Finite Group Databasebeta

← #24911732

Subgroups — #24911732

Order 10624.

672 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 139 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×#14581750C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D83C211yesmaximal, characteristic
3#19127638C211yesmaximal, characteristic, complemented
4#19127637C211yesmaximal, characteristic, complemented
5#19127627C211yesmaximal, characteristic, complemented
6#19127594C212yesmaximal, complemented
7#19127590C212yesmaximal, complemented
8#19127575C212yesmaximal, complemented
9#19127563C211yesmaximal, characteristic, complemented
10#19127556C212yesmaximal
11#397×C83C211yesmaximal, characteristic, complemented
12C24×D83C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581750—24no—
16C2×C4⋊Dic83C2211yescharacteristic, complemented
17C2×C4⋊Dic83C2212yescomplemented
18C2×Dic83⋊C4C2212yescomplemented
19C2×C22⋊Dic83C2211yescharacteristic
20C2×C22⋊Dic83C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C22⋊C4×D83—24no—
28C2×C4×Dic83C2212yescomplemented
29C83⋊#95C2212yescomplemented
30C83⋊#95C2211yescharacteristic, complemented
31C83⋊#95C2212yescomplemented
32C2×C4⋊C4×C83C2211yescharacteristic, complemented
33C2×C22⋊C4×C83C2212yes—
34C2×C22⋊C4×C83C2211yescharacteristic
35#95×C83C2211yescharacteristic, complemented
36#95×C83C2212yescomplemented
37C23×D83—22no—
38C23×D83—22no—
39C23×D83—22no—
40C23×D83—24no—
41C23×D83—42no—
42C23×D83C2311yescharacteristic
43C23×D83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C2×C4×D83—24no—
49C2×C4×D83—22no—
50C2×C4×D83—42no—
51C2×C4×D83D414yes—
52C4⋊Dic83—22no—
53C4⋊Dic83—42no—
54Dic83⋊C4—42no—
55C22⋊Dic83—22no—
56C22⋊Dic83—42no—
57(C2×D83)⋊C4—24no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—42no—
63C4×Dic83—24no—
64C24×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2312yes—
67C22×C4×C83C2312yes—
68C22×C4×C83C2311yescharacteristic
69C4⋊C4×C83—22no—
70C22⋊C4×C83—24no—
71C22⋊C4×C83—22no—
72C23×C83—21no—
73C23×C83—21no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83#4112yes—
77C23×C83C2411yescharacteristic
78C22×D83—21no—
79C22×D83—21no—
80C22×D83—22no—
81C22×D83—28no—
82C22×D83—21no—
83C22×D83—22no—
84C22×D83—21no—
85C22×D83—22no—
86C22×D83—48no—
87C22×D83—48no—
88C22×D83—44no—
89C22×D83—44no—
90C22×D83C2×D412yes—
91C22×D83#4112yes—
92C22×D83#4114yes—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83C2×D412yes—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83C2×D412yes—
113C4×D83—28no—
114C4×D83—44no—
115C22×C83—22no—
116C22×C83—24no—
117C22×C83—41no—
118C22×C83—41no—
119C22×C83—42no—
120C22×C83C22×D411yescharacteristic
121C22×C83C2×#4112yes—
122C22×C83C2×#4111yescharacteristic
123C22×C83C2×#4111yescharacteristic
124C22×C83H(5,𝔽2)12yes—
125C2×D83—22no—
126C2×D83—22no—
127C2×D83—24no—
128C2×D83—24no—
129C2×D83—22no—
130C2×D83—24no—
131C2×D83—416no—
132C2×D83—41no—
133C2×D83—41no—
134C2×D83—42no—
135C2×D83#12314yes—
136C4×C83—22no—
137C4×C83—42no—
138C4×C83—41no—
139C4×C83—42no—
140Dic83—22no—
141Dic83—41no—
142Dic83—42no—
143Dic83—42no—
144C2×C83—42no—
145C2×C83C2×#12311yescharacteristic
146C2×C83#52511yescharacteristic
147C2×C83#53511yescharacteristic
148C2×C83#55112yes—
149C2×C83#55112yes—
150D83—24no—
151D83—42no—
152#2482—831nomaximal, Sylow
153C83#248211yescharacteristic, complemented, Sylow
154C2×#117—831no—
155C22×C22⋊C4—831no—
156#399—831no—
157#399—831no—
158#397—831no—
159#397D8311yescharacteristic, complemented
160#388—832no—
161#388—832no—
162#385—832no—
163#381—831no—
164#379—832no—
165C25—831no—
166C23×C4—831no—
167C23×C4—831no—
168#117—1664no—
169C2×C4⋊C4—832no—
170C2×C4⋊C4—832no—
171C2×C4⋊C4—831no—
172C2×C4⋊C4C2×D8311yescharacteristic, complemented
173C2×C22⋊C4—832no—
174C2×C22⋊C4—832no—
175C2×C22⋊C4—831no—
176C2×C22⋊C4—831no—
177C2×C22⋊C4—831no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—832no—
180C2×C22⋊C4—832no—
181C2×C22⋊C4—1664no—
182C2×C22⋊C4C2×D8311yescharacteristic
183C2×C22⋊C4C2×D8312yes—
184C2×C42—832no—
185#95—832no—
186#95—831no—
187#95—832no—
188#95C2×D8311yescharacteristic, complemented
189#95C2×D8312yescomplemented
190C24—3322no—
191C24—831no—
192C24—831no—
193C24—1664no—
194C24—1662no—
195C24—1662no—
196C24—1662no—
197C24C22×D8311yescharacteristic
198C22×C4—3322no—
199C22×C4—831no—
200C22×C4—834no—
201C22×C4—832no—
202C22×C4—831no—
203C22×C4—832no—
204C22×C4—1662no—
205C22×C4—1664no—
206C22×C4C22×D8312yes—
207C22×C4C22×D8312yes—
208C22×C4C22×D8311yescharacteristic
209C22×C4C22×D8311yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—3322no—
212C4⋊C4—3322no—
213C4⋊C4—1662no—
214C22⋊C4—24no—
215C22⋊C4—22no—
216C22⋊C4—3322no—
217C22⋊C4—3322no—
218C22⋊C4—1662no—
219C22⋊C4—1664no—
220C22⋊C4—1664no—
221C22⋊C4—1664no—
222C22⋊C4—1662no—
223C22⋊C4—1662no—
224C42—1664no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—3328no—
230C23—3324no—
231C23—3324no—
232C23—3328no—
233C23—832no—
234C23—834no—
235C23—832no—
236C23—1668no—
237C23—1661no—
238C23—1662no—
239C23—1661no—
240C23—1662no—
241C23—1661no—
242C23—1661no—
243C23—1662no—
244C23#1305058912yes—
245C23C23×D8311yescharacteristic
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3324no—
256C2×C4—832no—
257C2×C4—1668no—
258C2×C4—1662no—
259C2×C4—1661no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4D4×D8312yes—
268C22—24no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—33216no—
274C22—3322no—
275C22—3321no—
276C22—3321no—
277C22—834no—
278C22—1664no—
279C22—1662no—
280C22—1664no—
281C22—1664no—
282C22—1662no—
283C22—1662no—
284C22C2×D4×D8311yescharacteristic
285C22#41×D8312yes—
286C22#41×D8311yescharacteristic
287C22C2×#1305058911yescharacteristic
288C22#1458183812yes—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—41no—
293C4—3322no—
294C4—3321no—
295C4—3322no—
296C4—1662no—
297C2—42no—
298C2—3322no—
299C2—1664no—
300C2#123×D8311yescharacteristic
301C2#1912821111yescharacteristic
302C2#1912826911yescharacteristic
303C2#1912837412yes—
304C2#1912837612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
prime radicalC24×D83C22
hypercenter#397D83
hypocenterC83#2482
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14581750C2semidirect
3. #19127638C2semidirect
4. #19127637C2semidirect
5. #19127627C2semidirect
6. #19127594C2semidirect
7. #19127590C2semidirect
8. #19127575C2semidirect
9. #19127563C2semidirect
11. #397×C83C2semidirect
16. C2×C4⋊Dic83C22semidirect
17. C2×C4⋊Dic83C22semidirect
18. C2×Dic83⋊C4C22semidirect
28. C2×C4×Dic83C22semidirect
29. C83⋊#95C22semidirect
30. C83⋊#95C22semidirect
31. C83⋊#95C22semidirect
32. C2×C4⋊C4×C83C22semidirect
35. #95×C83C22semidirect
36. #95×C83C22semidirect
153. C83#2482semidirect
159. #397D83semidirect
172. C2×C4⋊C4C2×D83semidirect
188. #95C2×D83semidirect
189. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817502
C2×C22⋊C4×D832
#191276382
#191276372
#191276272
#191275942
#191275902
#191275752
#191275632
#191275562
#397×C832
#248283