Finite Group Databasebeta

← #24917401

Subgroups — #24917401

Order 10624.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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×#14581763C211yesmaximal, characteristic
2#117×D83C211yesmaximal, characteristic, complemented
3#19128024C212yesmaximal, complemented
4#19128023C212yesmaximal, complemented
5#19128018C212yesmaximal, complemented
6#19128013C212yesmaximal, complemented
7#19127744C211yesmaximal, characteristic, complemented
8#19127743C211yesmaximal, characteristic, complemented
9#19127706C211yesmaximal, characteristic, complemented
10#19127664C211yesmaximal, characteristic, complemented
11#418×C83C211yesmaximal, characteristic, complemented
12C22×C22⋊D83—21no—
13C22×C4×D83C2211yescharacteristic
14C2×#13050574C2211yescharacteristic
15#14581763—21no—
16#14581763—21no—
17#14581763C2212yes—
18#14581763C2212yes—
19#14581751C2211yescharacteristic
20#14581750C2211yescharacteristic, complemented
21#14581750C2211yescharacteristic, complemented
22#14581749—21no—
23#14581748—21no—
24C2×C4⋊Dic83C2211yescharacteristic
25C4⋊C4×D83C2211yescharacteristic, complemented
26C4⋊C4×D83C2211yescharacteristic, complemented
27C2×C22⋊Dic83—21no—
28C22⋊C4×D83—21no—
29C42×D83—21no—
30#14581726C2212yescomplemented
31#14581725C2212yescomplemented
32#14581724C2212yescomplemented
33#14581723C2212yescomplemented
34#14581713C2212yescomplemented
35#14581713C2212yescomplemented
36(C2×D83)⋊C8C2212yescomplemented
37(C2×D83)⋊C8C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×M16×C83C2211yescharacteristic
40#117×C83C2211yescharacteristic
41#102×C83C2212yescomplemented
42#102×C83C2212yescomplemented
43C23×D83D411yescharacteristic
44C2×C22⋊D83—22no—
45C2×C22⋊D83—22no—
46C2×C22⋊D83—44no—
47C22×Dic83—21no—
48C22×Dic83D411yescharacteristic
49C2×C4×D83—22no—
50C2×C4×D83—22no—
51C2×C4×D83—21no—
52C2×C4×D83—21no—
53C2×C4×D83D411yescharacteristic
54C2×C4×D83D411yescharacteristic
55C2×C4×D83C2311yescharacteristic
56C2×C4×D83C2311yescharacteristic
57#13050574—22no—
58C2×C83⋊C8C2312yes—
59C4⋊Dic83—21no—
60C4⋊Dic83C2312yes—
61C4⋊Dic83C2311yescharacteristic
62C4⋊Dic83C2311yescharacteristic
63Dic83⋊C4—21no—
64Dic83⋊C4—21no—
65C22⋊Dic83—21no—
66C22⋊Dic83—22no—
67C22⋊Dic83—22no—
68(C2×D83)⋊C4—21no—
69(C2×D83)⋊C4—21no—
70C4×Dic83—21no—
71C4×Dic83—21no—
72C4×Dic83—21no—
73C24×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83—22no—
76C2×D4×C83—22no—
77C2×D4×C83C2312yes—
78C22×C4×C83C2311yescharacteristic
79M16×C83—22no—
80C2×C8×C83C2312yes—
81C4⋊C4×C83C2311yescharacteristic
82C4⋊C4×C83C2311yescharacteristic
83C22⋊C4×C83—21no—
84C42×C83—21no—
85C23×C83—22no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83—42no—
89C23×C83C2×D411yescharacteristic
90C22×D83—22no—
91C22×D83—22no—
92C22×D83—41no—
93C22×D83C2×D411yescharacteristic
94C22×D83C2×D411yescharacteristic
95D4×C83—22no—
96D4×C83—22no—
97D4×C83—42no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83C2×D411yescharacteristic
101C2×C4×C83#4112yes—
102C2×C4×C83#4112yes—
103C2×C4×C83C2411yescharacteristic
104C22⋊D83—44no—
105C22⋊D83—44no—
106C2×Dic83—22no—
107C2×Dic83—21no—
108C2×Dic83—21no—
109C2×Dic83—22no—
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—
119C4×D83—42no—
120C4×D83—42no—
121C8×C83—22no—
122C83⋊C8—22no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—84no—
129C22×C83#12312yes—
130C22×C83C22×D411yescharacteristic
131C2×D83—21no—
132C2×D83—21no—
133C2×D83—21no—
134C2×D83—21no—
135C2×D83—21no—
136C2×D83—44no—
137C2×D83AGL(1,ℤ/8)12yes—
138C4×C83—41no—
139C4×C83—41no—
140C4×C83#12312yes—
141C4×C83C2×#4111yescharacteristic
142C4×C83C2×#4111yescharacteristic
143Dic83—21no—
144Dic83—21no—
145Dic83—41no—
146Dic83—42no—
147Dic83—41no—
148C2×C83—22no—
149C2×C83—82no—
150C2×C83C2×#12311yescharacteristic
151C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C83#57511yescharacteristic
153D83—22no—
154D83—41no—
155#3094—831nomaximal, Sylow
156C83#309411yescharacteristic, complemented, Sylow
157C2×#121—831no—
158C2×#117—831no—
159#482—832no—
160#482—832no—
161#481—832no—
162#480—832no—
163#428—831no—
164#428—831no—
165#418—831no—
166#418D8311yescharacteristic, complemented
167#407—831no—
168C22×D4—1661no—
169C22×D4C2×D8311yescharacteristic
170C23×C4—831no—
171C2×M16—831no—
172C2×M16C2×D8311yescharacteristic
173#121—832no—
174#121—832no—
175#121—1661no—
176#121—1661no—
177#117—831no—
178#117—831no—
179#117—831no—
180#117—1661no—
181#117—1661no—
182#117C2×D8311yescharacteristic
183C2×C4⋊C4—831no—
184C2×C4⋊C4—831no—
185C2×C4⋊C4—831no—
186C2×C22⋊C4—1661no—
187C2×C22⋊C4—1661no—
188C2×C42—1661no—
189#107—832no—
190#107—832no—
191#106—832no—
192#106—832no—
193#102—832no—
194#102—832no—
195#102C2×D8312yescomplemented
196#102C2×D8312yescomplemented
197C22⋊C8—832no—
198C22⋊C8—832no—
199C24—21no—
200C24—831no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—3324no—
205C2×D4—1662no—
206C2×D4—1662no—
207C2×D4C22×D8312yes—
208C22×C4—831no—
209C22×C4—831no—
210C22×C4—831no—
211C22×C4—831no—
212C22×C4—831no—
213C22×C4—1661no—
214C22×C4—1661no—
215C22×C4—1661no—
216C22×C4—1662no—
217C22×C4—1662no—
218C22×C4C22×D8311yescharacteristic
219M16—22no—
220M16—1662no—
221C2×C8—832no—
222C2×C8C22×D8312yes—
223C4⋊C4—831no—
224C4⋊C4—831no—
225C4⋊C4—832no—
226C4⋊C4—1661no—
227C4⋊C4—1661no—
228C4⋊C4—1661no—
229C4⋊C4C22×D8311yescharacteristic
230C4⋊C4C22×D8311yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—1661no—
233C22⋊C4—1661no—
234C22⋊C4—1662no—
235C22⋊C4—1662no—
236C22⋊C4—1661no—
237C42—21no—
238C42—1661no—
239C42—1661no—
240C42—1661no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—3321no—
246C23—831no—
247C23—831no—
248C23—1662no—
249C23—1662no—
250C23D4×D8311yescharacteristic
251D4—22no—
252D4—22no—
253D4—42no—
254D4—3324no—
255D4—3324no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—3322no—
259C2×C4—3322no—
260C2×C4—3322no—
261C2×C4—831no—
262C2×C4—831no—
263C2×C4—1661no—
264C2×C4—1661no—
265C2×C4—1662no—
266C2×C4—1662no—
267C2×C4—1662no—
268C2×C4—1662no—
269C2×C4—1661no—
270C2×C4—1661no—
271C2×C4—1661no—
272C2×C4—1661no—
273C2×C4D4×D8311yescharacteristic
274C2×C4#1305058912yes—
275C2×C4#1305059012yes—
276C2×C4C23×D8311yescharacteristic
277C8—22no—
278C8—1662no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—3324no—
286C22—832no—
287C22—1661no—
288C22—1661no—
289C22—1661no—
290C22—1661no—
291C22—1661no—
292C22#1458177412yes—
293C22C2×D4×D8311yescharacteristic
294C4—41no—
295C4—41no—
296C4—3322no—
297C4—3321no—
298C4—3321no—
299C4—1661no—
300C4—1661no—
301C4#1458177412yes—
302C4C2×#1305058911yescharacteristic
303C4C2×#1305059011yescharacteristic
304C2—22no—
305C2—82no—
306C2—3321no—
307C2—1662no—
308C2C2×#1458177411yescharacteristic
309C2AGL(1,ℤ/8)×D8311yescharacteristic
310C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#418×C83C2
socleC22×C83C22×D4
prime radicalC2×#14581763C2
hypercenter#418D83
hypocenterC83#3094
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #117×D83C2semidirect
3. #19128024C2semidirect
4. #19128023C2semidirect
5. #19128018C2semidirect
6. #19128013C2semidirect
7. #19127744C2semidirect
8. #19127743C2semidirect
9. #19127706C2semidirect
10. #19127664C2semidirect
11. #418×C83C2semidirect
20. #14581750C22semidirect
21. #14581750C22semidirect
25. C4⋊C4×D83C22semidirect
26. C4⋊C4×D83C22semidirect
30. #14581726C22semidirect
31. #14581725C22semidirect
32. #14581724C22semidirect
33. #14581723C22semidirect
34. #14581713C22semidirect
35. #14581713C22semidirect
36. (C2×D83)⋊C8C22semidirect
37. (C2×D83)⋊C8C22semidirect
41. #102×C83C22semidirect
42. #102×C83C22semidirect
156. C83#3094semidirect
166. #418D83semidirect
195. #102C2×D83semidirect
196. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817632
#117×D832
#191280242
#191280232
#191280182
#191280132
#191277442
#191277432
#191277062
#191276642
#418×C832
#309483