195SD16—2362no—
196C2×C8—592no—
197C2×C8C22×D5912yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—591no—
201C4⋊C4—591no—
202C4⋊C4—1182no—
203C4⋊C4—1182no—
204C4⋊C4—1181no—
205C4⋊C4—1181no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4—1181no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—1181no—
213C22⋊C4—1181no—
214C22⋊C4—1181no—
215C42—591no—
216C42—591no—
217C42—591no—
218C42—1181no—
219C42C22×D5911yescharacteristic
220C23—21no—
221C23—591no—
222Q8—22no—
223Q8—22no—
224Q8—22no—
225Q8—1182no—
226Q8—1182no—
227Q8—2364no—
228D4—22no—
229D4—2362no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—591no—
233C2×C4—591no—
234C2×C4—591no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—1182no—
238C2×C4—1181no—
239C2×C4—1181no—
240C2×C4—1182no—
241C2×C4—2364no—
242C2×C4—2362no—
243C2×C4D4×D5911yescharacteristic
244C2×C4D4×D5911yescharacteristic
245C2×C4C23×D5911yescharacteristic
246C8—22no—
247C8—1182no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—592no—
252C22—1181no—
253C22—1181no—
254C22C2×D4×D5911yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—41no—
259C4—592no—
260C4—1181no—
261C4—1181no—
262C4—2361no—
263C4—2361no—
264C4—2362no—
265C4#1427377212yes—
266C2—81no—
267C2—1182no—
268C2#1683796511yescharacteristic
269C2#137×D5911yescharacteristic
270C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#497C2
socleC22×C59C22×D4
hypercenter#497D59
hypocenterC59#3218
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. C4⋊Q8×D59C2semidirect
3. #16837797C2semidirect
4. #16837771C2semidirect
5. #16837770C2semidirect
6. #16837610C2semidirect
7. #16837605C2semidirect
8. #16837556C2semidirect
10. #16837457C2semidirect
11. C59×#497C2semidirect
139. #497D59semidirect
147. C59#3218semidirect