Finite Group Databasebeta

← #24913841

Subgroups — #24913841

Order 10624.

688 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581779C211yesmaximal, characteristic, complemented
2#19127629C212yesmaximal, complemented
3#19127615C212yesmaximal, complemented
4#19127613C212yesmaximal, complemented
5#19127589C212yesmaximal, complemented
6#19127581C212yesmaximal, complemented
7#19127581C211yesmaximal, characteristic, complemented
8#19127557C212yesmaximal, complemented
9C2×#120×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic, complemented
11C23×Dic83C2211yescharacteristic, complemented
12#14581779—24no—
13C2×Dic83⋊C4C2212yescomplemented
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yescomplemented
21C2×C22⋊Dic83C2211yescharacteristic, complemented
22C2×C4×Dic83C2211yescharacteristic, complemented
23C83⋊#95C2212yescomplemented
24C83⋊#95C2212yescomplemented
25C83⋊#95C2212yescomplemented
26C83⋊#95C2212yescomplemented
27C25×C83C2211yescharacteristic
28C22×D4×C83C2212yes—
29C22×D4×C83C2211yescharacteristic
30#120×C83—24no—
31C2×C22⋊C4×C83C2212yes—
32C2×C22⋊C4×C83C2211yescharacteristic
33C2×Dic166—22no—
34C2×Dic166—41no—
35C2×Dic166—41no—
36C22×Dic83—41no—
37C22×Dic83—42no—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42Dic83⋊C4—42no—
43C22⋊Dic83—24no—
44C22⋊Dic83—24no—
45C22⋊Dic83—24no—
46C22⋊Dic83—42no—
47C22⋊Dic83—42no—
48C22⋊Dic83—42no—
49C22⋊Dic83—42no—
50C22⋊Dic83—41no—
51C4×Dic83—22no—
52C24×C83—22no—
53C24×C83—24no—
54C24×C83—41no—
55C24×C83—42no—
56C24×C83—41no—
57C24×C83C2311yescharacteristic
58C24×C83C2311yescharacteristic
59C24×C83C2312yes—
60C2×D4×C83—24no—
61C2×D4×C83—24no—
62C2×D4×C83—41no—
63C2×D4×C83—41no—
64C2×D4×C83—42no—
65C2×D4×C83D414yes—
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C22⋊C4×C83—22no—
69C22⋊C4×C83—24no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—21no—
75C23×C83—21no—
76C23×C83—22no—
77C23×C83—21no—
78C23×C83—21no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—48no—
85C23×C83—44no—
86C23×C83—44no—
87C23×C83—48no—
88C23×C83—44no—
89C23×C83C2×D412yes—
90C23×C83C2×D412yes—
91C23×C83#4114yes—
92C23×C83C2411yescharacteristic
93D4×C83—44no—
94D4×C83—48no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—22no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83C2×D412yes—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
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—44no—
114Dic166—44no—
115C22×C83—22no—
116C22×C83—22no—
117C22×C83—24no—
118C22×C83—24no—
119C22×C83—24no—
120C22×C83—24no—
121C22×C83—24no—
122C22×C83—416no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—41no—
129C22×C83—41no—
130C22×C83—41no—
131C22×C83—41no—
132C22×C83—42no—
133C22×C83C22×D411yescharacteristic
134C22×C83C2×#4112yes—
135C22×C83H(5,𝔽2)11yescharacteristic
136C22×C83H(5,𝔽2)11yescharacteristic
137C22×C83H(5,𝔽2)12yes—
138C4×C83—22no—
139C4×C83—42no—
140Dic83—42no—
141Dic83—42no—
142Dic83—42no—
143Dic83—42no—
144C2×C83—42no—
145C2×C83—42no—
146C2×C83—44no—
147C2×C83#53411yescharacteristic
148C2×C83#53812yes—
149C2×C83#54612yes—
150C2×C83#55111yescharacteristic
151C2×C83H(3,𝔽4)11yescharacteristic
152#2684—831nomaximal, Sylow
153C83#268411yescharacteristic, complemented, Sylow
154C2×#124—831no—
155C2×#120D8311yescharacteristic
156#397—832no—
157#394—832no—
158#393—832no—
159#388—832no—
160#386—831no—
161#386—832no—
162#379—832no—
163C25C2×D8311yescharacteristic
164C22×Q8—831no—
165C22×D4C2×D8312yes—
166C22×D4C2×D8311yescharacteristic
167C23×C4—831no—
168#124—1664no—
169#120—24no—
170C2×C4⋊C4—832no—
171C2×C22⋊C4—832no—
172C2×C22⋊C4—832no—
173C2×C22⋊C4—832no—
174C2×C22⋊C4—832no—
175C2×C22⋊C4—831no—
176C2×C22⋊C4—832no—
177C2×C22⋊C4—832no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4C2×D8312yes—
180C2×C22⋊C4C2×D8311yescharacteristic
181C2×C42—831no—
182#95—832no—
183#95—832no—
184#95—832no—
185#95—832no—
186C24—22no—
187C24—24no—
188C24—42no—
189C24—41no—
190C24—41no—
191C24C22×D8311yescharacteristic
192C24C22×D8311yescharacteristic
193C24C22×D8312yes—
194C2×Q8—3321no—
195C2×Q8—3321no—
196C2×Q8—1662no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4C22⋊D8314yes—
203C22×C4—3322no—
204C22×C4—3321no—
205C22×C4—832no—
206C22×C4—832no—
207C22×C4—832no—
208C22×C4—832no—
209C22×C4C22×D8312yes—
210C22×C4C22×D8311yescharacteristic
211C4⋊C4—3322no—
212C22⋊C4—22no—
213C22⋊C4—24no—
214C22⋊C4—3321no—
215C22⋊C4—3322no—
216C22⋊C4—3322no—
217C22⋊C4—3322no—
218C22⋊C4—3322no—
219C22⋊C4—1664no—
220C22⋊C4—1664no—
221C22⋊C4—1664no—
222C42—1662no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—48no—
238C23—44no—
239C23—44no—
240C23—48no—
241C23—44no—
242C23C2×C22⋊D8312yes—
243C23C2×C22⋊D8312yes—
244C23#1305058914yes—
245C23C23×D8311yescharacteristic
246Q8—3324no—
247D4—44no—
248D4—48no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—3324no—
255C2×C4—1662no—
256C2×C4—1662no—
257C2×C4—1662no—
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—24no—
271C22—24no—
272C22—22no—
273C22—22no—
274C22—24no—
275C22—416no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22C22×C22⋊D8311yescharacteristic
287C22C2×#1305058912yes—
288C22#1458183712yes—
289C22#1458183711yescharacteristic
290C22#1458183711yescharacteristic
291C4—22no—
292C4—42no—
293C4—3322no—
294C4—3322no—
295C4—3322no—
296C4—3322no—
297C2—44no—
298C2—42no—
299C2—42no—
300C2#1912826011yescharacteristic
301C2#1912828612yes—
302C2#1912834512yes—
303C2#1912838211yescharacteristic
304C2#1912844411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#120×C83C2
hypercenterC2×#120D83
hypocenterC83#2684
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#14581779C2semidirect
2. #19127629C2semidirect
3. #19127615C2semidirect
4. #19127613C2semidirect
5. #19127589C2semidirect
6. #19127581C2semidirect
7. #19127581C2semidirect
8. #19127557C2semidirect
10. C22×Dic166C22semidirect
11. C23×Dic83C22semidirect
13. C2×Dic83⋊C4C22semidirect
20. C2×C22⋊Dic83C22semidirect
21. C2×C22⋊Dic83C22semidirect
22. C2×C4×Dic83C22semidirect
23. C83⋊#95C22semidirect
24. C83⋊#95C22semidirect
25. C83⋊#95C22semidirect
26. C83⋊#95C22semidirect
153. C83#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#145817792
#191276292
#191276152
#191276132
#191275892
#191275812
#191275572
C2×#120×C832
#268483