Finite Group Databasebeta

← #24914510

Subgroups — #24914510

Order 10624.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 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#19127628C212yesmaximal, complemented
2#19127614C212yesmaximal, complemented
3#19127613C211yesmaximal, characteristic, complemented
4#19127594C212yesmaximal, complemented
5#19127593C212yesmaximal, complemented
6#19127591C212yesmaximal, complemented
7#19127556C212yesmaximal
8#19127556C211yesmaximal, characteristic
9#397×C83C211yesmaximal, characteristic, complemented
10C24×D83C2211yescharacteristic
11C22×Dic166C2211yescharacteristic, complemented
12C2×C4⋊Dic83C2212yescomplemented
13C2×C22⋊Dic83C2211yescharacteristic
14C2×C22⋊Dic83C2212yes—
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×C4×Dic83C2212yescomplemented
22C83⋊#95C2212yescomplemented
23C83⋊#95C2212yescomplemented
24C83⋊#95C2211yescharacteristic, complemented
25C83⋊#95C2212yescomplemented
26C2×C4⋊C4×C83C2211yescharacteristic, complemented
27C2×C22⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2211yescharacteristic
29#95×C83C2212yescomplemented
30#95×C83C2211yescharacteristic, complemented
31C23×D83—22no—
32C23×D83—24no—
33C23×D83—41no—
34C23×D83—42no—
35C23×D83—41no—
36C23×D83C2312yes—
37C2×Dic166—41no—
38C2×Dic166—42no—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—42no—
43C22⋊Dic83—41no—
44C22⋊Dic83—42no—
45(C2×D83)⋊C4—24no—
46(C2×D83)⋊C4—24no—
47(C2×D83)⋊C4—42no—
48(C2×D83)⋊C4—42no—
49(C2×D83)⋊C4—42no—
50(C2×D83)⋊C4—42no—
51C4×Dic83—42no—
52C24×C83C2311yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2312yes—
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2312yes—
57C4⋊C4×C83—41no—
58C22⋊C4×C83—24no—
59C22⋊C4×C83—41no—
60C23×C83—22no—
61C23×C83—21no—
62C23×C83—21no—
63C23×C83—22no—
64C23×C83#4112yes—
65C23×C83C2411yescharacteristic
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—22no—
72C22×D83—44no—
73C22×D83—48no—
74C22×D83—48no—
75C22×D83—44no—
76C22×D83—44no—
77C22×D83#4114yes—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—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—22no—
99C22×C83—24no—
100C22×C83—42no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83C2×#4112yes—
104C22×C83C2×#4111yescharacteristic
105C22×C83H(5,𝔽2)11yescharacteristic
106C22×C83H(5,𝔽2)11yescharacteristic
107C22×C83H(5,𝔽2)12yes—
108C2×D83—24no—
109C2×D83—24no—
110C2×D83—24no—
111C2×D83—416no—
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—42no—
115C2×D83—42no—
116C4×C83—42no—
117C4×C83—42no—
118C4×C83—41no—
119C4×C83—41no—
120Dic83—42no—
121Dic83—42no—
122Dic83—42no—
123C2×C83—42no—
124C2×C83#53811yescharacteristic
125C2×C83#53812yes—
126C2×C83#55112yes—
127C2×C83#55111yescharacteristic
128C2×C83H(3,𝔽4)11yescharacteristic
129D83—44no—
130#2736—831nomaximal, Sylow
131C83#273611yescharacteristic, complemented, Sylow
132#397—832no—
133#397D8311yescharacteristic, complemented
134#393—831no—
135#393—832no—
136#388—832no—
137#388—832no—
138#388—832no—
139#379—831no—
140#379—832no—
141C25—831no—
142C22×Q8—831no—
143C2×C4⋊C4—832no—
144C2×C4⋊C4C2×D8311yescharacteristic, complemented
145C2×C22⋊C4—831no—
146C2×C22⋊C4—832no—
147C2×C22⋊C4—832no—
148C2×C22⋊C4—832no—
149C2×C22⋊C4—832no—
150C2×C22⋊C4—832no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C22⋊C4C2×D8312yes—
154C2×C22⋊C4C2×D8311yescharacteristic
155C2×C42—832no—
156#95—832no—
157#95—831no—
158#95—832no—
159#95—832no—
160#95C2×D8312yescomplemented
161#95C2×D8311yescharacteristic, complemented
162C24—3321no—
163C24—3321no—
164C24—3322no—
165C24—832no—
166C24—1664no—
167C24—1662no—
168C24C22×D8311yescharacteristic
169C2×Q8—3321no—
170C2×Q8—3322no—
171C22×C4—832no—
172C22×C4—832no—
173C22×C4—832no—
174C22×C4C22×D8311yescharacteristic
175C22×C4C22×D8312yes—
176C22×C4C22×D8311yescharacteristic
177C22×C4C22×D8312yes—
178C4⋊C4—41no—
179C4⋊C4—3322no—
180C22⋊C4—24no—
181C22⋊C4—41no—
182C22⋊C4—3322no—
183C22⋊C4—3322no—
184C22⋊C4—3322no—
185C22⋊C4—3322no—
186C22⋊C4—3321no—
187C22⋊C4—3322no—
188C22⋊C4—1664no—
189C22⋊C4—1664no—
190C42—3322no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23—3324no—
196C23—3328no—
197C23—3324no—
198C23—3328no—
199C23—3324no—
200C23—834no—
201C23—1662no—
202C23—1662no—
203C23—1662no—
204C23—1662no—
205C23—1662no—
206C23—1662no—
207C23#1305058912yes—
208C23C23×D8311yescharacteristic
209Q8—3324no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
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—22no—
231C22—41no—
232C22—41no—
233C22—42no—
234C22—33216no—
235C22—3322no—
236C22—3322no—
237C22—3322no—
238C22—3322no—
239C22—1664no—
240C22—1664no—
241C22—1664no—
242C22#41×D8312yes—
243C22C2×#1305058911yescharacteristic
244C22#1458183711yescharacteristic
245C22#1458183812yes—
246C22#1458183811yescharacteristic
247C4—42no—
248C4—41no—
249C4—41no—
250C4—42no—
251C4—3322no—
252C4—3322no—
253C4—3322no—
254C2—42no—
255C2—3324no—
256C2#1912828811yescharacteristic
257C2#1912829012yes—
258C2#1912837412yes—
259C2#1912837911yescharacteristic
260C2#1912844311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC24×D83C22
hypercenter#397D83
hypocenterC83#2736
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. #19127628C2semidirect
2. #19127614C2semidirect
3. #19127613C2semidirect
4. #19127594C2semidirect
5. #19127593C2semidirect
6. #19127591C2semidirect
9. #397×C83C2semidirect
11. C22×Dic166C22semidirect
12. C2×C4⋊Dic83C22semidirect
21. C2×C4×Dic83C22semidirect
22. C83⋊#95C22semidirect
23. C83⋊#95C22semidirect
24. C83⋊#95C22semidirect
25. C83⋊#95C22semidirect
26. C2×C4⋊C4×C83C22semidirect
29. #95×C83C22semidirect
30. #95×C83C22semidirect
131. C83#2736semidirect
133. #397D83semidirect
144. C2×C4⋊C4C2×D83semidirect
160. #95C2×D83semidirect
161. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191276282
#191276142
#191276132
#191275942
#191275932
#191275912
#191275562
#397×C832
#273683