Finite Group Databasebeta

← #24917393

Subgroups — #24917393

Order 10624.

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

normal subgroups only

This group has 414 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
1#19128208C211yesmaximal, characteristic
2C2×#14581750C211yesmaximal, characteristic, complemented
3#19128041C212yesmaximal, complemented
4#19128040C212yesmaximal, complemented
5#19128024C212yesmaximal, complemented
6#19128023C212yesmaximal, complemented
7#19127743C211yesmaximal, characteristic, complemented
8#19127741C211yesmaximal, characteristic, complemented
9#19127702C211yesmaximal, characteristic, complemented
10#19127664C211yesmaximal, characteristic, complemented
11#417×C83C211yesmaximal, characteristic, complemented
12C22×C4×D83C2211yescharacteristic
13C2×#13050574C2211yescharacteristic
14#14581773—21no—
15#14581768C2211yescharacteristic
16#14581768C2211yescharacteristic
17#14581763C2211yescharacteristic
18#14581763C2211yescharacteristic
19#14581760—21no—
20C4×C22⋊D83—21no—
21C4×C22⋊D83—21no—
22#14581751C2211yescharacteristic
23#14581750—21no—
24#14581750—21no—
25#14581750C2212yes—
26#14581750C2212yes—
27C2×C4⋊Dic83C2211yescharacteristic
28C2×(C2×D83)⋊C4—21no—
29C2×C4×Dic83—21no—
30#14581726C2212yescomplemented
31#14581725C2212yescomplemented
32#14581724C2212yescomplemented
33#14581723C2212yescomplemented
34#14581716C2212yescomplemented
35#14581713C2212yescomplemented
36(C2×D83)⋊C8C2212yes—
37(C2×D83)⋊C8C2212yes—
38C2×#41×C83C2211yescharacteristic
39C2×M16×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2211yescharacteristic
41#103×C83C2212yescomplemented
42#102×C83C2212yescomplemented
43C23×D83D411yescharacteristic
44C2×C22⋊D83—21no—
45C2×C22⋊D83—21no—
46C22×Dic83—21no—
47C22×Dic83D411yescharacteristic
48C2×C4×D83—22no—
49C2×C4×D83—22no—
50C2×C4×D83D411yescharacteristic
51C2×C4×D83D411yescharacteristic
52C2×C4×D83C2311yescharacteristic
53C2×C4×D83C2311yescharacteristic
54#13050574—22no—
55C2×C83⋊C8C2312yes—
56C4⋊Dic83—21no—
57C4⋊Dic83C2311yescharacteristic
58C4⋊Dic83C2311yescharacteristic
59C4⋊Dic83C2312yes—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62C22⋊Dic83—21no—
63C22⋊Dic83—21no—
64C22⋊Dic83—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—22no—
68(C2×D83)⋊C4—22no—
69C4×Dic83—21no—
70C4×Dic83—22no—
71C4×Dic83—22no—
72#41×C83—22no—
73#41×C83—22no—
74C2×Q8×C83C2311yescharacteristic
75C2×D4×C83—21no—
76C2×D4×C83C2311yescharacteristic
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83C2311yescharacteristic
80M16×C83—22no—
81C2×C8×C83C2312yes—
82C4⋊C4×C83—21no—
83C4⋊C4×C83C2312yes—
84C23×C83—21no—
85C23×C83C2×D411yescharacteristic
86C22×D83—22no—
87C22×D83—41no—
88C22×D83—41no—
89C22×D83C2×D411yescharacteristic
90C22×D83C2×D411yescharacteristic
91Q8×C83—21no—
92Q8×C83—21no—
93D4×C83—21no—
94D4×C83—21no—
95D4×C83—42no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—42no—
99C2×C4×C83—42no—
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83#4112yes—
102C2×C4×C83#4112yes—
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—42no—
105C22⋊D83—42no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—21no—
109C2×Dic83—21no—
110C2×Dic83—42no—
111C2×Dic83C2×D411yescharacteristic
112C2×Dic83C2×D411yescharacteristic
113C4×D83—22no—
114C4×D83—22no—
115C4×D83—21no—
116C4×D83—21no—
117C4×D83—21no—
118C4×D83—21no—
119C8×C83—22no—
120C83⋊C8—22no—
121C22×C83—21no—
122C22×C83—21no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83—41no—
126C22×C83#12312yes—
127C22×C83C22×D411yescharacteristic
128C2×D83—21no—
129C2×D83—21no—
130C2×D83—21no—
131C2×D83—21no—
132C2×D83—21no—
133C2×D83—21no—
134C2×D83—44no—
135C4×C83—41no—
136C4×C83—42no—
137C4×C83#12312yes—
138C4×C83C2×#4111yescharacteristic
139C4×C83C2×#4111yescharacteristic
140Dic83—21no—
141Dic83—21no—
142Dic83—42no—
143Dic83—41no—
144Dic83—41no—
145C2×C83—22no—
146C2×C83—81no—
147C2×C83C2×#12311yescharacteristic
148C2×C83#57511yescharacteristic
149C2×C83#57511yescharacteristic
150D83—41no—
151D83—41no—
152#3093—831nomaximal, Sylow
153C83#309311yescharacteristic, complemented, Sylow
154#525—831no—
155C2×#117—831no—
156#485—832no—
157#485—832no—
158#482—832no—
159#482—832no—
160#428—831no—
161#428—831no—
162#417—831no—
163#417D8311yescharacteristic, complemented
164#407—831no—
165C2×#41C2×D8311yescharacteristic
166C23×C4—831no—
167C2×M16—831no—
168C2×M16C2×D8311yescharacteristic
169#123—1661no—
170C22⋊Q8—831no—
171C22⋊Q8—831no—
172#121—831no—
173#121—831no—
174#120—1661no—
175C4×D4—1661no—
176C4×D4—1661no—
177#117—831no—
178#117—832no—
179#117—832no—
180#117—1661no—
181#117—1661no—
182C2×C4⋊C4—831no—
183C2×C4⋊C4C2×D8311yescharacteristic
184C2×C22⋊C4—1661no—
185C2×C42—1661no—
186#107—832no—
187#107—832no—
188#106—832no—
189#106—832no—
190#103—832no—
191#103C2×D8312yescomplemented
192#102—832no—
193#102C2×D8312yescomplemented
194C22⋊C8—832no—
195C22⋊C8—832no—
196C24—831no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D8311yescharacteristic
200C2×D4—21no—
201C2×D4—1661no—
202C2×D4—1661no—
203C2×D4C22×D8311yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—831no—
207C22×C4—831no—
208C22×C4—831no—
209C22×C4—831no—
210C22×C4—831no—
211C22×C4—1661no—
212C22×C4—1662no—
213C22×C4—1662no—
214C22×C4C22×D8311yescharacteristic
215M16—22no—
216M16—1662no—
217C2×C8—832no—
218C2×C8C22×D8312yes—
219C4⋊C4—21no—
220C4⋊C4—832no—
221C4⋊C4—831no—
222C4⋊C4—831no—
223C4⋊C4—1661no—
224C4⋊C4—1661no—
225C4⋊C4—1661no—
226C4⋊C4C22×D8312yes—
227C22⋊C4—1662no—
228C22⋊C4—1662no—
229C22⋊C4—1661no—
230C22⋊C4—1661no—
231C22⋊C4—1661no—
232C22⋊C4—1661no—
233C22⋊C4—1661no—
234C42—1662no—
235C42—1662no—
236C42—1661no—
237C23—21no—
238C23—3321no—
239C23—3321no—
240C23—831no—
241C23—831no—
242C23—1662no—
243C23D4×D8311yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—3322no—
250D4—3322no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—3322no—
256C2×C4—831no—
257C2×C4—831no—
258C2×C4—1662no—
259C2×C4—1661no—
260C2×C4—1661no—
261C2×C4—1662no—
262C2×C4—1662no—
263C2×C4—1662no—
264C2×C4—1661no—
265C2×C4—1661no—
266C2×C4—1661no—
267C2×C4—1661no—
268C2×C4D4×D8311yescharacteristic
269C2×C4#1305058912yes—
270C2×C4#1305059012yes—
271C2×C4C23×D8311yescharacteristic
272C8—22no—
273C8—1662no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—3324no—
280C22—1661no—
281C22—1661no—
282C22—1661no—
283C22—1661no—
284C22—1661no—
285C22—1661no—
286C22#1458177412yes—
287C22C2×D4×D8311yescharacteristic
288C4—41no—
289C4—42no—
290C4—3321no—
291C4—3322no—
292C4—3321no—
293C4—1661no—
294C4—1661no—
295C4#1458177412yes—
296C4C2×#1305058911yescharacteristic
297C4C2×#1305059011yescharacteristic
298C2—22no—
299C2—81no—
300C2—3321no—
301C2—3321no—
302C2C2×#1458177411yescharacteristic
303C2#1912849711yescharacteristic
304C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#417×C83C2
socleC22×C83C22×D4
prime radical#19128208C2
hypercenter#417D83
hypocenterC83#3093
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#14581750C2semidirect
3. #19128041C2semidirect
4. #19128040C2semidirect
5. #19128024C2semidirect
6. #19128023C2semidirect
7. #19127743C2semidirect
8. #19127741C2semidirect
9. #19127702C2semidirect
10. #19127664C2semidirect
11. #417×C83C2semidirect
30. #14581726C22semidirect
31. #14581725C22semidirect
32. #14581724C22semidirect
33. #14581723C22semidirect
34. #14581716C22semidirect
35. #14581713C22semidirect
41. #103×C83C22semidirect
42. #102×C83C22semidirect
153. C83#3093semidirect
163. #417D83semidirect
191. #103C2×D83semidirect
193. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282082
C2×#145817502
#191280412
#191280402
#191280242
#191280232
#191277432
#191277412
#191277022
#191276642
#417×C832
#309383