Finite Group Databasebeta

← #24914509

Subgroups — #24914509

Order 10624.

560 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 125 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#19127629C212yesmaximal, complemented
2#19127626C211yesmaximal, characteristic, complemented
3#19127613C212yesmaximal, complemented
4#19127613C211yesmaximal, characteristic, complemented
5#19127595C212yesmaximal, complemented
6#19127594C212yesmaximal, complemented
7#19127589C212yesmaximal, complemented
8#19127557C212yesmaximal
9#379×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic, complemented
11C2×C4⋊Dic83C2211yescharacteristic, complemented
12C2×Dic83⋊C4C2212yescomplemented
13C2×C22⋊Dic83C2212yes—
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C4×Dic83C2212yescomplemented
20C83⋊#95C2212yescomplemented
21C83⋊#95C2212yescomplemented
22C83⋊#95C2212yescomplemented
23C83⋊#95C2211yescharacteristic, complemented
24C83⋊#95C2212yescomplemented
25C83⋊#95C2211yescharacteristic, complemented
26C25×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2211yescharacteristic
31C2×Dic166—42no—
32C2×Dic166—41no—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C4⋊Dic83—41no—
38Dic83⋊C4—42no—
39C22⋊Dic83—24no—
40C22⋊Dic83—24no—
41C22⋊Dic83—42no—
42C22⋊Dic83—42no—
43C22⋊Dic83—42no—
44C22⋊Dic83—42no—
45C4×Dic83—42no—
46C24×C83—24no—
47C24×C83—22no—
48C24×C83—41no—
49C24×C83—42no—
50C24×C83—41no—
51C24×C83C2312yes—
52C24×C83C2311yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2312yes—
56C22⋊C4×C83—24no—
57C22⋊C4×C83—41no—
58C22⋊C4×C83—42no—
59C22⋊C4×C83—41no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83—22no—
63C23×C83—21no—
64C23×C83—21no—
65C23×C83—22no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—48no—
71C23×C83—44no—
72C23×C83—48no—
73C23×C83—44no—
74C23×C83—44no—
75C23×C83#4114yes—
76C23×C83#4112yes—
77C23×C83C2411yescharacteristic
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97Dic166—44no—
98C22×C83—24no—
99C22×C83—22no—
100C22×C83—24no—
101C22×C83—24no—
102C22×C83—24no—
103C22×C83—416no—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—41no—
109C22×C83—41no—
110C22×C83—42no—
111C22×C83C2×#4112yes—
112C22×C83C2×#4111yescharacteristic
113C22×C83H(5,𝔽2)11yescharacteristic
114C22×C83H(5,𝔽2)12yes—
115C22×C83H(5,𝔽2)11yescharacteristic
116C4×C83—42no—
117C4×C83—41no—
118C4×C83—41no—
119Dic83—42no—
120Dic83—42no—
121Dic83—42no—
122Dic83—42no—
123C2×C83—44no—
124C2×C83—42no—
125C2×C83#53812yes—
126C2×C83#53811yescharacteristic
127C2×C83#55112yes—
128C2×C83#55111yescharacteristic
129C2×C83H(3,𝔽4)11yescharacteristic
130#2736—831nomaximal, Sylow
131C83#273611yescharacteristic, complemented, Sylow
132#397—831no—
133#397—832no—
134#393—831no—
135#393—832no—
136#388—832no—
137#388—832no—
138#388—832no—
139#379—832no—
140#379D8311yescharacteristic
141C25C2×D8311yescharacteristic
142C22×Q8—831no—
143C2×C4⋊C4—832no—
144C2×C4⋊C4—831no—
145C2×C22⋊C4—832no—
146C2×C22⋊C4—832no—
147C2×C22⋊C4—832no—
148C2×C22⋊C4—832no—
149C2×C22⋊C4—832no—
150C2×C22⋊C4—832no—
151C2×C22⋊C4C2×D8312yes—
152C2×C22⋊C4C2×D8311yescharacteristic
153C2×C22⋊C4C2×D8312yes—
154C2×C22⋊C4C2×D8311yescharacteristic
155C2×C42—832no—
156#95—832no—
157#95—831no—
158#95—832no—
159#95—832no—
160#95—832no—
161#95—831no—
162C24—24no—
163C24—22no—
164C24—42no—
165C24—41no—
166C24—41no—
167C24C22×D8312yes—
168C24C22×D8311yescharacteristic
169C2×Q8—3322no—
170C2×Q8—3321no—
171C22×C4—832no—
172C22×C4—832no—
173C22×C4—832no—
174C22×C4—832no—
175C22×C4C22×D8311yescharacteristic
176C22×C4C22×D8311yescharacteristic
177C22×C4C22×D8312yes—
178C4⋊C4—3322no—
179C4⋊C4—3321no—
180C22⋊C4—24no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—3322no—
185C22⋊C4—3322no—
186C22⋊C4—3322no—
187C22⋊C4—3322no—
188C22⋊C4—1664no—
189C22⋊C4—1664no—
190C42—3322no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—48no—
202C23—44no—
203C23—48no—
204C23—44no—
205C23—44no—
206C23#1305058914yes—
207C23#1305059012yes—
208C23C23×D8311yescharacteristic
209Q8—3324no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4—1662no—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—24no—
233C22—22no—
234C22—416no—
235C22—42no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22C2×#1305058912yes—
243C22C2×#1305059011yescharacteristic
244C22#1458183712yes—
245C22#1458183711yescharacteristic
246C22#1458183711yescharacteristic
247C4—41no—
248C4—41no—
249C4—42no—
250C4—3322no—
251C4—3322no—
252C4—3322no—
253C4—3322no—
254C2—44no—
255C2—42no—
256C2#1912828612yes—
257C2#1912828911yescharacteristic
258C2#1912837712yes—
259C2#1912838211yescharacteristic
260C2#1912844411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#379×C83C2
prime radicalC25×C83C22
hypercenter#379D83
hypocenterC83#2736
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #19127629C2semidirect
2. #19127626C2semidirect
3. #19127613C2semidirect
4. #19127613C2semidirect
5. #19127595C2semidirect
6. #19127594C2semidirect
7. #19127589C2semidirect
10. C22×Dic166C22semidirect
11. C2×C4⋊Dic83C22semidirect
12. C2×Dic83⋊C4C22semidirect
19. C2×C4×Dic83C22semidirect
20. C83⋊#95C22semidirect
21. C83⋊#95C22semidirect
22. C83⋊#95C22semidirect
23. C83⋊#95C22semidirect
24. C83⋊#95C22semidirect
25. C83⋊#95C22semidirect
131. C83#2736semidirect

Maximal subgroups

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

subgroupindex
#191276292
#191276262
#191276132
#191275952
#191275942
#191275892
#191275572
#379×C832
#273683