Finite Group Databasebeta

← #21434559

Subgroups — #21434559

Order 7552.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14273759C212yesmaximal, complemented
2C2×#14273754C212yesmaximal, complemented
3#16837953C211yesmaximal, characteristic, complemented
4#124×D59C211yesmaximal, characteristic, complemented
5#16837794C211yesmaximal, characteristic, complemented
6#16837792C211yesmaximal, characteristic, complemented
7#16837791C211yesmaximal, characteristic, complemented
8#16837787C212yesmaximal, complemented
9#16837783C212yesmaximal, complemented
10#16837354C211yesmaximal, characteristic, complemented
11C59×#497C211yesmaximal, characteristic, complemented
12C2×#11745689C2212yes—
13C2×Q8×D59C2211yescharacteristic
14C2×D4×D59C2211yescharacteristic
15#14273759—28no—
16#14273754—28no—
17C2×#11745672C2212yescomplemented
18C2×#11745660C2212yescomplemented
19C2×#11745671C2212yes—
20C2×#11745671C2212yescomplemented
21C2×D472C2212yes—
22C2×#11745669C2212yes—
23C2×#11745659C2212yes—
24#14273724C2211yescharacteristic, complemented
25#14273716C2211yescharacteristic
26#14273716C2212yescomplemented
27#14273715C2211yescharacteristic
28#14273714C2211yescharacteristic
29#14273713—22no—
30#14273702—22no—
31#14273689—22no—
32C22⋊C4×D59—22no—
33C42×D59C2211yescharacteristic
34#14273638C2211yescharacteristic, complemented
35C8×Dic59C2212yescomplemented
36C2×Q16×C59C2212yescomplemented
37C2×SD16×C59C2212yes—
38C4⋊Q8×C59C2211yescharacteristic, complemented
39#124×C59C2211yescharacteristic
40#97×C59C2211yescharacteristic, complemented
41C23×D59—21no—
42C2×Q8×C59C2311yescharacteristic
43C2×Q8×C59C2312yes—
44C2×D4×C59C2311yescharacteristic
45C2×C22⋊D59—21no—
46#11745689—24no—
47#11745689—24no—
48Q8×D59—24no—
49D4×D59—24no—
50C2×D236—22no—
51C2×D236—22no—
52C2×D236C2311yescharacteristic
53C2×D236C2312yes—
54C2×C4×D59—21no—
55C2×C4×D59—22no—
56C2×C4×D59D412yes—
57C2×C4×D59C2311yescharacteristic
58C2×Dic118—21no—
59C2×Dic118C2311yescharacteristic
60Q16×C59—24no—
61SD16×C59—24no—
62C2×C8×C59C2312yes—
63C4⋊C4×C59—22no—
64C22⋊C4×C59—22no—
65C42×C59C2311yescharacteristic
66C22⋊Dic59—22no—
67#11745672—24no—
68#11745671—24no—
69#11745671—24no—
70#11745669—24no—
71(C2×D59)⋊C4—22no—
72(C2×D59)⋊C4—22no—
73(C2×D59)⋊C4—24no—
74C4×Dic59D412yes—
75C4×Dic59C2311yescharacteristic
76C2×C59⋊C8C2312yes—
77D472—24no—
78#11745660—24no—
79#11745659D418yescomplemented
80C23×C59—21no—
81C22×D59—21no—
82C22×D59—22no—
83C22×D59—44no—
84C22×D59—42no—
85C22×D59C2×D411yescharacteristic
86Q8×C59—22no—
87Q8×C59—22no—
88Q8×C59—22no—
89D4×C59—22no—
90C2×C4×C59—21no—
91C2×C4×C59—22no—
92C2×C4×C59C2×D412yes—
93C2×C4×C59C2411yescharacteristic
94C22⋊D59—42no—
95C2×Dic59—21no—
96C2×Dic59C2×D412yes—
97C2×Dic59C2×D411yescharacteristic
98D236—22no—
99D236—22no—
100D236—22no—
101D236—44no—
102D236—44no—
103C4×D59—42no—
104C4×D59—42no—
105C4×D59—44no—
106C4×D59C2×D414yes—
107Dic118—22no—
108Dic118—42no—
109C8×C59C2×D414yes—
110C59⋊C8C2×D414yes—
111C22×C59—41no—
112C22×C59—42no—
113C22×C59C22×D411yescharacteristic
114C2×D59—22no—
115C2×D59—42no—
116C2×D59—41no—
117C2×D59—44no—
118C2×D59—42no—
119C2×D59—84no—
120C2×D59#12712yes—
121C4×C59—22no—
122C4×C59—41no—
123C4×C59—42no—
124C4×C59C22×D412yes—
125Dic59—22no—
126Dic59—41no—
127Dic59#12712yes—
128#3153—591nomaximal, Sylow
129C2×C59—81no—
130C2×C59C2×#12711yescharacteristic
131C2×C59#57512yes—
132D59—22no—
133D59—82no—
134D59—81no—
135C2×#137—592no—
136C2×AGL(1,ℤ/8)—592no—
137#532—591no—
138C2×#124—591no—
139#497—591no—
140#497D5911yescharacteristic, complemented
141#496—591no—
142#496—591no—
143#495—592no—
144#495—592no—
145C4×M16—591no—
146C59#315311yescharacteristic, complemented, Sylow
147C2×#41—592no—
148C22×Q8—591no—
149C22×D4—591no—
150#137—1188no—
151AGL(1,ℤ/8)—1188no—
152C2×Q16—592no—
153C2×Q16C2×D5912yescomplemented
154C2×SD16—592no—
155C2×SD16—592no—
156C2×SD16—592no—
157C2×SD16C2×D5912yes—
158C2×D8—592no—
159C2×D8—592no—
160C2×M16—592no—
161C4⋊Q8C2×D5911yescharacteristic, complemented
162#127—591no—
163#124—591no—
164#124—592no—
165#124—591no—
166#124—591no—
167#124—1182no—
168#124C2×D5911yescharacteristic
169#121—1182no—
170C4×D4—1182no—
171C2×C22⋊C4—1182no—
172C2×C42—591no—
173#97—591no—
174#97C2×D5911yescharacteristic, complemented
175C4×C8—592no—
176C24—1181no—
177#41—1184no—
178#41—1184no—
179C2×Q8—591no—
180C2×Q8—1181no—
181C2×Q8—1184no—
182C2×Q8C22×D5911yescharacteristic
183C2×Q8C22×D5912yes—
184C2×D4—591no—
185C2×D4—592no—
186C2×D4—1182no—
187C2×D4—1184no—
188C2×D4—1181no—
189C2×D4—1182no—
190C2×D4C22×D5911yescharacteristic
191C22×C4—591no—
192C22×C4—592no—
193C22×C4—1182no—
194C22×C4—1181no—
195Q16—24no—
196Q16—1184no—
197SD16—24no—
198SD16—1184no—
199SD16—1184no—
200SD16—1184no—
201D8—1184no—
202D8—1184no—
203M16—598no—
204C2×C8—592no—
205C2×C8C22×D5912yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—1182no—
209C22⋊C4—1184no—
210C22⋊C4—1182no—
211C22⋊C4—1182no—
212C42—592no—
213C42—591no—
214C42C22×D5911yescharacteristic
215C23—21no—
216C23—591no—
217C23—1181no—
218C23—1182no—
219C23—2362no—
220C23—2364no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—1182no—
225Q8—2362no—
226D4—22no—
227D4—1182no—
228D4—1182no—
229D4—1182no—
230D4—2364no—
231D4—2364no—
232D4—2362no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—591no—
236C2×C4—594no—
237C2×C4—592no—
238C2×C4—1181no—
239C2×C4—2362no—
240C2×C4—2364no—
241C2×C4—2362no—
242C2×C4D4×D5912yes—
243C2×C4C23×D5911yescharacteristic
244C8—594no—
245C8D4×D5914yes—
246C22—41no—
247C22—42no—
248C22—592no—
249C22—1182no—
250C22—4724no—
251C22—2362no—
252C22—2361no—
253C22—2364no—
254C22—2362no—
255C22C2×D4×D5911yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—592no—
260C4—1182no—
261C4—2361no—
262C4C2×D4×D5912yes—
263C2—81no—
264C2—1182no—
265C2—4721no—
266C2—4722no—
267C2#127×D5911yescharacteristic
268C2#1683820112yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#497C2
socleC22×C59C22×D4
hypercenter#497D59
hypocenterC59#3153
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#14273759C2semidirect
2. C2×#14273754C2semidirect
3. #16837953C2semidirect
4. #124×D59C2semidirect
5. #16837794C2semidirect
6. #16837792C2semidirect
7. #16837791C2semidirect
8. #16837787C2semidirect
9. #16837783C2semidirect
10. #16837354C2semidirect
11. C59×#497C2semidirect
17. C2×#11745672C22semidirect
18. C2×#11745660C22semidirect
20. C2×#11745671C22semidirect
24. #14273724C22semidirect
26. #14273716C22semidirect
34. #14273638C22semidirect
35. C8×Dic59C22semidirect
36. C2×Q16×C59C22semidirect
38. C4⋊Q8×C59C22semidirect
40. #97×C59C22semidirect
79. #11745659D4semidirect
140. #497D59semidirect
146. C59#3153semidirect
153. C2×Q16C2×D59semidirect
161. C4⋊Q8C2×D59semidirect
174. #97C2×D59semidirect