Finite Group Databasebeta

← #21435313

Subgroups — #21435313

Order 7552.

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

normal subgroups only

This group has 362 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#16837811C211yesmaximal, characteristic, complemented
4#16837808C211yesmaximal, characteristic, complemented
5#16837761C211yesmaximal, characteristic, complemented
6#16837731C212yesmaximal, complemented
7#16837714C212yesmaximal
8#16837695C212yesmaximal, complemented
9#16837690C212yesmaximal, complemented
10#16837457C211yesmaximal, characteristic, complemented
11C59×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D59—21no—
13#14273727C2211yescharacteristic
14#14273726—21no—
15#14273726C2212yes—
16#14273725—21no—
17#14273720—21no—
18#14273719C2211yescharacteristic
19#14273709—21no—
20#14273703C2211yescharacteristic
21#14273698C2211yescharacteristic
22Q8×Dic59C2211yescharacteristic
23D4×Dic59C2211yescharacteristic
24(C22⋊D59)⋊C4—21no—
25#14273683—21no—
26C4⋊C4×D59—21no—
27C4⋊C4×D59C2212yes—
28C42×D59C2211yescharacteristic
29#14273661C2212yes—
30#14273660C2212yes—
31#14273659C2212yes—
32#14273658C2212yes—
33Dic59⋊C8C2212yes—
34#14273651C2212yes—
35#14273648C2212yes—
36(C2×D59)⋊C8C2212yes—
37#14273638C2211yescharacteristic
38C4⋊Q8×C59C2211yescharacteristic
39#124×C59C2211yescharacteristic
40#103×C59C2212yes—
41#102×C59C2212yes—
42#97×C59C2211yescharacteristic
43C2×Q8×C59—21no—
44C2×Q8×C59C2311yescharacteristic
45C2×D4×C59C2311yescharacteristic
46C2×C22⋊D59—21no—
47C22×Dic59—21no—
48Q8×D59—44no—
49C2×C4×D59—22no—
50C2×C4×D59D411yescharacteristic
51C2×C4×D59D411yescharacteristic
52C2×C4×D59C2311yescharacteristic
53C2×Dic118—21no—
54C2×Dic118—22no—
55C2×C8×C59C2312yes—
56C4⋊C4×C59—21no—
57C4⋊C4×C59C2312yes—
58C22⋊C4×C59—21no—
59C22⋊C4×C59—21no—
60C42×C59C2311yescharacteristic
61C22⋊Dic59—21no—
62C22⋊Dic59—21no—
63(C2×D59)⋊C4—21no—
64C4⋊Dic59—21no—
65C4⋊Dic59—21no—
66C4⋊Dic59C2312yes—
67C4⋊Dic59C2311yescharacteristic
68C4⋊Dic59C2311yescharacteristic
69Dic59⋊C4—21no—
70Dic59⋊C4—21no—
71Dic59⋊C4—22no—
72Dic59⋊C4—22no—
73Dic59⋊C4—21no—
74C4×Dic59—21no—
75C4×Dic59D411yescharacteristic
76C4×Dic59D411yescharacteristic
77C4×Dic59C2311yescharacteristic
78C2×C59⋊C8C2312yes—
79C23×C59—21no—
80C22×D59C2×D411yescharacteristic
81Q8×C59—22no—
82Q8×C59—42no—
83D4×C59—22no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59C2×D411yescharacteristic
87C2×C4×C59C2×D411yescharacteristic
88C2×C4×C59C2411yescharacteristic
89C22⋊D59—42no—
90C2×Dic59—21no—
91C2×Dic59—21no—
92C2×Dic59—22no—
93C2×Dic59—42no—
94C2×Dic59C2×D411yescharacteristic
95C2×Dic59C2×D411yescharacteristic
96C2×Dic59C2×D411yescharacteristic
97C4×D59—22no—
98C4×D59—22no—
99C4×D59—22no—
100C4×D59—44no—
101Dic118—44no—
102Dic118—42no—
103C8×C59—22no—
104C59⋊C8—22no—
105C22×C59—41no—
106C22×C59—41no—
107C22×C59—41no—
108C22×C59C22×D411yescharacteristic
109C2×D59—21no—
110C2×D59—21no—
111C2×D59AGL(1,ℤ/8)12yes—
112C4×C59—21no—
113C4×C59—21no—
114C4×C59—41no—
115C4×C59—42no—
116C4×C592−1+412yes—
117Dic59—21no—
118Dic59—21no—
119Dic59—42no—
120Dic59—41no—
121Dic59—41no—
122Dic59#13712yes—
123#3238—591nomaximal, Sylow
124C2×C59—81no—
125C2×C59#53611yescharacteristic
126C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C59C2×#13711yescharacteristic
128D59—22no—
129C2×C4⋊Q8—591no—
130#529—591no—
131#501—591no—
132#501—591no—
133#489—591no—
134#489D5911yescharacteristic, complemented
135#483—592no—
136#480—592no—
137#477—592no—
138#476—592no—
139#432—591no—
140C59#323811yescharacteristic, complemented, Sylow
141C22×Q8—1181no—
142C4⋊Q8—592no—
143C4⋊Q8—591no—
144C4⋊Q8—1181no—
145C4⋊Q8—1181no—
146C4⋊Q8C2×D5911yescharacteristic
147#126—1181no—
148#125—591no—
149#124C2×D5911yescharacteristic
150#123—1181no—
151C22⋊Q8—591no—
152#121—591no—
153C4×Q8—591no—
154C4×D4—591no—
155C4×D4—1181no—
156#117—1181no—
157C2×C4⋊C4—592no—
158C2×C4⋊C4—1181no—
159C2×C42—591no—
160#107—592no—
161#107—592no—
162#106—592no—
163#106—592no—
164C4⋊C8—592no—
165#103—592no—
166#103C2×D5912yes—
167#102—592no—
168#102C2×D5912yes—
169C22⋊C8—592no—
170#97—591no—
171#97C2×D5911yescharacteristic
172C2×Q8—21no—
173C2×Q8—1182no—
174C2×Q8—1181no—
175C2×Q8—2364no—
176C2×Q8C22×D5911yescharacteristic
177C2×D4—1181no—
178C2×D4C22×D5911yescharacteristic
179C22×C4—591no—
180C22×C4—591no—
181C22×C4—591no—
182C22×C4—1182no—
183C22×C4—1181no—
184C2×C8—592no—
185C2×C8C22×D5912yes—
186C4⋊C4—21no—
187C4⋊C4—592no—
188C4⋊C4—591no—
189C4⋊C4—591no—
190C4⋊C4—1182no—
191C4⋊C4—1182no—
192C4⋊C4—1181no—
193C4⋊C4—1181no—
194C4⋊C4—1181no—
195C4⋊C4—1181no—
196C4⋊C4—1181no—
197C4⋊C4C22×D5912yes—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—1181no—
201C22⋊C4—1181no—
202C22⋊C4—1181no—
203C42—591no—
204C42—591no—
205C42—591no—
206C42—1181no—
207C42C22×D5911yescharacteristic
208C23—21no—
209C23—591no—
210Q8—22no—
211Q8—42no—
212Q8—2362no—
213Q8—2364no—
214D4—22no—
215D4—2362no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—591no—
219C2×C4—591no—
220C2×C4—591no—
221C2×C4—1182no—
222C2×C4—1182no—
223C2×C4—1182no—
224C2×C4—1181no—
225C2×C4—1181no—
226C2×C4—1182no—
227C2×C4—2364no—
228C2×C4—2362no—
229C2×C4D4×D5911yescharacteristic
230C2×C4D4×D5911yescharacteristic
231C2×C4C23×D5911yescharacteristic
232C8—22no—
233C8—1182no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—592no—
238C22—1181no—
239C22—1181no—
240C22C2×D4×D5911yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—592no—
246C4—1181no—
247C4—1181no—
248C4—2362no—
249C4—2361no—
250C4—2361no—
251C4#1427377412yes—
252C2—81no—
253C2—1182no—
254C2#1683797211yescharacteristic
255C2AGL(1,ℤ/8)×D5911yescharacteristic
256C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#489C2
socleC22×C59C22×D4
hypercenter#489D59
hypocenterC59#3238
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D59C2semidirect
3. #16837811C2semidirect
4. #16837808C2semidirect
5. #16837761C2semidirect
6. #16837731C2semidirect
8. #16837695C2semidirect
9. #16837690C2semidirect
10. #16837457C2semidirect
11. C59×#489C2semidirect
134. #489D59semidirect
140. C59#3238semidirect