Finite Group Databasebeta

← #24918335

Subgroups — #24918335

Order 10624.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128272C211yesmaximal, characteristic, complemented
2#19128042C212yesmaximal, complemented
3#19128022C212yesmaximal, complemented
4#19127945C212yesmaximal, complemented
5#19127928C212yesmaximal, complemented
6#19127852C212yesmaximal, complemented
7#19127839C212yesmaximal, complemented
8#19127668C211yesmaximal, characteristic, complemented
9#535×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic, complemented
11C2×#13050578C2212yes—
12C2×#13050578C2212yes—
13C2×#13050574C2211yescharacteristic, complemented
14C22×C83⋊C8C2211yescharacteristic, complemented
15#14581790—21no—
16#14581779—21no—
17#14581773—21no—
18#14581773—21no—
19#14581771C2212yes—
20#14581771C2212yescomplemented
21#14581751C2211yescharacteristic, complemented
22#14581725C2212yescomplemented
23#14581723C2212yescomplemented
24#14581715C2212yescomplemented
25#14581715C2212yescomplemented
26#14581713C2212yes—
27#14581713C2212yes—
28C22⋊(C83⋊C8)C2212yes—
29C22⋊(C83⋊C8)C2212yescomplemented
30C2×#41×C83C2211yescharacteristic
31C22×D4×C83C2211yescharacteristic
32#127×C83—21no—
33#124×C83—21no—
34#121×C83C2212yes—
35#121×C83C2212yes—
36#120×C83—21no—
37#120×C83—21no—
38#117×C83C2211yescharacteristic
39C2×Dic166—21no—
40C2×Dic166—41no—
41C2×Dic166—41no—
42C2×Dic166C2312yes—
43C22×Dic83—21no—
44#13050578—42no—
45#13050578—42no—
46#13050574—41no—
47C2×C83⋊C8—41no—
48C2×C83⋊C8C2312yes—
49C2×C83⋊C8C2312yes—
50C4⋊Dic83C2312yes—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53C22⋊Dic83—22no—
54C22⋊Dic83—22no—
55C22⋊Dic83—21no—
56C4×Dic83—21no—
57C24×C83—21no—
58#41×C83—41no—
59#41×C83—41no—
60C2×Q8×C83D411yescharacteristic
61C2×D4×C83—22no—
62C2×D4×C83—22no—
63C2×D4×C83—21no—
64C2×D4×C83—41no—
65C2×D4×C83—41no—
66C2×D4×C83D411yescharacteristic
67C2×D4×C83D411yescharacteristic
68C2×D4×C83D411yescharacteristic
69C2×D4×C83C2312yes—
70C22×C4×C83C2311yescharacteristic
71C22×C4×C83C2311yescharacteristic
72C22×C4×C83C2311yescharacteristic
73C4⋊C4×C83C2312yes—
74C22⋊C4×C83—21no—
75C22⋊C4×C83—22no—
76C22⋊C4×C83—22no—
77C42×C83—21no—
78C23×C83—22no—
79C23×C83—42no—
80C23×C83—82no—
81C23×C83C2×D411yescharacteristic
82C23×C83C2×D411yescharacteristic
83C23×C83C2×D411yescharacteristic
84Q8×C83—41no—
85D4×C83—22no—
86D4×C83—22no—
87D4×C83—41no—
88D4×C83—41no—
89D4×C83—41no—
90D4×C83—42no—
91D4×C83—82no—
92D4×C83—82no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83C2×D411yescharacteristic
102C2×C4×C83C2×D411yescharacteristic
103C2×C4×C83C2411yescharacteristic
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—42no—
107Dic166—22no—
108Dic166—22no—
109Dic166—42no—
110C83⋊C8—22no—
111C83⋊C8—22no—
112C22×C83—21no—
113C22×C83—21no—
114C22×C83—21no—
115C22×C83—22no—
116C22×C83—22no—
117C22×C83—22no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—84no—
122C22×C83C22×D411yescharacteristic
123C4×C83—21no—
124C4×C83—21no—
125C4×C83—21no—
126C4×C83—42no—
127C4×C83H(5,𝔽2)11yescharacteristic
128C4×C83H(5,𝔽2)11yescharacteristic
129Dic83—42no—
130Dic83—42no—
131C2×C83—41no—
132C2×C83—41no—
133C2×C83—41no—
134C2×C83—82no—
135C2×C83#53411yescharacteristic
136C2×C83#57712yes—
137#3200—831nomaximal, Sylow
138C83#320011yescharacteristic, complemented, Sylow
139#536—831no—
140#535D8311yescharacteristic
141#485—832no—
142#482—832no—
143#468—832no—
144#465—832no—
145#450—832no—
146#448—832no—
147#408—831no—
148C2×#41C2×D8311yescharacteristic
149C22×Q8—831no—
150C22×D4C2×D8311yescharacteristic
151C2×SD16—832no—
152C2×SD16—832no—
153C2×M16—831no—
154C22×C8—831no—
155C4⋊Q8—1661no—
156#127—21no—
157#124—21no—
158#124—1661no—
159#123—1661no—
160#123—1661no—
161C22⋊Q8—832no—
162C22⋊Q8—832no—
163#121C2×D8312yes—
164#121C2×D8312yes—
165#120—21no—
166#120—21no—
167#117—831no—
168#117C2×D8311yescharacteristic
169#107—832no—
170#106—832no—
171#103—832no—
172#103—832no—
173#102—832no—
174#102—832no—
175C22⋊C8—832no—
176C22⋊C8—832no—
177C24—21no—
178#41—41no—
179#41—41no—
180C2×Q8—3321no—
181C2×Q8—3321no—
182C2×Q8—832no—
183C2×Q8—1661no—
184C2×Q8C22⋊D8311yescharacteristic
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C22⋊D8311yescharacteristic
191C2×D4C22⋊D8311yescharacteristic
192C2×D4C22⋊D8311yescharacteristic
193C2×D4C22×D8312yes—
194C22×C4—1661no—
195C22×C4C22×D8311yescharacteristic
196C22×C4C22×D8311yescharacteristic
197C22×C4C22×D8311yescharacteristic
198SD16—3322no—
199SD16—3322no—
200M16—3321no—
201C2×C8—3321no—
202C2×C8—832no—
203C2×C8—832no—
204C4⋊C4—832no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C4⋊C4C22×D8312yes—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—1661no—
212C22⋊C4—1662no—
213C22⋊C4—1662no—
214C42—21no—
215C42—1661no—
216C23—22no—
217C23—42no—
218C23—82no—
219C23C2×C22⋊D8311yescharacteristic
220C23C2×C22⋊D8311yescharacteristic
221C23C2×C22⋊D8311yescharacteristic
222Q8—41no—
223Q8—3322no—
224Q8—1662no—
225Q8—1662no—
226D4—22no—
227D4—22no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—42no—
232D4—82no—
233D4—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—3322no—
242C2×C4—1662no—
243C2×C4—1662no—
244C2×C4C2×C22⋊D8311yescharacteristic
245C2×C4C2×C22⋊D8311yescharacteristic
246C2×C4C2×C22⋊D8311yescharacteristic
247C2×C4C23×D8311yescharacteristic
248C8—1662no—
249C8—1662no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—84no—
260C22C22×C22⋊D8311yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—3322no—
266C4—3322no—
267C4#1458183711yescharacteristic
268C4#1458183711yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—82no—
273C2#1912826011yescharacteristic
274C2#1912851212yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#535×C83C2
socleC22×C83C22×D4
hypercenter#535D83
hypocenterC83#3200
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #19128272C2semidirect
2. #19128042C2semidirect
3. #19128022C2semidirect
4. #19127945C2semidirect
5. #19127928C2semidirect
6. #19127852C2semidirect
7. #19127839C2semidirect
8. #19127668C2semidirect
10. C22×Dic166C22semidirect
13. C2×#13050574C22semidirect
14. C22×C83⋊C8C22semidirect
20. #14581771C22semidirect
21. #14581751C22semidirect
22. #14581725C22semidirect
23. #14581723C22semidirect
24. #14581715C22semidirect
25. #14581715C22semidirect
29. C22⋊(C83⋊C8)C22semidirect
138. C83#3200semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#191282722
#191280422
#191280222
#191279452
#191279282
#191278522
#191278392
#191276682
#535×C832
#320083