Finite Group Databasebeta

← #24915461

Subgroups — #24915461

Order 10624.

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

normal subgroups only

This group has 712 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
1C2×#14581762C212yesmaximal, complemented
2#19127630C212yesmaximal, complemented
3#19127619C212yesmaximal, complemented
4#19127618C212yesmaximal, complemented
5#19127616C212yesmaximal, complemented
6#19127611C212yesmaximal, complemented
7#19127610C211yesmaximal, characteristic, complemented
8#19127556C211yesmaximal, characteristic, complemented
9#397×C83C211yesmaximal, characteristic, complemented
10C24×D83C2211yescharacteristic
11C22×D332C2212yes—
12C22×D332C2212yes—
13C22×D332C2211yescharacteristic, complemented
14C22×C22⋊D83C2212yes—
15#14581762—28no—
16C2×Dic83⋊C4C2212yescomplemented
17C2×C22⋊Dic83C2211yescharacteristic
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yescomplemented
23C2×(C2×D83)⋊C4C2212yescomplemented
24C2×(C2×D83)⋊C4C2212yescomplemented
25C83⋊#95C2211yescharacteristic, complemented
26C83⋊#95C2212yescomplemented
27C2×C4⋊C4×C83C2211yescharacteristic, complemented
28C2×C22⋊C4×C83C2212yes—
29C2×C22⋊C4×C83C2211yescharacteristic
30#95×C83C2212yescomplemented
31#95×C83C2211yescharacteristic, complemented
32C23×D83—42no—
33C23×D83—42no—
34C23×D83—41no—
35C23×D83—41no—
36C23×D83D414yescomplemented
37C23×D83C2312yes—
38C23×D83C2312yes—
39C2×D332—24no—
40C2×D332—24no—
41C2×D332—42no—
42C2×D332—42no—
43C2×D332—42no—
44C2×D332—42no—
45C2×D332—41no—
46C2×D332—42no—
47C2×C22⋊D83—24no—
48C2×C22⋊D83—42no—
49C2×C22⋊D83—42no—
50C22×Dic83C2312yes—
51C22×Dic83C2312yes—
52Dic83⋊C4—42no—
53C22⋊Dic83—41no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—24no—
56(C2×D83)⋊C4—42no—
57(C2×D83)⋊C4—42no—
58(C2×D83)⋊C4—42no—
59(C2×D83)⋊C4—42no—
60(C2×D83)⋊C4—42no—
61C24×C83C2311yescharacteristic
62C22×C4×C83C2312yes—
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—41no—
67C22⋊C4×C83—24no—
68C22⋊C4×C83—41no—
69C23×C83—21no—
70C23×C83—21no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83C2×D412yes—
74C23×C83C2411yescharacteristic
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—22no—
78C22×D83—22no—
79C22×D83—22no—
80C22×D83—22no—
81C22×D83—22no—
82C22×D83—22no—
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—44no—
89C22×D83—48no—
90C22×D83—44no—
91C22×D83—48no—
92C22×D83—44no—
93C22×D83C2×D414yes—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—22no—
104C22⋊D83—48no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111D332—48no—
112D332—48no—
113D332—44no—
114C22×C83—24no—
115C22×C83—22no—
116C22×C83—41no—
117C22×C83—42no—
118C22×C83—41no—
119C22×C83C22×D411yescharacteristic
120C22×C83H(5,𝔽2)11yescharacteristic
121C22×C83H(5,𝔽2)12yes—
122C22×C83H(5,𝔽2)11yescharacteristic
123C22×C83H(5,𝔽2)12yes—
124C2×D83—24no—
125C2×D83—24no—
126C2×D83—24no—
127C2×D83—416no—
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—42no—
137C2×D83—42no—
138C2×D83—42no—
139C4×C83—42no—
140C4×C83—42no—
141C4×C83—41no—
142C4×C83—41no—
143Dic83—42no—
144Dic83—42no—
145C2×C83—42no—
146C2×C83#53411yescharacteristic
147C2×C83#53412yes—
148C2×C83#56011yescharacteristic
149C2×C83#56012yes—
150C2×C83H(3,𝔽4)11yescharacteristic
151D83—44no—
152D83—82no—
153#2812—831nomaximal, Sylow
154C83#281211yescharacteristic, complemented, Sylow
155C2×#120—832no—
156#397—832no—
157#397D8311yescharacteristic, complemented
158#394—832no—
159#394—832no—
160#394—832no—
161#392—831no—
162#392—832no—
163#379—831no—
164C25—831no—
165C22×D4—832no—
166C22×D4—831no—
167C22×D4—832no—
168C22×D4—832no—
169#120—1668no—
170C2×C4⋊C4—832no—
171C2×C4⋊C4C2×D8311yescharacteristic, complemented
172C2×C22⋊C4—831no—
173C2×C22⋊C4—832no—
174C2×C22⋊C4—832no—
175C2×C22⋊C4—832no—
176C2×C22⋊C4—832no—
177C2×C22⋊C4—832no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—832no—
180C2×C22⋊C4C2×D8312yes—
181C2×C22⋊C4C2×D8311yescharacteristic
182#95—831no—
183#95—832no—
184#95C2×D8312yescomplemented
185#95C2×D8311yescharacteristic, complemented
186C24—3321no—
187C24—3321no—
188C24—3322no—
189C24—3322no—
190C24—834no—
191C24—832no—
192C24—832no—
193C24C22×D8311yescharacteristic
194C2×D4—3321no—
195C2×D4—3322no—
196C2×D4—3322no—
197C2×D4—3322no—
198C2×D4—3322no—
199C2×D4—3322no—
200C2×D4—3322no—
201C2×D4—3322no—
202C2×D4—1664no—
203C2×D4—1664no—
204C2×D4—1664no—
205C22×C4—832no—
206C22×C4—832no—
207C22×C4C22×D8311yescharacteristic
208C22×C4C22×D8312yes—
209C22×C4C22×D8312yes—
210C22×C4C22×D8311yescharacteristic
211C4⋊C4—41no—
212C4⋊C4—3322no—
213C22⋊C4—24no—
214C22⋊C4—41no—
215C22⋊C4—3322no—
216C22⋊C4—3322no—
217C22⋊C4—3322no—
218C22⋊C4—3322no—
219C22⋊C4—3322no—
220C22⋊C4—3321no—
221C22⋊C4—1664no—
222C22⋊C4—1664no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—3324no—
228C23—3328no—
229C23—3324no—
230C23—3328no—
231C23—3324no—
232C23—834no—
233C23—1662no—
234C23—1662no—
235C23—1662no—
236C23—1662no—
237C23—1662no—
238C23—1662no—
239C23—1662no—
240C23—1662no—
241C23—1662no—
242C23—1662no—
243C23—1662no—
244C23—1662no—
245C23—1662no—
246C23D4×D8312yes—
247C23C23×D8311yescharacteristic
248D4—3324no—
249D4—3328no—
250D4—3328no—
251D4—3328no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4—1662no—
268C22—22no—
269C22—24no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22—33216no—
274C22—3322no—
275C22—3322no—
276C22—3322no—
277C22—3322no—
278C22—3322no—
279C22—3322no—
280C22—3322no—
281C22—3322no—
282C22—3322no—
283C22—3322no—
284C22—3322no—
285C22—1664no—
286C22—1664no—
287C22—1664no—
288C22C2×D4×D8311yescharacteristic
289C22#1458183711yescharacteristic
290C22#1458183812yes—
291C22#1458183811yescharacteristic
292C22#1458183812yes—
293C4—41no—
294C4—41no—
295C4—42no—
296C4—42no—
297C4—3322no—
298C4—3322no—
299C2—42no—
300C2—3324no—
301C2—6642no—
302C2#1912825911yescharacteristic
303C2#1912826212yes—
304C2#1912844111yescharacteristic
305C2#1912844212yes—
306C2#1912844311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#397×C83C2
hypercenter#397D83
hypocenterC83#2812
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#14581762C2semidirect
2. #19127630C2semidirect
3. #19127619C2semidirect
4. #19127618C2semidirect
5. #19127616C2semidirect
6. #19127611C2semidirect
7. #19127610C2semidirect
8. #19127556C2semidirect
9. #397×C83C2semidirect
13. C22×D332C22semidirect
16. C2×Dic83⋊C4C22semidirect
22. C2×(C2×D83)⋊C4C22semidirect
23. C2×(C2×D83)⋊C4C22semidirect
24. C2×(C2×D83)⋊C4C22semidirect
25. C83⋊#95C22semidirect
26. C83⋊#95C22semidirect
27. C2×C4⋊C4×C83C22semidirect
30. #95×C83C22semidirect
31. #95×C83C22semidirect
36. C23×D83D4semidirect
154. C83#2812semidirect
157. #397D83semidirect
171. C2×C4⋊C4C2×D83semidirect
184. #95C2×D83semidirect
185. #95C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817622
#191276302
#191276192
#191276182
#191276162
#191276112
#191276102
#191275562
#397×C832
#281283