Finite Group Databasebeta

← #24913665

Subgroups — #24913665

Order 10624.

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

normal subgroups only

This group has 816 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×#14581764C212yesmaximal, complemented
2C2×#14581760C212yesmaximal, complemented
3#19127637C211yesmaximal, characteristic, complemented
4#19127630C212yesmaximal, complemented
5#19127617C212yesmaximal, complemented
6#19127581C212yesmaximal, complemented
7#19127578C212yesmaximal, complemented
8#19127556C211yesmaximal, characteristic, complemented
9#392×C83C211yesmaximal, characteristic, complemented
10C24×D83C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12C22×C22⋊D83C2212yes—
13C23×Dic83C2212yescomplemented
14#14581764—28no—
15#14581760—28no—
16C2×C4⋊Dic83C2211yescharacteristic, complemented
17C2×Dic83⋊C4C2212yescomplemented
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×C22⋊Dic83C2212yescomplemented
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C83⋊#95C2211yescharacteristic, complemented
26C83⋊#95C2212yescomplemented
27C83⋊#95C2212yescomplemented
28C22×D4×C83C2212yes—
29C22×D4×C83C2211yescharacteristic, complemented
30C2×C22⋊C4×C83C2211yescharacteristic
31C2×C22⋊C4×C83C2212yescomplemented
32#95×C83C2211yescharacteristic, complemented
33C23×D83—22no—
34C23×D83—24no—
35C23×D83—42no—
36C23×D83—41no—
37C23×D83—41no—
38C23×D83C2312yes—
39C2×C22⋊D83—24no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—42no—
42C2×C22⋊D83—42no—
43C2×C22⋊D83—42no—
44C2×C22⋊D83D418yescomplemented
45C22×Dic83—24no—
46C22×Dic83—42no—
47C22×Dic83—42no—
48C22×Dic83C2312yes—
49C22×Dic83C2312yes—
50C22×Dic83C2312yes—
51C4⋊Dic83—41no—
52Dic83⋊C4—24no—
53C22⋊Dic83—24no—
54C22⋊Dic83—24no—
55C22⋊Dic83—41no—
56(C2×D83)⋊C4—24no—
57(C2×D83)⋊C4—24no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4—42no—
60C24×C83C2312yes—
61C24×C83C2311yescharacteristic
62C2×D4×C83—24no—
63C2×D4×C83—24no—
64C2×D4×C83—24no—
65C2×D4×C83—42no—
66C2×D4×C83—41no—
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2312yes—
69C22×C4×C83C2311yescharacteristic
70C22⋊C4×C83—41no—
71C22⋊C4×C83—42no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—21no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—21no—
81C23×C83—22no—
82C23×C83C2×D414yes—
83C23×C83#4112yes—
84C23×C83C2411yescharacteristic
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—22no—
89C22×D83—22no—
90C22×D83—22no—
91C22×D83—44no—
92C22×D83—48no—
93C22×D83—48no—
94C22×D83—44no—
95C22×D83—44no—
96C22×D83C2×D414yes—
97D4×C83—44no—
98D4×C83—48no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—22no—
102C2×C4×C83—21no—
103C2×C4×C83—22no—
104C2×C4×C83—22no—
105C2×C4×C83—21no—
106C22⋊D83—48no—
107C22⋊D83—48no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—22no—
112C2×Dic83—22no—
113C2×Dic83—22no—
114C2×Dic83—22no—
115C2×Dic83—22no—
116C2×Dic83—48no—
117C2×Dic83C2×D414yes—
118C22×C83—24no—
119C22×C83—22no—
120C22×C83—24no—
121C22×C83—24no—
122C22×C83—24no—
123C22×C83—42no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—41no—
127C22×C83—42no—
128C22×C83—41no—
129C22×C83—42no—
130C22×C83C22×D412yes—
131C22×C83C2×#4111yescharacteristic
132C22×C83H(5,𝔽2)12yes—
133C22×C83H(5,𝔽2)11yescharacteristic
134C22×C83H(5,𝔽2)11yescharacteristic
135C2×D83—24no—
136C2×D83—24no—
137C2×D83—24no—
138C2×D83—416no—
139C2×D83—42no—
140C2×D83—42no—
141C2×D83—42no—
142C2×D83—42no—
143C4×C83—41no—
144C4×C83—42no—
145C4×C83—41no—
146Dic83—24no—
147Dic83—42no—
148Dic83—42no—
149C2×C83—42no—
150C2×C83—44no—
151C2×C83#53412yes—
152C2×C83#53811yescharacteristic
153C2×C83D4211yescharacteristic
154C2×C83#54612yes—
155C2×C83#56011yescharacteristic
156D83—44no—
157#2668—831nomaximal, Sylow
158C83#266811yescharacteristic, complemented, Sylow
159C2×#121—832no—
160C2×#120—832no—
161#399—831no—
162#397—832no—
163#394—832no—
164#392D8311yescharacteristic, complemented
165#386—832no—
166#386—832no—
167#379—831no—
168C25—831no—
169C22×D4—832no—
170C22×D4—832no—
171C22×D4C2×D8312yes—
172C22×D4C2×D8311yescharacteristic, complemented
173C23×C4—832no—
174#121—1668no—
175#120—1668no—
176C2×C4⋊C4—831no—
177C2×C4⋊C4—832no—
178C2×C22⋊C4—831no—
179C2×C22⋊C4—832no—
180C2×C22⋊C4—832no—
181C2×C22⋊C4—832no—
182C2×C22⋊C4—832no—
183C2×C22⋊C4—832no—
184C2×C22⋊C4—832no—
185C2×C22⋊C4C2×D8311yescharacteristic
186C2×C22⋊C4C2×D8312yescomplemented
187#95—832no—
188#95—831no—
189#95—832no—
190#95C2×D8311yescharacteristic, complemented
191C24—3321no—
192C24—3322no—
193C24—3321no—
194C24—832no—
195C24—1664no—
196C24—1662no—
197C24C22×D8312yes—
198C24C22×D8311yescharacteristic
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—3322no—
205C2×D4—3322no—
206C2×D4—3322no—
207C2×D4—838no—
208C2×D4—1664no—
209C2×D4—1664no—
210C22×C4—3322no—
211C22×C4—3322no—
212C22×C4—832no—
213C22×C4—832no—
214C22×C4—832no—
215C22×C4—1664no—
216C22×C4C22×D8312yes—
217C22×C4C22×D8311yescharacteristic
218C22×C4C22×D8311yescharacteristic
219C4⋊C4—3321no—
220C4⋊C4—1664no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—3322no—
224C22⋊C4—3321no—
225C22⋊C4—1664no—
226C22⋊C4—1664no—
227C22⋊C4—1664no—
228C22⋊C4—1664no—
229C22⋊C4—1664no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—3324no—
241C23—3328no—
242C23—3324no—
243C23—3328no—
244C23—3324no—
245C23—834no—
246C23—1662no—
247C23—1662no—
248C23—1662no—
249C23—1662no—
250C23—1662no—
251C23—1662no—
252C23D4×D8314yes—
253C23#1305058912yes—
254C23C23×D8311yescharacteristic
255D4—44no—
256D4—48no—
257D4—3328no—
258D4—3328no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—3328no—
267C2×C4—834no—
268C2×C4—1662no—
269C2×C4—1662no—
270C2×C4—1662no—
271C2×C4—1662no—
272C2×C4—1662no—
273C2×C4—1662no—
274C2×C4—1662no—
275C2×C4—1662no—
276C22—24no—
277C22—24no—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—41no—
282C22—42no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—33216no—
289C22—3322no—
290C22—3322no—
291C22—3322no—
292C22—3322no—
293C22—1664no—
294C22—1664no—
295C22—1664no—
296C22C2×D4×D8312yes—
297C22C2×#1305058911yescharacteristic
298C22#1458183711yescharacteristic
299C22#1458183712yes—
300C22#1458183811yescharacteristic
301C4—41no—
302C4—42no—
303C4—41no—
304C4—3322no—
305C4—3322no—
306C4—1664no—
307C2—42no—
308C2—44no—
309C2—3324no—
310C2#1912825912yes—
311C2#1912828811yescharacteristic
312C2#1912833311yescharacteristic
313C2#1912833812yes—
314C2#1912843811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#392×C83C2
hypercenter#392D83
hypocenterC83#2668
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14581764C2semidirect
2. C2×#14581760C2semidirect
3. #19127637C2semidirect
4. #19127630C2semidirect
5. #19127617C2semidirect
6. #19127581C2semidirect
7. #19127578C2semidirect
8. #19127556C2semidirect
9. #392×C83C2semidirect
13. C23×Dic83C22semidirect
16. C2×C4⋊Dic83C22semidirect
17. C2×Dic83⋊C4C22semidirect
20. C2×C22⋊Dic83C22semidirect
25. C83⋊#95C22semidirect
26. C83⋊#95C22semidirect
27. C83⋊#95C22semidirect
29. C22×D4×C83C22semidirect
31. C2×C22⋊C4×C83C22semidirect
32. #95×C83C22semidirect
44. C2×C22⋊D83D4semidirect
158. C83#2668semidirect
164. #392D83semidirect
172. C22×D4C2×D83semidirect
186. C2×C22⋊C4C2×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×#145817642
C2×#145817602
#191276372
#191276302
#191276172
#191275812
#191275782
#191275562
#392×C832
#266883