Finite Group Databasebeta

← #21434572

Subgroups — #21434572

Order 7552.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#14273760C211yesmaximal, characteristic, complemented
2C2×#14273755C211yesmaximal, characteristic, complemented
3C2×#14273749C211yesmaximal, characteristic, complemented
4C22×#11745660C211yesmaximal, characteristic, complemented
5#16837978C211yesmaximal, characteristic, complemented
6#16837971C211yesmaximal, characteristic, complemented
7#16837796C212yesmaximal, complemented
8#16837791C212yesmaximal, complemented
9#16837786C212yesmaximal, complemented
10#16837775C212yesmaximal, complemented
11C59×#405C211yesmaximal, characteristic, complemented
12C2×#11745685C2211yescharacteristic
13C2×#11745685C2211yescharacteristic
14C22×Dic118C2211yescharacteristic
15C22×D236C2211yescharacteristic
16#14273760—24no—
17#14273755—24no—
18#14273749—24no—
19C2×Dic236C2211yescharacteristic, complemented
20C2×Dic236C2212yescomplemented
21C2×#11745660—24no—
22C2×#11745660C2212yes—
23C2×#11745660C2212yes—
24C2×#11745660C2212yes—
25C2×#11745660C2211yescharacteristic, complemented
26C2×#11745660C2211yescharacteristic, complemented
27C2×#11745660C2212yescomplemented
28C2×D472C2212yes—
29C2×D472C2211yescharacteristic, complemented
30#14273727C2212yescomplemented
31#14273724C2212yes—
32#14273715C2212yes—
33#14273715C2212yescomplemented
34#14273711—22no—
35#14273707—22no—
36#14273702—22no—
37#14273697—22no—
38C2×M16×C59C2211yescharacteristic
39C22×C8×C59C2211yescharacteristic
40#117×C59C2211yescharacteristic
41#97×C59C2212yescomplemented
42C4×C8×C59C2212yescomplemented
43C23×D59—21no—
44C22×C4×C59C2311yescharacteristic
45C2×C22⋊D59—21no—
46C2×C22⋊D59—21no—
47C22×Dic59—21no—
48#11745685—22no—
49#11745685—22no—
50#11745685—22no—
51#11745685—22no—
52C2×D236—22no—
53C2×D236—22no—
54C2×D236—21no—
55C2×D236—24no—
56C2×D236C2312yes—
57C2×D236C2311yescharacteristic
58C2×D236C2311yescharacteristic
59C2×C4×D59—21no—
60C2×C4×D59—21no—
61C2×Dic118—22no—
62C2×Dic118—22no—
63C2×Dic118—21no—
64C2×Dic118C2311yescharacteristic
65C2×Dic118C2312yes—
66C2×Dic118C2311yescharacteristic
67M16×C59D414yescomplemented
68C2×C8×C59D414yescomplemented
69C2×C8×C59C2311yescharacteristic
70C2×C8×C59C2312yes—
71C2×C8×C59C2311yescharacteristic
72C4⋊C4×C59D412yes—
73C22⋊C4×C59D412yes—
74C42×C59C2312yes—
75(C2×D59)⋊C4—24no—
76(C2×D59)⋊C4—24no—
77C4⋊Dic59—24no—
78Dic236—22no—
79Dic236—24no—
80D472—24no—
81D472—22no—
82#11745660—24no—
83#11745660—24no—
84#11745660—24no—
85#11745660—22no—
86#11745660—22no—
87#11745660—24no—
88C23×C59C2×D411yescharacteristic
89C22×D59—21no—
90C22×D59—22no—
91C22×D59—21no—
92C22×D59—42no—
93C22×D59—82no—
94C2×C4×C59C2×D414yes—
95C2×C4×C59C2×D411yescharacteristic
96C2×C4×C59C2×D412yes—
97C2×C4×C59C2×D412yes—
98C2×C4×C59C2411yescharacteristic
99C22⋊D59—42no—
100C22⋊D59—42no—
101C2×Dic59—21no—
102C2×Dic59—22no—
103C2×Dic59—21no—
104C2×Dic59—42no—
105D236—22no—
106D236—21no—
107D236—22no—
108D236—21no—
109D236—21no—
110D236—21no—
111D236—42no—
112D236—84no—
113C4×D59—42no—
114C4×D59—42no—
115Dic118—22no—
116Dic118—21no—
117Dic118—22no—
118Dic118—21no—
119Dic118—21no—
120Dic118—21no—
121Dic118—42no—
122C8×C59C2×D412yes—
123C8×C59C2×D412yes—
124C8×C59C2×D414yes—
125C22×C59—22no—
126C22×C59#12712yes—
127C22×C59C22×D411yescharacteristic
128C2×D59—41no—
129C2×D59—41no—
130C2×D59—42no—
131C2×D59—42no—
132C2×D59—44no—
133C2×D59—42no—
134C2×D59—84no—
135C4×C59—24no—
136C4×C59#12712yes—
137C4×C59C22×D411yescharacteristic
138C4×C59C22×D411yescharacteristic
139Dic59—42no—
140Dic59—41no—
141Dic59—41no—
142#3154—591nomaximal, Sylow
143C2×C59—22no—
144C2×C59C2×#12711yescharacteristic
145C2×C59#57712yes—
146D59—81no—
147D59—81no—
148D59—82no—
149C2×#137—591no—
150C2×AGL(1,ℤ/8)—591no—
151C2×#135—591no—
152C22×SD16—591no—
153#536—591no—
154#535—591no—
155#497—592no—
156#496—592no—
157#495—592no—
158#492—592no—
159#405D5911yescharacteristic, complemented
160C59#315411yescharacteristic, complemented, Sylow
161C2×#41—591no—
162C2×#41—591no—
163C22×Q8—591no—
164C22×D4—591no—
165#137—1184no—
166AGL(1,ℤ/8)—1184no—
167#135—1184no—
168C2×Q16—591no—
169C2×Q16—592no—
170C2×SD16—592no—
171C2×SD16—592no—
172C2×SD16—591no—
173C2×SD16—591no—
174C2×SD16—592no—
175C2×SD16—592no—
176C2×SD16—1184no—
177C2×D8—591no—
178C2×D8—592no—
179C2×M16C2×D5911yescharacteristic
180C22×C8C2×D5911yescharacteristic
181C4⋊Q8—592no—
182#127—592no—
183#124—592no—
184#124—592no—
185#123—1182no—
186C22⋊Q8—1182no—
187#121—1182no—
188#120—1182no—
189#117C2×D5911yescharacteristic
190#97C2×D5912yescomplemented
191C4×C8C2×D5912yescomplemented
192C24—1181no—
193#41—1182no—
194#41—1182no—
195#41—1182no—
196#41—1182no—
197C2×Q8—591no—
198C2×Q8—592no—
199C2×Q8—591no—
200C2×Q8—1181no—
201C2×Q8—1182no—
202C2×Q8—1182no—
203C2×D4—591no—
204C2×D4—592no—
205C2×D4—591no—
206C2×D4—1181no—
207C2×D4—1181no—
208C2×D4—1182no—
209C2×D4—1182no—
210C2×D4—1184no—
211C2×D4—1181no—
212C22×C4—1181no—
213C22×C4—1181no—
214C22×C4—1181no—
215C22×C4C22×D5911yescharacteristic
216Q16—1182no—
217Q16—1184no—
218SD16—1184no—
219SD16—1184no—
220SD16—1182no—
221SD16—1182no—
222SD16—1184no—
223SD16—1184no—
224D8—1182no—
225D8—1184no—
226M16D23614yescomplemented
227C2×C8D23614yescomplemented
228C2×C8C22×D5911yescharacteristic
229C2×C8C22×D5912yes—
230C2×C8C22×D5911yescharacteristic
231C4⋊C4—1184no—
232C4⋊C4D23612yes—
233C22⋊C4—1184no—
234C22⋊C4—1184no—
235C22⋊C4D23612yes—
236C42C22×D5912yes—
237C23—1181no—
238C23—1182no—
239C23—1181no—
240C23—4722no—
241C23—2362no—
242C23C2×D23611yescharacteristic
243Q8—1182no—
244Q8—1181no—
245Q8—1182no—
246Q8—1181no—
247Q8—1181no—
248Q8—1181no—
249Q8—2362no—
250D4—1181no—
251D4—1182no—
252D4—1181no—
253D4—1181no—
254D4—1182no—
255D4—1181no—
256D4—4724no—
257D4—2362no—
258D4—2362no—
259D4—2362no—
260C2×C4—1181no—
261C2×C4—1182no—
262C2×C4—1181no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4—2362no—
266C2×C4C2×D23614yes—
267C2×C4C2×D23611yescharacteristic
268C2×C4C2×D23612yes—
269C2×C4C2×D23612yes—
270C2×C4C23×D5911yescharacteristic
271C8C2×D23612yes—
272C8C2×D23612yes—
273C8C2×D23614yes—
274C22—22no—
275C22—4724no—
276C22—2361no—
277C22—2361no—
278C22—2362no—
279C22—2362no—
280C22—2364no—
281C22—2362no—
282C22#1427372412yes—
283C22C22×D23611yescharacteristic
284C4—24no—
285C4—2361no—
286C4—2361no—
287C4—2362no—
288C4#1427372412yes—
289C4C22×D23611yescharacteristic
290C4C22×D23611yescharacteristic
291C2—22no—
292C2—4722no—
293C2—4721no—
294C2—4721no—
295C2C2×#1427372411yescharacteristic
296C2#1683821412yes—
subgroupG/H
centerC22C22×D236
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#405C2
socleC22×C59C22×D4
hypercenter#405D59
hypocenterC59#3154
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#14273760C2semidirect
2. C2×#14273755C2semidirect
3. C2×#14273749C2semidirect
4. C22×#11745660C2semidirect
5. #16837978C2semidirect
6. #16837971C2semidirect
7. #16837796C2semidirect
8. #16837791C2semidirect
9. #16837786C2semidirect
10. #16837775C2semidirect
11. C59×#405C2semidirect
19. C2×Dic236C22semidirect
20. C2×Dic236C22semidirect
25. C2×#11745660C22semidirect
26. C2×#11745660C22semidirect
27. C2×#11745660C22semidirect
29. C2×D472C22semidirect
30. #14273727C22semidirect
33. #14273715C22semidirect
41. #97×C59C22semidirect
42. C4×C8×C59C22semidirect
67. M16×C59D4semidirect
68. C2×C8×C59D4semidirect
159. #405D59semidirect
160. C59#3154semidirect
190. #97C2×D59semidirect
191. C4×C8C2×D59semidirect
226. M16D236semidirect
227. C2×C8D236semidirect