Finite Group Databasebeta

← #24910790

Subgroups — #24910790

Order 10624.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C22⋊D83C211yesmaximal, characteristic
2C2×#14581775C212yesmaximal, complemented
3#19127616C214yesmaximal, complemented
4#19127580C212yesmaximal, complemented
5#19127573C212yesmaximal, complemented
6#19127557C211yesmaximal, characteristic, complemented
7#19127556C212yesmaximal, complemented
8C22×C22⋊C4×C83C211yesmaximal, characteristic, complemented
9C24×D83C2211yescharacteristic
10C22×C22⋊D83—28no—
11C22×C22⋊D83—24no—
12C22×C22⋊D83C2212yes—
13C22×C22⋊D83C2212yes—
14C23×Dic83C2211yescharacteristic
15#14581775—28no—
16C2×Dic83⋊C4C2214yescomplemented
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2212yescomplemented
19C2×C22⋊Dic83C2212yescomplemented
20C2×(C2×D83)⋊C4C2214yescomplemented
21C2×(C2×D83)⋊C4C2214yescomplemented
22C83⋊#95C2214yescomplemented
23C25×C83C2211yescharacteristic
24C23×C4×C83C2211yescharacteristic, complemented
25C23×C4×C83C2211yescharacteristic, complemented
26C2×C22⋊C4×C83—24no—
27C2×C22⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2212yescomplemented
29C23×D83—22no—
30C23×D83—24no—
31C23×D83—24no—
32C23×D83—42no—
33C23×D83C2312yes—
34C2×C22⋊D83—232no—
35C2×C22⋊D83—24no—
36C2×C22⋊D83—24no—
37C2×C22⋊D83—24no—
38C2×C22⋊D83—24no—
39C2×C22⋊D83—24no—
40C2×C22⋊D83D418yes—
41C22×Dic83—22no—
42C22×Dic83—24no—
43C22×Dic83C2314yes—
44C22×Dic83C2312yes—
45Dic83⋊C4—28no—
46C22⋊Dic83—24no—
47C22⋊Dic83—24no—
48C22⋊Dic83—42no—
49(C2×D83)⋊C4—28no—
50(C2×D83)⋊C4—44no—
51C24×C83—24no—
52C24×C83—24no—
53C24×C83—42no—
54C24×C83D414yes—
55C24×C83C2311yescharacteristic
56C24×C83C2311yescharacteristic
57C24×C83C2311yescharacteristic
58C22×C4×C83—24no—
59C22×C4×C83—22no—
60C22×C4×C83—22no—
61C22×C4×C83—42no—
62C22×C4×C83C2312yes—
63C22×C4×C83C2312yes—
64C22⋊C4×C83—24no—
65C22⋊C4×C83—24no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—28no—
71C23×C83—28no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—44no—
76C23×C83—416no—
77C23×C83C2×D414yes—
78C23×C83C2×D412yes—
79C23×C83C2×D412yes—
80C23×C83C2×D412yes—
81C23×C83#4114yes—
82C23×C83C2411yescharacteristic
83C22×D83—28no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—24no—
87C22×D83—24no—
88C22×D83—416no—
89C22×D83—44no—
90C22×D83—44no—
91C22×D83C2×D414yes—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—24no—
95C2×C4×C83—24no—
96C2×C4×C83—44no—
97C2×C4×C83—44no—
98C22⋊D83—216no—
99C22⋊D83—216no—
100C2×Dic83—28no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—24no—
104C2×Dic83—24no—
105C2×Dic83—24no—
106C2×Dic83C2×D414yes—
107C22×C83—22no—
108C22×C83—22no—
109C22×C83—22no—
110C22×C83—22no—
111C22×C83—28no—
112C22×C83—28no—
113C22×C83—22no—
114C22×C83—22no—
115C22×C83—22no—
116C22×C83—416no—
117C22×C83—44no—
118C22×C83#12014yes—
119C22×C83#12314yescomplemented
120C22×C83C22×D412yes—
121C22×C83C22×D411yescharacteristic
122C22×C83C2×#4112yes—
123C22×C83H(5,𝔽2)12yes—
124C2×D83—24no—
125C2×D83—28no—
126C2×D83—416no—
127C2×D83—42no—
128C2×D83—42no—
129C2×D83—44no—
130C4×C83—42no—
131C4×C83—42no—
132Dic83—24no—
133Dic83—44no—
134C2×C83—24no—
135C2×C83—24no—
136C2×C83—42no—
137C2×C83C2×#12011yescharacteristic
138C2×C83C2×#12311yescharacteristic
139C2×C83#53411yescharacteristic
140C2×C83#54614yes—
141D83—44no—
142#2405—831nomaximal, Sylow
143C83#240511yescharacteristic, complemented, Sylow
144C23×D4—831no—
145C2×#123—832no—
146C22×C22⋊C4D8311yescharacteristic, complemented
147#394—834no—
148#386—832no—
149#385—832no—
150#379—831no—
151#379—832no—
152C25—831no—
153C25C2×D8311yescharacteristic
154C22×D4—832no—
155C22×D4—832no—
156C22×D4—1668no—
157C22×D4—1664no—
158C23×C4—831no—
159C23×C4C2×D8311yescharacteristic, complemented
160C23×C4C2×D8311yescharacteristic, complemented
161#123—1668no—
162C2×C4⋊C4—834no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—834no—
166C2×C22⋊C4—832no—
167C2×C22⋊C4—832no—
168C2×C22⋊C4—834no—
169C2×C22⋊C4C2×D8312yes—
170C2×C22⋊C4C2×D8312yescomplemented
171#95—834no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—3322no—
176C24—832no—
177C24—1664no—
178C24—1664no—
179C24—1662no—
180C24C22⋊D8314yes—
181C24C22×D8311yescharacteristic
182C24C22×D8311yescharacteristic
183C24C22×D8311yescharacteristic
184C2×D4—838no—
185C2×D4—1664no—
186C2×D4—1664no—
187C2×D4—16632no—
188C2×D4—1664no—
189C2×D4—1664no—
190C2×D4—1664no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—832no—
196C22×C4—834no—
197C22×C4—1664no—
198C22×C4—1662no—
199C22×C4C22×D8312yes—
200C22×C4C22×D8312yes—
201C4⋊C4—1668no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—3324no—
205C22⋊C4—3322no—
206C22⋊C4—1668no—
207C22⋊C4—1664no—
208C22⋊C4—1664no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—3324no—
221C23—33216no—
222C23—3324no—
223C23—834no—
224C23—1668no—
225C23—1662no—
226C23—1662no—
227C23—1664no—
228C23—1664no—
229C23D4×D8314yes—
230C23C2×C22⋊D8312yes—
231C23C2×C22⋊D8312yes—
232C23C2×C22⋊D8312yes—
233C23#1305059014yes—
234C23C23×D8311yescharacteristic
235D4—16616no—
236D4—16616no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—834no—
244C2×C4—1664no—
245C2×C4—1664no—
246C2×C4—1664no—
247C2×C4—1668no—
248C2×C4—1662no—
249C2×C4—1662no—
250C22—28no—
251C22—28no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—33216no—
262C22—3324no—
263C22—3322no—
264C22—3322no—
265C22—1668no—
266C22—1664no—
267C22#1458176014yes—
268C22#1458177514yescomplemented
269C22C2×D4×D8312yes—
270C22C22×C22⋊D8311yescharacteristic
271C22C2×#1305059012yes—
272C22#1458183712yes—
273C4—42no—
274C4—42no—
275C4—3324no—
276C4—1664no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—3324no—
281C2C2×#1458176011yescharacteristic
282C2C2×#1458177511yescharacteristic
283C2#1912826011yescharacteristic
284C2#1912834114yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC22×C22⋊C4×C83C2
prime radicalC23×C22⋊D83C2
hypercenterC22×C22⋊C4D83
hypocenterC83#2405
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#14581775C2semidirect
3. #19127616C2semidirect
4. #19127580C2semidirect
5. #19127573C2semidirect
6. #19127557C2semidirect
7. #19127556C2semidirect
8. C22×C22⋊C4×C83C2semidirect
16. C2×Dic83⋊C4C22semidirect
18. C2×C22⋊Dic83C22semidirect
19. C2×C22⋊Dic83C22semidirect
20. C2×(C2×D83)⋊C4C22semidirect
21. C2×(C2×D83)⋊C4C22semidirect
22. C83⋊#95C22semidirect
24. C23×C4×C83C22semidirect
25. C23×C4×C83C22semidirect
28. C2×C22⋊C4×C83C22semidirect
119. C22×C83#123semidirect
143. C83#2405semidirect
146. C22×C22⋊C4D83semidirect
159. C23×C4C2×D83semidirect
160. C23×C4C2×D83semidirect
170. C2×C22⋊C4C2×D83semidirect
268. C22#14581775semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D832
C2×#145817752
#191276162
#191275802
#191275732
#191275572
#191275562
C22×C22⋊C4×C832
#240583