Finite Group Databasebeta

← #24912617

Subgroups — #24912617

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×#14581768C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D83C211yesmaximal, characteristic
3#19127628C211yesmaximal, characteristic, complemented
4#19127626C211yesmaximal, characteristic, complemented
5#19127612C211yesmaximal, characteristic, complemented
6#19127590C212yesmaximal, complemented
7#19127589C212yesmaximal, complemented
8#19127563C212yesmaximal, complemented
9#19127556C212yesmaximal
10#19127548C211yesmaximal, characteristic, complemented
11#393×C83C211yesmaximal, characteristic, complemented
12C24×D83C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581768—24no—
16C2×C4⋊Dic83C2211yescharacteristic, complemented
17C2×Dic83⋊C4C2212yescomplemented
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2212yes—
26C22⋊C4×D83—24no—
27C2×C4×Dic83C2212yescomplemented
28C83⋊#95C2212yescomplemented
29C83⋊#95C2212yescomplemented
30C83⋊#95C2212yescomplemented
31C83⋊#95C2211yescharacteristic, complemented
32C22×Q8×C83C2211yescharacteristic, complemented
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35#95×C83C2212yescomplemented
36#95×C83C2211yescharacteristic, complemented
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—22no—
49C2×C4×D83—24no—
50C2×C4×D83—42no—
51C2×C4×D83D414yes—
52C4⋊Dic83—22no—
53Dic83⋊C4—24no—
54C22⋊Dic83—24no—
55C22⋊Dic83—22no—
56(C2×D83)⋊C4—24no—
57(C2×D83)⋊C4—24no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—42no—
62C4×Dic83—42no—
63C24×C83C2311yescharacteristic
64C2×Q8×C83—22no—
65C2×Q8×C83—42no—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2312yes—
68C22×C4×C83C2312yes—
69C22×C4×C83C2311yescharacteristic
70C22⋊C4×C83—22no—
71C22⋊C4×C83—42no—
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—
93Q8×C83—44no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
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×C83—22no—
103C2×C4×C83C2×D412yes—
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×Dic83—22no—
113C2×Dic83C2×D412yes—
114C4×D83—44no—
115C4×D83—44no—
116C22×C83—22no—
117C22×C83—24no—
118C22×C83—41no—
119C22×C83—41no—
120C22×C83—42no—
121C22×C83C22×D411yescharacteristic
122C22×C83C2×#4111yescharacteristic
123C22×C83C2×#4112yes—
124C22×C83C2×#4111yescharacteristic
125C22×C83H(5,𝔽2)12yes—
126C2×D83—22no—
127C2×D83—22no—
128C2×D83—24no—
129C2×D83—24no—
130C2×D83—22no—
131C2×D83—24no—
132C2×D83—416no—
133C2×D83—41no—
134C2×D83—41no—
135C2×D83—42no—
136C2×D83#12414yes—
137C4×C83—22no—
138C4×C83—41no—
139C4×C83—42no—
140C4×C83—42no—
141Dic83—22no—
142Dic83—42no—
143Dic83—41no—
144Dic83—42no—
145C2×C83—42no—
146C2×C83#52511yescharacteristic
147C2×C83C2×#12411yescharacteristic
148C2×C83#53411yescharacteristic
149C2×C83#55112yes—
150C2×C83#55112yes—
151D83—24no—
152D83—42no—
153#2561—831nomaximal, Sylow
154C83#256111yescharacteristic, complemented, Sylow
155C2×C22⋊Q8—831no—
156C22×C22⋊C4—831no—
157#397—831no—
158#397—831no—
159#393—831no—
160#393D8311yescharacteristic, complemented
161#388—832no—
162#388—832no—
163#381—832no—
164#379—832no—
165C4×C22⋊C4—831no—
166C25—831no—
167C22×Q8C2×D8311yescharacteristic, complemented
168C23×C4—831no—
169C23×C4—831no—
170C22⋊Q8—1664no—
171C2×C4⋊C4—832no—
172C2×C4⋊C4—831no—
173C2×C22⋊C4—832no—
174C2×C22⋊C4—831no—
175C2×C22⋊C4—832no—
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—831no—
186#95—832no—
187#95—832no—
188#95—832no—
189#95C2×D8311yescharacteristic, complemented
190#95C2×D8312yescomplemented
191C24—3322no—
192C24—831no—
193C24—831no—
194C24—1664no—
195C24—1662no—
196C24—1662no—
197C24—1662no—
198C24C22×D8311yescharacteristic
199C2×Q8—22no—
200C2×Q8—42no—
201C22×C4—3322no—
202C22×C4—834no—
203C22×C4—831no—
204C22×C4—831no—
205C22×C4—832no—
206C22×C4—832no—
207C22×C4—1662no—
208C22×C4—1664no—
209C22×C4C22×D8311yescharacteristic
210C22×C4C22×D8312yes—
211C22×C4C22×D8312yes—
212C22×C4C22×D8311yescharacteristic
213C4⋊C4—1664no—
214C4⋊C4—1662no—
215C22⋊C4—22no—
216C22⋊C4—42no—
217C22⋊C4—3322no—
218C22⋊C4—1662no—
219C22⋊C4—1664no—
220C22⋊C4—1664no—
221C22⋊C4—1664no—
222C22⋊C4—1662no—
223C22⋊C4—1664no—
224C22⋊C4—1662no—
225C42—3322no—
226C23—21no—
227C23—22no—
228C23—21no—
229C23—22no—
230C23—3328no—
231C23—3324no—
232C23—3324no—
233C23—3328no—
234C23—832no—
235C23—834no—
236C23—832no—
237C23—1668no—
238C23—1661no—
239C23—1662no—
240C23—1661no—
241C23—1662no—
242C23—1661no—
243C23—1661no—
244C23—1662no—
245C23#1305059112yes—
246C23C23×D8311yescharacteristic
247Q8—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—3324no—
258C2×C4—3324no—
259C2×C4—832no—
260C2×C4—1661no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4D4×D8312yes—
270C22—24no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—33216no—
276C22—3322no—
277C22—3321no—
278C22—3321no—
279C22—834no—
280C22—1664no—
281C22—1662no—
282C22—1664no—
283C22—1664no—
284C22—1662no—
285C22—1662no—
286C22C2×D4×D8311yescharacteristic
287C22#41×D8312yes—
288C22#41×D8311yescharacteristic
289C22C2×#1305059111yescharacteristic
290C22#1458183712yes—
291C4—22no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—3322no—
296C4—3322no—
297C4—3321no—
298C4—1662no—
299C2—42no—
300C2—3322no—
301C2—1664no—
302C2#1912821411yescharacteristic
303C2#124×D8311yescharacteristic
304C2#1912826111yescharacteristic
305C2#1912837812yes—
306C2#1912838112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#393×C83C2
prime radicalC24×D83C22
hypercenter#393D83
hypocenterC83#2561
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14581768C2semidirect
3. #19127628C2semidirect
4. #19127626C2semidirect
5. #19127612C2semidirect
6. #19127590C2semidirect
7. #19127589C2semidirect
8. #19127563C2semidirect
10. #19127548C2semidirect
11. #393×C83C2semidirect
16. C2×C4⋊Dic83C22semidirect
17. C2×Dic83⋊C4C22semidirect
27. C2×C4×Dic83C22semidirect
28. C83⋊#95C22semidirect
29. C83⋊#95C22semidirect
30. C83⋊#95C22semidirect
31. C83⋊#95C22semidirect
32. C22×Q8×C83C22semidirect
35. #95×C83C22semidirect
36. #95×C83C22semidirect
154. C83#2561semidirect
160. #393D83semidirect
167. C22×Q8C2×D83semidirect
189. #95C2×D83semidirect
190. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
C2×C22⋊C4×D832
#191276282
#191276262
#191276122
#191275902
#191275892
#191275632
#191275562
#191275482
#393×C832
#256183