Finite Group Databasebeta

← #21435318

Subgroups — #21435318

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#16837937C211yesmaximal, characteristic
3#16837812C211yesmaximal, characteristic, complemented
4#16837764C211yesmaximal, characteristic, complemented
5#16837762C211yesmaximal, characteristic, complemented
6#16837734C212yesmaximal, complemented
7#16837717C212yesmaximal
8#16837696C212yesmaximal, complemented
9#16837691C212yesmaximal, complemented
10#16837457C211yesmaximal, characteristic, complemented
11C59×#501C211yesmaximal, characteristic, complemented
12C2×Q8×D59—21no—
13#14273727C2211yescharacteristic
14#14273726—21no—
15#14273726C2212yes—
16#14273725—21no—
17#14273722—21no—
18#14273715C2211yescharacteristic
19#14273712—21no—
20#14273707C2211yescharacteristic
21#14273702C2211yescharacteristic
22#14273692C2211yescharacteristic
23#14273689—21no—
24#14273689C2211yescharacteristic
25#14273685—21no—
26C4⋊C4×D59—21no—
27C4⋊C4×D59C2212yes—
28C42×D59C2211yescharacteristic
29#14273660C2212yes—
30#14273658C2212yes—
31Dic59⋊C8C2212yes—
32#14273652C2212yes—
33#14273650C2212yes—
34#14273649C2212yes—
35#14273647C2212yes—
36(C2×D59)⋊C8C2212yes—
37#14273638C2211yescharacteristic
38C4⋊Q8×C59C2211yescharacteristic
39#125×C59C2211yescharacteristic
40#107×C59C2212yes—
41#106×C59C2212yes—
42#97×C59C2211yescharacteristic
43C2×Q8×C59—21no—
44Q8×D59—44no—
45C2×D236—21no—
46C2×D236C2311yescharacteristic
47C2×C4×D59—22no—
48C2×C4×D59—21no—
49C2×C4×D59D411yescharacteristic
50C2×C4×D59D411yescharacteristic
51C2×C4×D59C2311yescharacteristic
52C2×Dic118—21no—
53C2×Dic118—22no—
54C2×Dic118C2311yescharacteristic
55C2×C8×C59C2312yes—
56C4⋊C4×C59—21no—
57C4⋊C4×C59—21no—
58C4⋊C4×C59—21no—
59C4⋊C4×C59C2311yescharacteristic
60C4⋊C4×C59C2312yes—
61C4⋊C4×C59C2311yescharacteristic
62C42×C59C2311yescharacteristic
63(C2×D59)⋊C4—21no—
64(C2×D59)⋊C4—21no—
65(C2×D59)⋊C4—21no—
66(C2×D59)⋊C4—21no—
67(C2×D59)⋊C4—21no—
68C4⋊Dic59—21no—
69C4⋊Dic59—21no—
70C4⋊Dic59C2312yes—
71Dic59⋊C4—22no—
72Dic59⋊C4—22no—
73Dic59⋊C4—21no—
74C4×Dic59—21no—
75C4×Dic59D411yescharacteristic
76C4×Dic59D411yescharacteristic
77C4×Dic59C2311yescharacteristic
78C2×C59⋊C8C2312yes—
79C22×D59—21no—
80C22×D59C2×D411yescharacteristic
81Q8×C59—42no—
82C2×C4×C59—21no—
83C2×C4×C59—22no—
84C2×C4×C59—21no—
85C2×C4×C59C2×D411yescharacteristic
86C2×C4×C59C2×D411yescharacteristic
87C2×C4×C59C2411yescharacteristic
88C2×Dic59—21no—
89C2×Dic59—22no—
90C2×Dic59C2×D411yescharacteristic
91C2×Dic59C2×D411yescharacteristic
92C2×Dic59C2×D411yescharacteristic
93D236—22no—
94D236—42no—
95C4×D59—22no—
96C4×D59—22no—
97C4×D59—22no—
98C4×D59—44no—
99C4×D59—42no—
100Dic118—22no—
101Dic118—44no—
102Dic118—42no—
103C8×C59—22no—
104C59⋊C8—22no—
105C22×C59C22×D411yescharacteristic
106C2×D59—21no—
107C2×D59—21no—
108C2×D59—41no—
109C2×D59—41no—
110C2×D59—41no—
111C2×D59AGL(1,ℤ/8)12yes—
112C4×C59—21no—
113C4×C59—21no—
114C4×C59—42no—
115C4×C59—41no—
116C4×C59—41no—
117C4×C592−1+412yes—
118Dic59—21no—
119Dic59—21no—
120Dic59—41no—
121Dic59—42no—
122Dic59#13712yes—
123#3238—591nomaximal, Sylow
124C2×C59#53611yescharacteristic
125C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C59C2×#13711yescharacteristic
127D59—22no—
128D59—81no—
129C2×C4⋊Q8—591no—
130#529—591no—
131#501—591no—
132#501D5911yescharacteristic, complemented
133#489—591no—
134#489—591no—
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#125C2×D5911yescharacteristic
149#124—591no—
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#107C2×D5912yes—
162#106—592no—
163#106C2×D5912yes—
164C4⋊C8—592no—
165#103—592no—
166#103—592no—
167#102—592no—
168#102—592no—
169C22⋊C8—592no—
170#97—591no—
171#97C2×D5911yescharacteristic
172C2×Q8—21no—
173C2×Q8—591no—
174C2×Q8—1182no—
175C2×Q8—1181no—
176C2×Q8—2364no—
177C2×D4—591no—
178C2×D4—1181no—
179C22×C4—591no—
180C22×C4—591no—
181C22×C4—591no—
182C22×C4—1181no—
183C22×C4—1182no—
184C2×C8—592no—
185C2×C8C22×D5912yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—592no—
190C4⋊C4—1182no—
191C4⋊C4—1182no—
192C4⋊C4—1181no—
193C4⋊C4—1181no—
194C4⋊C4—1181no—
195C4⋊C4C22×D5912yes—
196C4⋊C4C22×D5911yescharacteristic
197C4⋊C4C22×D5911yescharacteristic
198C22⋊C4—1181no—
199C22⋊C4—1181no—
200C22⋊C4—1181no—
201C22⋊C4—1181no—
202C22⋊C4—1181no—
203C42—591no—
204C42—591no—
205C42—591no—
206C42—1181no—
207C42C22×D5911yescharacteristic
208C23—591no—
209C23—1181no—
210Q8—42no—
211Q8—1182no—
212Q8—2362no—
213Q8—2364no—
214D4—1182no—
215D4—2362no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—591no—
220C2×C4—591no—
221C2×C4—591no—
222C2×C4—1182no—
223C2×C4—1182no—
224C2×C4—1182no—
225C2×C4—1181no—
226C2×C4—1182no—
227C2×C4—2362no—
228C2×C4—2364no—
229C2×C4D4×D5911yescharacteristic
230C2×C4D4×D5911yescharacteristic
231C2×C4C23×D5911yescharacteristic
232C8—22no—
233C8—1182no—
234C22—592no—
235C22—1181no—
236C22—1181no—
237C22—2361no—
238C22—2361no—
239C22—2361no—
240C22C2×D4×D5911yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—592no—
247C4—1181no—
248C4—1181no—
249C4—2362no—
250C4—2361no—
251C4#1427377512yes—
252C2—1182no—
253C2—4721no—
254C2#1683797711yescharacteristic
255C2AGL(1,ℤ/8)×D5911yescharacteristic
256C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#501C2
socleC22×C59C22×D4
hypercenter#501D59
hypocenterC59#3238
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D59C2semidirect
3. #16837812C2semidirect
4. #16837764C2semidirect
5. #16837762C2semidirect
6. #16837734C2semidirect
8. #16837696C2semidirect
9. #16837691C2semidirect
10. #16837457C2semidirect
11. C59×#501C2semidirect
132. #501D59semidirect
140. C59#3238semidirect