Finite Group Databasebeta

← #24913839

Subgroups — #24913839

Order 10624.

688 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14581760C211yesmaximal, characteristic
2#19127629C212yesmaximal, complemented
3#19127616C212yesmaximal, complemented
4#19127612C212yesmaximal, complemented
5#19127592C212yesmaximal, complemented
6#19127581C211yesmaximal, characteristic, complemented
7#19127578C212yesmaximal, complemented
8#19127556C212yesmaximal, complemented
9C2×#124×C83C211yesmaximal, characteristic, complemented
10C24×D83C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12C23×Dic83C2211yescharacteristic, complemented
13#14581760—24no—
14C2×Dic83⋊C4C2212yescomplemented
15C2×C22⋊Dic83C2211yescharacteristic
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2211yescharacteristic, complemented
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C83⋊#95C2212yescomplemented
24C83⋊#95C2212yescomplemented
25C83⋊#95C2212yescomplemented
26C83⋊#95C2212yescomplemented
27C22×Q8×C83C2211yescharacteristic, complemented
28C22×D4×C83C2211yescharacteristic
29#124×C83—24no—
30C2×C22⋊C4×C83C2212yes—
31C2×C22⋊C4×C83C2212yescomplemented
32C2×C42×C83C2211yescharacteristic, complemented
33C23×D83—22no—
34C23×D83—24no—
35C23×D83—41no—
36C23×D83—41no—
37C23×D83—42no—
38C23×D83C2312yes—
39C2×C22⋊D83—24no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—42no—
42C22×Dic83—41no—
43C22×Dic83—42no—
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46C22×Dic83C2312yes—
47Dic83⋊C4—42no—
48C22⋊Dic83—22no—
49C22⋊Dic83—42no—
50C22⋊Dic83—41no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—24no—
53(C2×D83)⋊C4—42no—
54(C2×D83)⋊C4—42no—
55(C2×D83)⋊C4—42no—
56C24×C83C2311yescharacteristic
57C24×C83C2311yescharacteristic
58C2×Q8×C83—22no—
59C2×Q8×C83—41no—
60C2×Q8×C83—41no—
61C2×D4×C83—41no—
62C2×D4×C83—41no—
63C2×D4×C83D414yes—
64C22×C4×C83C2312yes—
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C22⋊C4×C83—24no—
68C22⋊C4×C83—24no—
69C42×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—21no—
73C23×C83—21no—
74C23×C83—21no—
75C23×C83—21no—
76C23×C83—22no—
77C23×C83—22no—
78C23×C83C2×D412yes—
79C23×C83C2×D412yes—
80C23×C83C2411yescharacteristic
81C22×D83—22no—
82C22×D83—22no—
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—44no—
88C22×D83—48no—
89C22×D83—48no—
90C22×D83—44no—
91C22×D83—44no—
92C22×D83#4114yes—
93Q8×C83—44no—
94D4×C83—44no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
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—
104C22⋊D83—48no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—44no—
115C22×C83—22no—
116C22×C83—22no—
117C22×C83—24no—
118C22×C83—24no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83C22×D411yescharacteristic
126C22×C83C2×#4112yes—
127C22×C83H(5,𝔽2)11yescharacteristic
128C22×C83H(5,𝔽2)11yescharacteristic
129C22×C83H(5,𝔽2)12yes—
130C2×D83—24no—
131C2×D83—24no—
132C2×D83—24no—
133C2×D83—416no—
134C2×D83—42no—
135C2×D83—42no—
136C2×D83—42no—
137C2×D83—42no—
138C4×C83—22no—
139C4×C83—42no—
140C4×C83—42no—
141Dic83—42no—
142Dic83—42no—
143Dic83—42no—
144C2×C83—42no—
145C2×C83—42no—
146C2×C83#53411yescharacteristic
147C2×C83#53812yes—
148C2×C83#54612yes—
149C2×C83#55111yescharacteristic
150C2×C83H(3,𝔽4)11yescharacteristic
151D83—44no—
152#2684—831nomaximal, Sylow
153C83#268411yescharacteristic, complemented, Sylow
154C2×#124D8311yescharacteristic, complemented
155C2×#120—831no—
156#397—832no—
157#394—832no—
158#393—832no—
159#388—832no—
160#386—831no—
161#386—832no—
162#379—832no—
163C25—831no—
164C22×Q8C2×D8311yescharacteristic, complemented
165C22×D4—832no—
166C22×D4C2×D8311yescharacteristic
167C23×C4—831no—
168#124—24no—
169#120—1664no—
170C2×C4⋊C4—832no—
171C2×C22⋊C4—832no—
172C2×C22⋊C4—831no—
173C2×C22⋊C4—832no—
174C2×C22⋊C4—832no—
175C2×C22⋊C4—832no—
176C2×C22⋊C4—831no—
177C2×C22⋊C4—832no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4C2×D8312yes—
180C2×C22⋊C4C2×D8312yescomplemented
181C2×C42C2×D8311yescharacteristic, complemented
182#95—832no—
183#95—832no—
184#95—832no—
185#95—832no—
186C24—3321no—
187C24—3321no—
188C24—3322no—
189C24—832no—
190C24—1664no—
191C24—1662no—
192C24C22×D8311yescharacteristic
193C24C22×D8311yescharacteristic
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—3322no—
200C2×D4—1664no—
201C2×D4—1664no—
202C2×D4C22⋊D8314yes—
203C22×C4—3321no—
204C22×C4—3322no—
205C22×C4—832no—
206C22×C4—832no—
207C22×C4—832no—
208C22×C4C22×D8312yes—
209C22×C4C22×D8312yes—
210C22×C4C22×D8311yescharacteristic
211C4⋊C4—3322no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—3322no—
215C22⋊C4—3322no—
216C22⋊C4—3322no—
217C22⋊C4—3321no—
218C22⋊C4—3322no—
219C22⋊C4—1664no—
220C22⋊C4—1662no—
221C22⋊C4—1664no—
222C42—22no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—3324no—
232C23—3328no—
233C23—3324no—
234C23—3328no—
235C23—3324no—
236C23—834no—
237C23—1662no—
238C23—1662no—
239C23—1662no—
240C23—1662no—
241C23—1662no—
242C23—1662no—
243C23C2×C22⋊D8312yes—
244C23C2×C22⋊D8312yes—
245C23C23×D8311yescharacteristic
246Q8—44no—
247D4—44no—
248D4—3328no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—3324no—
258C2×C4—1662no—
259C2×C4—1662no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4C2×C22⋊D8312yes—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—33216no—
279C22—3322no—
280C22—3322no—
281C22—3322no—
282C22—3322no—
283C22—1664no—
284C22—1664no—
285C22—1664no—
286C22C22×C22⋊D8311yescharacteristic
287C22#41×D8312yes—
288C22#1458183711yescharacteristic
289C22#1458183711yescharacteristic
290C22#1458183812yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—3322no—
295C4—3322no—
296C4—3322no—
297C2—42no—
298C2—42no—
299C2—3324no—
300C2#1912826011yescharacteristic
301C2#1912828512yes—
302C2#1912833912yes—
303C2#1912838111yescharacteristic
304C2#1912844311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#124×C83C2
prime radicalC2×#14581760C2
hypercenterC2×#124D83
hypocenterC83#2684
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #19127629C2semidirect
3. #19127616C2semidirect
4. #19127612C2semidirect
5. #19127592C2semidirect
6. #19127581C2semidirect
7. #19127578C2semidirect
8. #19127556C2semidirect
9. C2×#124×C83C2semidirect
12. C23×Dic83C22semidirect
14. C2×Dic83⋊C4C22semidirect
17. C2×C22⋊Dic83C22semidirect
23. C83⋊#95C22semidirect
24. C83⋊#95C22semidirect
25. C83⋊#95C22semidirect
26. C83⋊#95C22semidirect
27. C22×Q8×C83C22semidirect
31. C2×C22⋊C4×C83C22semidirect
32. C2×C42×C83C22semidirect
153. C83#2684semidirect
154. C2×#124D83semidirect
164. C22×Q8C2×D83semidirect
180. C2×C22⋊C4C2×D83semidirect
181. C2×C42C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817602
#191276292
#191276162
#191276122
#191275922
#191275812
#191275782
#191275562
C2×#124×C832
#268483