Finite Group Databasebeta

← #24917397

Subgroups — #24917397

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#19128213C211yesmaximal, characteristic
2C2×#14581750C211yesmaximal, characteristic, complemented
3#19128045C212yesmaximal, complemented
4#19128044C212yesmaximal, complemented
5#19128028C212yesmaximal, complemented
6#19128027C212yesmaximal, complemented
7#19127744C211yesmaximal, characteristic, complemented
8#19127703C211yesmaximal, characteristic, complemented
9#19127700C211yesmaximal, characteristic, complemented
10#19127664C211yesmaximal, characteristic, complemented
11#428×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C2×#13050574C2211yescharacteristic
15#14581776—21no—
16#14581772C2211yescharacteristic
17#14581772C2211yescharacteristic
18#14581767C2211yescharacteristic
19#14581767C2211yescharacteristic
20#14581762—21no—
21C4×D332—21no—
22C4×D332—21no—
23#14581750—21no—
24#14581750—21no—
25#14581750C2212yes—
26#14581750C2212yes—
27C2×C4⋊Dic83C2211yescharacteristic
28C2×(C2×D83)⋊C4—21no—
29C2×C4×Dic83—21no—
30#14581725C2212yescomplemented
31#14581723C2212yescomplemented
32#14581717C2212yescomplemented
33#14581715C2212yescomplemented
34#14581714C2212yescomplemented
35#14581712C2212yescomplemented
36(C2×D83)⋊C8C2212yes—
37(C2×D83)⋊C8C2212yes—
38C2×M16×C83C2211yescharacteristic
39#117×C83C2211yescharacteristic
40C2×C4⋊C4×C83C2211yescharacteristic
41#107×C83C2212yescomplemented
42#106×C83C2212yescomplemented
43C23×D83D411yescharacteristic
44#13050590—22no—
45#13050590—22no—
46C2×Dic166C2311yescharacteristic
47C2×D332—21no—
48C2×D332—21no—
49C2×D332C2311yescharacteristic
50C2×C22⋊D83—21no—
51C22×Dic83—21no—
52C22×Dic83D411yescharacteristic
53C2×C4×D83—22no—
54C2×C4×D83—22no—
55C2×C4×D83—21no—
56C2×C4×D83D411yescharacteristic
57C2×C4×D83D411yescharacteristic
58C2×C4×D83C2311yescharacteristic
59C2×C4×D83C2311yescharacteristic
60#13050574—22no—
61C2×C83⋊C8C2312yes—
62C4⋊Dic83—21no—
63C4⋊Dic83—21no—
64C4⋊Dic83—21no—
65C4⋊Dic83C2312yes—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68(C2×D83)⋊C4—22no—
69(C2×D83)⋊C4—22no—
70(C2×D83)⋊C4—21no—
71(C2×D83)⋊C4—21no—
72C4×Dic83—22no—
73C4×Dic83—22no—
74C22×C4×C83—21no—
75C22×C4×C83C2311yescharacteristic
76M16×C83—22no—
77C2×C8×C83C2312yes—
78C4⋊C4×C83—21no—
79C4⋊C4×C83C2311yescharacteristic
80C4⋊C4×C83C2311yescharacteristic
81C4⋊C4×C83C2312yes—
82C22⋊C4×C83—21no—
83C42×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—22no—
86C22×D83—21no—
87C22×D83—41no—
88C22×D83—41no—
89C22×D83C2×D411yescharacteristic
90C22×D83C2×D411yescharacteristic
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—42no—
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83#4112yes—
97C2×C4×C83#4112yes—
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—42no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—42no—
104C2×Dic83C2×D411yescharacteristic
105C2×Dic83C2×D411yescharacteristic
106D332—21no—
107D332—21no—
108D332—42no—
109D332—42no—
110C4×D83—22no—
111C4×D83—22no—
112C4×D83—21no—
113C4×D83—21no—
114C4×D83—21no—
115C4×D83—21no—
116C4×D83—42no—
117Dic166—21no—
118Dic166—21no—
119C8×C83—22no—
120C83⋊C8—22no—
121C22×C83—21no—
122C22×C83—21no—
123C22×C83#12312yes—
124C22×C83C22×D411yescharacteristic
125C2×D83—21no—
126C2×D83—21no—
127C2×D83—21no—
128C2×D83—21no—
129C2×D83—21no—
130C2×D83—21no—
131C2×D83—44no—
132C2×D83—41no—
133C2×D83—41no—
134C2×D83—41no—
135C4×C83—41no—
136C4×C83—41no—
137C4×C83—42no—
138C4×C83#12312yes—
139C4×C83C2×#4111yescharacteristic
140C4×C83C2×#4111yescharacteristic
141Dic83—21no—
142Dic83—21no—
143Dic83—41no—
144Dic83—42no—
145C2×C83—22no—
146C2×C83C2×#12311yescharacteristic
147C2×C83#57511yescharacteristic
148C2×C83#57511yescharacteristic
149D83—41no—
150D83—41no—
151D83—81no—
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#428D8311yescharacteristic, complemented
162#417—831no—
163#417—831no—
164#407—831no—
165C2×#41—831no—
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—832no—
178#117—832no—
179#117—1661no—
180#117—1661no—
181#117C2×D8311yescharacteristic
182C2×C4⋊C4—831no—
183C2×C4⋊C4C2×D8311yescharacteristic
184C2×C22⋊C4—1661no—
185C2×C42—1661no—
186#107—832no—
187#107C2×D8312yescomplemented
188#106—832no—
189#106C2×D8312yescomplemented
190#103—832no—
191#103—832no—
192#102—832no—
193#102—832no—
194C22⋊C8—832no—
195C22⋊C8—832no—
196C24—831no—
197#41—1662no—
198#41—1662no—
199C2×Q8—831no—
200C2×D4—831no—
201C2×D4—1661no—
202C2×D4—1661no—
203C2×D4—1661no—
204C22×C4—21no—
205C22×C4—831no—
206C22×C4—831no—
207C22×C4—831no—
208C22×C4—831no—
209C22×C4—831no—
210C22×C4—1661no—
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—1661no—
222C4⋊C4—1661no—
223C4⋊C4—1661no—
224C4⋊C4C22×D8312yes—
225C4⋊C4C22×D8311yescharacteristic
226C4⋊C4C22×D8311yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—1661no—
229C22⋊C4—1661no—
230C22⋊C4—1662no—
231C22⋊C4—1662no—
232C22⋊C4—1661no—
233C22⋊C4—1661no—
234C42—21no—
235C42—1662no—
236C42—1662no—
237C23—3321no—
238C23—3321no—
239C23—831no—
240C23—831no—
241C23—1661no—
242C23—1662no—
243C23D4×D8311yescharacteristic
244Q8—1661no—
245Q8—1661no—
246D4—3322no—
247D4—3322no—
248D4—3322no—
249D4—1661no—
250D4—1661no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—831no—
258C2×C4—831no—
259C2×C4—1662no—
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#1305059012yes—
270C2×C4#1305059112yes—
271C2×C4C23×D8311yescharacteristic
272C8—22no—
273C8—1662no—
274C22—21no—
275C22—21no—
276C22—3324no—
277C22—3321no—
278C22—3321no—
279C22—3321no—
280C22—1661no—
281C22—1661no—
282C22—1661no—
283C22—1661no—
284C22—1661no—
285C22—1661no—
286C22#1458177712yes—
287C22C2×D4×D8311yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—3321no—
292C4—3322no—
293C4—1661no—
294C4—1661no—
295C4#1458177712yes—
296C4C2×#1305059011yescharacteristic
297C4C2×#1305059111yescharacteristic
298C2—22no—
299C2—3321no—
300C2—3321no—
301C2—6641no—
302C2C2×#1458177711yescharacteristic
303C2#1912849711yescharacteristic
304C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#428×C83C2
socleC22×C83C22×D4
prime radical#19128213C2
hypercenter#428D83
hypocenterC83#3093
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#14581750C2semidirect
3. #19128045C2semidirect
4. #19128044C2semidirect
5. #19128028C2semidirect
6. #19128027C2semidirect
7. #19127744C2semidirect
8. #19127703C2semidirect
9. #19127700C2semidirect
10. #19127664C2semidirect
11. #428×C83C2semidirect
30. #14581725C22semidirect
31. #14581723C22semidirect
32. #14581717C22semidirect
33. #14581715C22semidirect
34. #14581714C22semidirect
35. #14581712C22semidirect
41. #107×C83C22semidirect
42. #106×C83C22semidirect
153. C83#3093semidirect
161. #428D83semidirect
187. #107C2×D83semidirect
189. #106C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282132
C2×#145817502
#191280452
#191280442
#191280282
#191280272
#191277442
#191277032
#191277002
#191276642
#428×C832
#309383