Finite Group Databasebeta

← #21435122

Subgroups — #21435122

Order 7552.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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#16837934C211yesmaximal, characteristic
3#16837793C212yesmaximal, complemented
4#16837769C212yesmaximal, complemented
5#16837613C211yesmaximal, characteristic, complemented
6#16837609C211yesmaximal, characteristic, complemented
7#16837607C211yesmaximal, characteristic, complemented
8#16837559C212yesmaximal, complemented
9#16837539C212yesmaximal
10#16837458C211yesmaximal, characteristic, complemented
11C59×#461C211yesmaximal, characteristic, complemented
12C2×Q8×D59C2212yes—
13C2×Dic236C2212yes—
14C2×#11745672C2212yes—
15C2×#11745670C2212yes—
16#14273727C2211yescharacteristic
17#14273726—21no—
18#14273726—21no—
19#14273725C2212yes—
20#14273720—21no—
21#14273718C2211yescharacteristic
22#14273714C2211yescharacteristic
23#14273709—21no—
24#14273707C2211yescharacteristic
25#14273698C2211yescharacteristic
26C4×Dic118C2211yescharacteristic
27C4×C22⋊D59—21no—
28#14273686—21no—
29C4⋊C4×D59—21no—
30C4⋊C4×D59—21no—
31C42×D59C2211yescharacteristic
32C4⋊(C59⋊C8)C2211yescharacteristic
33#14273652C2212yes—
34#14273651C2212yes—
35#14273650C2212yes—
36(C2×D59)⋊C8C2212yes—
37#14273637C2212yes—
38C2×SD16×C59C2212yes—
39C4⋊Q8×C59C2211yescharacteristic
40C4×D4×C59C2211yescharacteristic
41C4⋊C8×C59C2211yescharacteristic
42#103×C59C2212yes—
43C2×Q8×C59C2312yes—
44C2×D4×C59C2311yescharacteristic
45C22×C4×C59—21no—
46C2×C22⋊D59—21no—
47Q8×D59—42no—
48Q8×D59—42no—
49C2×C4×D59—22no—
50C2×C4×D59D411yescharacteristic
51C2×C4×D59D411yescharacteristic
52C2×C4×D59C2311yescharacteristic
53C2×Dic118—22no—
54C2×Dic118C2312yes—
55C2×Dic118C2311yescharacteristic
56SD16×C59—42no—
57C2×C8×C59C2312yes—
58C4⋊C4×C59—21no—
59C4⋊C4×C59—21no—
60C4⋊C4×C59C2311yescharacteristic
61C22⋊C4×C59—21no—
62C42×C59C2311yescharacteristic
63C22⋊Dic59—21no—
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⋊Dic59—21no—
73C4⋊Dic59C2311yescharacteristic
74Dic59⋊C4—21no—
75Dic59⋊C4—22no—
76Dic59⋊C4—22no—
77Dic59⋊C4—21no—
78C4×Dic59—21no—
79C4×Dic59D411yescharacteristic
80C4×Dic59D411yescharacteristic
81C4×Dic59C2311yescharacteristic
82C2×C59⋊C8C2312yes—
83Dic236—42no—
84C23×C59—21no—
85C22×D59C2×D411yescharacteristic
86Q8×C59—22no—
87Q8×C59—22no—
88D4×C59—21no—
89D4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—21no—
92C2×C4×C59—42no—
93C2×C4×C59C2×D411yescharacteristic
94C2×C4×C59C2×D411yescharacteristic
95C2×C4×C59C2411yescharacteristic
96C22⋊D59—42no—
97C2×Dic59—21no—
98C2×Dic59—21no—
99C2×Dic59—22no—
100C2×Dic59C2×D411yescharacteristic
101C2×Dic59C2×D411yescharacteristic
102C2×Dic59C2×D411yescharacteristic
103C4×D59—22no—
104C4×D59—22no—
105C4×D59—21no—
106C4×D59—21no—
107C4×D59—44no—
108Dic118—21no—
109Dic118—22no—
110Dic118—22no—
111Dic118—21no—
112Dic118—44no—
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#13712yes—
122C4×C59—21no—
123C4×C59—42no—
124C4×C59—41no—
125C4×C59#13712yes—
126C4×C59H(5,𝔽2)11yescharacteristic
127C4×C59H(5,𝔽2)11yescharacteristic
128Dic59—21no—
129Dic59—21no—
130Dic59—21no—
131Dic59—42no—
132Dic59—41no—
133Dic59—41no—
134#3218—591nomaximal, Sylow
135C2×C59—81no—
136C2×C59#53511yescharacteristic
137C2×C59C2×#13711yescharacteristic
138C2×C59C2×#13711yescharacteristic
139D59—22no—
140C2×C4⋊Q8—591no—
141#529—591no—
142#497—592no—
143#491—592no—
144#462—591no—
145#462—591no—
146#461—591no—
147#461D5911yescharacteristic, complemented
148#451—592no—
149#448—592no—
150#432—591no—
151C59#321811yescharacteristic, complemented, Sylow
152C22×Q8—592no—
153C2×Q16—592no—
154C2×Q16—592no—
155C2×SD16—592no—
156C2×SD16C2×D5912yes—
157C4⋊Q8—592no—
158C4⋊Q8—591no—
159C4⋊Q8—1181no—
160C4⋊Q8—1181no—
161C4⋊Q8C2×D5911yescharacteristic
162#126—1181no—
163#125—591no—
164#124—591no—
165#123—1181no—
166C22⋊Q8—591no—
167#121—591no—
168C4×Q8—591no—
169C4×D4—1181no—
170C4×D4C2×D5911yescharacteristic
171#117—1181no—
172C2×C4⋊C4—1181no—
173C2×C4⋊C4—1181no—
174C2×C42—591no—
175C4⋊C8—591no—
176C4⋊C8C2×D5911yescharacteristic
177#103—592no—
178#103—592no—
179#103—592no—
180#103C2×D5912yes—
181C22⋊C8—592no—
182#97—592no—
183C2×Q8—592no—
184C2×Q8—591no—
185C2×Q8—1182no—
186C2×Q8—2362no—
187C2×Q8—2362no—
188C2×Q8C22×D5912yes—
189C2×D4—1181no—
190C2×D4C22×D5911yescharacteristic
191C22×C4—21no—
192C22×C4—591no—
193C22×C4—591no—
194C22×C4—591no—
195C22×C4—1182no—
196Q16—2362no—
197Q16—2362no—
198SD16—42no—
199SD16—2362no—
200C2×C8—592no—
201C2×C8C22×D5912yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—591no—
205C4⋊C4—1182no—
206C4⋊C4—1182no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4—1181no—
210C4⋊C4—1181no—
211C4⋊C4—1181no—
212C4⋊C4—1181no—
213C4⋊C4C22×D5911yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1181no—
216C22⋊C4—1181no—
217C22⋊C4—1181no—
218C22⋊C4—1181no—
219C42—591no—
220C42—591no—
221C42—591no—
222C42—1181no—
223C42C22×D5911yescharacteristic
224C23—21no—
225C23—591no—
226Q8—22no—
227Q8—22no—
228Q8—1181no—
229Q8—1182no—
230Q8—1182no—
231Q8—1181no—
232Q8—2364no—
233D4—21no—
234D4—21no—
235D4—2362no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—591no—
240C2×C4—591no—
241C2×C4—591no—
242C2×C4—1182no—
243C2×C4—1182no—
244C2×C4—1181no—
245C2×C4—1181no—
246C2×C4—1181no—
247C2×C4—1181no—
248C2×C4—1182no—
249C2×C4—2364no—
250C2×C4D4×D5911yescharacteristic
251C2×C4D4×D5911yescharacteristic
252C2×C4C23×D5911yescharacteristic
253C8—22no—
254C8—1182no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—592no—
259C22—1181no—
260C22—1181no—
261C22C2×D4×D5911yescharacteristic
262C4—21no—
263C4—41no—
264C4—42no—
265C4—1181no—
266C4—1181no—
267C4—1181no—
268C4—2361no—
269C4—2362no—
270C4—2361no—
271C4#1427375612yes—
272C4#1427377211yescharacteristic
273C4#1427377311yescharacteristic
274C2—81no—
275C2—1182no—
276C2#1683796811yescharacteristic
277C2#137×D5911yescharacteristic
278C2C2×#1427375611yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#461C2
socleC22×C59C22×D4
hypercenter#461D59
hypocenterC59#3218
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D59C2semidirect
3. #16837793C2semidirect
4. #16837769C2semidirect
5. #16837613C2semidirect
6. #16837609C2semidirect
7. #16837607C2semidirect
8. #16837559C2semidirect
10. #16837458C2semidirect
11. C59×#461C2semidirect
147. #461D59semidirect
151. C59#3218semidirect