Finite Group Databasebeta

← #21435246

Subgroups — #21435246

Order 7552.

360 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 123 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#16837956C211yesmaximal, characteristic
2C4⋊Q8×D59C211yesmaximal, characteristic
3#16837806C212yesmaximal, complemented
4#16837751C212yesmaximal, complemented
5#16837739C212yesmaximal
6#16837736C212yesmaximal
7#16837698C211yesmaximal, characteristic, complemented
8#16837697C211yesmaximal, characteristic, complemented
9#16837694C211yesmaximal, characteristic, complemented
10#16837454C211yesmaximal, characteristic
11C59×#477C211yesmaximal, characteristic, complemented
12C2×Q8×D59—21no—
13#14273727C2211yescharacteristic
14#14273726—21no—
15#14273726C2212yes—
16#14273726C2211yescharacteristic
17#14273725—21no—
18#14273725C2211yescharacteristic
19#14273717—21no—
20#14273707C2211yescharacteristic
21#14273705—21no—
22#14273703C2211yescharacteristic
23C4×Dic118—21no—
24C4×Dic118C2211yescharacteristic
25#14273684—21no—
26C4⋊C4×D59—21no—
27C4⋊C4×D59C2212yes—
28C42×D59C2211yescharacteristic
29#14273661C2212yes—
30#14273660C2212yes—
31#14273660C2212yes—
32C4⋊(C59⋊C8)C2211yescharacteristic
33#14273652C2212yes—
34#14273651C2212yes—
35#14273650C2212yes—
36(C2×D59)⋊C8C2212yes—
37C8×Dic59C2212yes—
38C4⋊Q8×C59C2211yescharacteristic
39C4×Q8×C59C2211yescharacteristic
40#107×C59C2212yes—
41C4⋊C8×C59C2211yescharacteristic
42#103×C59C2212yes—
43C2×Q8×C59—21no—
44C2×Q8×C59C2311yescharacteristic
45Q8×D59—44no—
46C2×C4×D59—22no—
47C2×C4×D59D411yescharacteristic
48C2×C4×D59D411yescharacteristic
49C2×C4×D59C2311yescharacteristic
50C2×Dic118—21no—
51C2×Dic118—21no—
52C2×Dic118—22no—
53C2×Dic118C2311yescharacteristic
54C2×C8×C59C2312yes—
55C4⋊C4×C59—21no—
56C4⋊C4×C59—21no—
57C4⋊C4×C59C2312yes—
58C4⋊C4×C59C2311yescharacteristic
59C42×C59—21no—
60C42×C59C2311yescharacteristic
61(C2×D59)⋊C4—21no—
62(C2×D59)⋊C4—21no—
63C4⋊Dic59—21no—
64C4⋊Dic59—21no—
65C4⋊Dic59C2312yes—
66C4⋊Dic59C2311yescharacteristic
67Dic59⋊C4—21no—
68Dic59⋊C4—22no—
69Dic59⋊C4—22no—
70Dic59⋊C4—21no—
71Dic59⋊C4—21no—
72Dic59⋊C4—21no—
73Dic59⋊C4—21no—
74C4×Dic59—21no—
75C4×Dic59D411yescharacteristic
76C4×Dic59D411yescharacteristic
77C4×Dic59C2311yescharacteristic
78C2×C59⋊C8C2312yes—
79C22×D59C2×D411yescharacteristic
80Q8×C59—21no—
81Q8×C59—21no—
82Q8×C59—42no—
83C2×C4×C59—21no—
84C2×C4×C59—22no—
85C2×C4×C59—21no—
86C2×C4×C59C2×D411yescharacteristic
87C2×C4×C59C2×D411yescharacteristic
88C2×C4×C59C2411yescharacteristic
89C2×Dic59—21no—
90C2×Dic59—21no—
91C2×Dic59—22no—
92C2×Dic59C2×D411yescharacteristic
93C2×Dic59C2×D411yescharacteristic
94C2×Dic59C2×D411yescharacteristic
95C4×D59—21no—
96C4×D59—21no—
97C4×D59—22no—
98C4×D59—44no—
99C4×D59Q1614yes—
100Dic118—21no—
101Dic118—21no—
102Dic118—42no—
103Dic118—42no—
104Dic118—44no—
105C8×C59—22no—
106C59⋊C8—22no—
107C22×C59C22×D411yescharacteristic
108C2×D59—21no—
109C2×D59—21no—
110C2×D59C2×Q1612yes—
111C4×C59—21no—
112C4×C59—41no—
113C4×C59—42no—
114C4×C59—41no—
115C4×C59C2×Q1612yes—
116C4×C592−1+411yescharacteristic
117C4×C592−1+411yescharacteristic
118Dic59—21no—
119Dic59—41no—
120Dic59—42no—
121Dic59—41no—
122Dic59C2×Q1612yes—
123Dic59AGL(1,ℤ/8)12yes—
124#3230—591nomaximal, Sylow
125C2×C59#53611yescharacteristic
126C2×C59C22×Q1611yescharacteristic
127C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
128D59—22no—
129#533—591no—
130C2×C4⋊Q8—591no—
131#500—592no—
132#487—592no—
133#484—592no—
134#484—592no—
135#477—591no—
136#477—591no—
137#477—591no—
138#477D5911yescharacteristic, complemented
139#431—591no—
140C59#323011yescharacteristic, complemented, Sylow
141C22×Q8—1181no—
142C4⋊Q8—592no—
143C4⋊Q8—591no—
144C4⋊Q8—591no—
145C4⋊Q8—591no—
146C4⋊Q8—1181no—
147C4⋊Q8—1181no—
148C4⋊Q8C2×D5911yescharacteristic
149#125—1181no—
150C22⋊Q8—591no—
151C22⋊Q8—591no—
152C22⋊Q8—1181no—
153C4×Q8—591no—
154C4×Q8—1181no—
155C4×Q8C2×D5911yescharacteristic
156#117—1181no—
157C2×C4⋊C4—592no—
158C2×C4⋊C4—1181no—
159C2×C42—591no—
160#107—592no—
161#107—592no—
162#107—592no—
163#107C2×D5912yes—
164C4⋊C8—591no—
165C4⋊C8C2×D5911yescharacteristic
166#103—592no—
167#103—592no—
168#103—592no—
169#103C2×D5912yes—
170C22⋊C8—592no—
171C4×C8—592no—
172C2×Q8—21no—
173C2×Q8—591no—
174C2×Q8—1181no—
175C2×Q8—1182no—
176C2×Q8—1181no—
177C2×Q8—2364no—
178C2×Q8C22×D5911yescharacteristic
179C22×C4—591no—
180C22×C4—591no—
181C22×C4—591no—
182C22×C4—1182no—
183C2×C8—592no—
184C2×C8C22×D5912yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—592no—
188C4⋊C4—591no—
189C4⋊C4—1182no—
190C4⋊C4—1182no—
191C4⋊C4—1181no—
192C4⋊C4—1181no—
193C4⋊C4—1181no—
194C4⋊C4—1181no—
195C4⋊C4—1181no—
196C4⋊C4—1181no—
197C4⋊C4—1181no—
198C4⋊C4C22×D5912yes—
199C4⋊C4C22×D5911yescharacteristic
200C22⋊C4—1181no—
201C22⋊C4—1181no—
202C42—21no—
203C42—591no—
204C42—591no—
205C42—591no—
206C42—1181no—
207C42C22×D5911yescharacteristic
208C23—591no—
209Q8—21no—
210Q8—21no—
211Q8—42no—
212Q8—1181no—
213Q8—1181no—
214Q8—2362no—
215Q8—2362no—
216Q8—2364no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—591no—
221C2×C4—591no—
222C2×C4—594no—
223C2×C4—591no—
224C2×C4—1182no—
225C2×C4—1181no—
226C2×C4—1181no—
227C2×C4—1181no—
228C2×C4—1181no—
229C2×C4—1182no—
230C2×C4—2364no—
231C2×C4D4×D5911yescharacteristic
232C2×C4D4×D5911yescharacteristic
233C2×C4C23×D5911yescharacteristic
234C8—22no—
235C8—1182no—
236C22—592no—
237C22—1181no—
238C22—1181no—
239C22C2×D4×D5911yescharacteristic
240C4—21no—
241C4—42no—
242C4—41no—
243C4—41no—
244C4—592no—
245C4—592no—
246C4—1181no—
247C4—2362no—
248C4—2361no—
249C4—2361no—
250C4Q16×D5912yes—
251C4#1427377411yescharacteristic
252C4#1427377511yescharacteristic
253C2—1182no—
254C2#1683797511yescharacteristic
255C2C2×Q16×D5911yescharacteristic
256C2AGL(1,ℤ/8)×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#477C2
socleC22×C59C22×D4
hypercenter#477D59
hypocenterC59#3230
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #16837806C2semidirect
4. #16837751C2semidirect
7. #16837698C2semidirect
8. #16837697C2semidirect
9. #16837694C2semidirect
11. C59×#477C2semidirect
138. #477D59semidirect
140. C59#3230semidirect