Finite Group Databasebeta

← #21435020

Subgroups — #21435020

Order 7552.

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

normal subgroups only

This group has 428 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#16837949C211yesmaximal, characteristic
2C4⋊Q8×D59C211yesmaximal, characteristic, complemented
3#16837773C212yesmaximal, complemented
4#16837751C212yesmaximal, complemented
5#16837608C211yesmaximal, characteristic, complemented
6#16837607C211yesmaximal, characteristic, complemented
7#16837604C211yesmaximal, characteristic, complemented
8#16837542C212yesmaximal, complemented
9#16837539C212yesmaximal, complemented
10#16837454C211yesmaximal, characteristic, complemented
11C59×#461C211yesmaximal, characteristic, complemented
12C2×#11745685—21no—
13C2×Q8×D59C2212yes—
14C2×#11745660C2212yes—
15C2×#11745670C2212yes—
16C2×#11745671C2212yes—
17#14273727C2211yescharacteristic
18#14273726—21no—
19#14273726—21no—
20#14273726C2211yescharacteristic
21#14273725C2212yes—
22#14273723C2211yescharacteristic
23#14273713—21no—
24#14273702C2211yescharacteristic
25#14273700—21no—
26#14273698C2211yescharacteristic
27C4×D236C2211yescharacteristic
28C4×C22⋊D59—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—
37C8×Dic59C2212yes—
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—
47C2×C22⋊D59—21no—
48C2×C22⋊D59—21no—
49Q8×D59—42no—
50Q8×D59—42no—
51#11745685—44no—
52C2×D236—21no—
53C2×D236C2311yescharacteristic
54C2×C4×D59—22no—
55C2×C4×D59—21no—
56C2×C4×D59D411yescharacteristic
57C2×C4×D59D411yescharacteristic
58C2×C4×D59C2311yescharacteristic
59C2×Dic118—21no—
60C2×Dic118—22no—
61C2×Dic118C2312yes—
62SD16×C59—42no—
63C2×C8×C59C2312yes—
64C4⋊C4×C59—21no—
65C4⋊C4×C59—21no—
66C4⋊C4×C59C2311yescharacteristic
67C22⋊C4×C59—21no—
68C42×C59C2311yescharacteristic
69C22⋊Dic59—21no—
70#11745671—42no—
71#11745670—42no—
72(C2×D59)⋊C4—21no—
73(C2×D59)⋊C4—21no—
74C4⋊Dic59—21no—
75C4⋊Dic59—21no—
76C4⋊Dic59C2311yescharacteristic
77Dic59⋊C4—22no—
78Dic59⋊C4—22no—
79Dic59⋊C4—21no—
80C4×Dic59D411yescharacteristic
81C4×Dic59D411yescharacteristic
82C4×Dic59C2311yescharacteristic
83C2×C59⋊C8C2312yes—
84#11745660—42no—
85C23×C59—21no—
86C22×D59—21no—
87C22×D59C2×D411yescharacteristic
88Q8×C59—22no—
89Q8×C59—22no—
90D4×C59—21no—
91D4×C59—21no—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—42no—
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59C2×D411yescharacteristic
97C2×C4×C59C2411yescharacteristic
98C22⋊D59—42no—
99C22⋊D59—42no—
100C22⋊D59—42no—
101C2×Dic59—22no—
102C2×Dic59—21no—
103C2×Dic59C2×D411yescharacteristic
104C2×Dic59C2×D411yescharacteristic
105C2×Dic59C2×D411yescharacteristic
106D236—21no—
107D236—21no—
108D236—42no—
109C4×D59—21no—
110C4×D59—21no—
111C4×D59—22no—
112C4×D59—44no—
113C4×D59—42no—
114C4×D59SD1614yescomplemented
115Dic118—22no—
116Dic118—22no—
117Dic118—44no—
118Dic118—42no—
119C8×C59—22no—
120C59⋊C8—22no—
121C22×C59—41no—
122C22×C59—41no—
123C22×C59—41no—
124C22×C59C22×D411yescharacteristic
125C2×D59—21no—
126C2×D59—21no—
127C2×D59—41no—
128C2×D59—41no—
129C2×D59—41no—
130C2×D59C2×SD1612yes—
131C4×C59—21no—
132C4×C59—42no—
133C4×C59—41no—
134C4×C59C2×SD1612yes—
135C4×C59H(5,𝔽2)11yescharacteristic
136C4×C59H(5,𝔽2)11yescharacteristic
137Dic59—21no—
138Dic59—42no—
139Dic59—41no—
140Dic59C2×SD1612yes—
141Dic59#13712yes—
142#3206—591nomaximal, Sylow
143C2×C59—81no—
144C2×C59#53511yescharacteristic
145C2×C59C22×SD1611yescharacteristic
146C2×C59C2×#13711yescharacteristic
147D59—22no—
148D59—81no—
149#532—591no—
150C2×C4⋊Q8—591no—
151#492—592no—
152#487—592no—
153#461—591no—
154#461—591no—
155#461—591no—
156#461D5911yescharacteristic, complemented
157#448—592no—
158#448—592no—
159#431—591no—
160C59#320611yescharacteristic, complemented, Sylow
161C2×#41—1181no—
162C22×Q8—592no—
163C2×SD16—592no—
164C2×SD16—592no—
165C2×SD16—592no—
166C2×SD16C2×D5912yes—
167C4⋊Q8—591no—
168C4⋊Q8—592no—
169C4⋊Q8—591no—
170C4⋊Q8—1181no—
171C4⋊Q8—1181no—
172C4⋊Q8C2×D5911yescharacteristic
173#127—591no—
174#124—1181no—
175#121—591no—
176#121—591no—
177#121—1181no—
178C4×D4—591no—
179C4×D4—1181no—
180C4×D4C2×D5911yescharacteristic
181C2×C4⋊C4—1181no—
182C2×C4⋊C4—1181no—
183C2×C42—591no—
184C4⋊C8—591no—
185C4⋊C8C2×D5911yescharacteristic
186#103—592no—
187#103—592no—
188#103—592no—
189#103C2×D5912yes—
190C22⋊C8—592no—
191C4×C8—592no—
192#41—2364no—
193C2×Q8—592no—
194C2×Q8—1181no—
195C2×Q8—1182no—
196C2×Q8—2362no—
197C2×Q8—2362no—
198C2×Q8C22×D5912yes—
199C2×D4—591no—
200C2×D4—1181no—
201C2×D4—1181no—
202C2×D4—1181no—
203C2×D4—1181no—
204C2×D4C22×D5911yescharacteristic
205C22×C4—21no—
206C22×C4—591no—
207C22×C4—591no—
208C22×C4—591no—
209C22×C4—1182no—
210C22×C4—1181no—
211SD16—42no—
212SD16—2362no—
213SD16—2362no—
214SD16—2362no—
215C2×C8—592no—
216C2×C8C22×D5912yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—591no—
220C4⋊C4—1182no—
221C4⋊C4—1182no—
222C4⋊C4—1181no—
223C4⋊C4—1181no—
224C4⋊C4—1181no—
225C4⋊C4C22×D5911yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1181no—
228C22⋊C4—1181no—
229C22⋊C4—1181no—
230C42—591no—
231C42—591no—
232C42—591no—
233C42C22×D5911yescharacteristic
234C23—21no—
235C23—591no—
236C23—1181no—
237Q8—22no—
238Q8—22no—
239Q8—1182no—
240Q8—1182no—
241Q8—2362no—
242Q8—2364no—
243D4—21no—
244D4—21no—
245D4—1181no—
246D4—1181no—
247D4—2362no—
248D4—2362no—
249D4—2362no—
250D4—2362no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—591no—
255C2×C4—591no—
256C2×C4—594no—
257C2×C4—591no—
258C2×C4—1182no—
259C2×C4—1181no—
260C2×C4—1181no—
261C2×C4—1181no—
262C2×C4—1182no—
263C2×C4—2364no—
264C2×C4—2362no—
265C2×C4D4×D5911yescharacteristic
266C2×C4D4×D5911yescharacteristic
267C2×C4C23×D5911yescharacteristic
268C8—22no—
269C8—1182no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—592no—
274C22—1181no—
275C22—1181no—
276C22—2361no—
277C22—2361no—
278C22—2361no—
279C22C2×D4×D5911yescharacteristic
280C4—21no—
281C4—41no—
282C4—42no—
283C4—592no—
284C4—592no—
285C4—1181no—
286C4—2361no—
287C4—2362no—
288C4SD16×D5912yes—
289C4#1427377211yescharacteristic
290C4#1427377311yescharacteristic
291C2—81no—
292C2—1182no—
293C2—4721no—
294C2#1683796811yescharacteristic
295C2C2×SD16×D5911yescharacteristic
296C2#137×D5911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#461C2
socleC22×C59C22×D4
hypercenter#461D59
hypocenterC59#3206
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
2. C4⋊Q8×D59C2semidirect
3. #16837773C2semidirect
4. #16837751C2semidirect
5. #16837608C2semidirect
6. #16837607C2semidirect
7. #16837604C2semidirect
8. #16837542C2semidirect
9. #16837539C2semidirect
10. #16837454C2semidirect
11. C59×#461C2semidirect
114. C4×D59SD16semidirect
156. #461D59semidirect
160. C59#3206semidirect