Finite Group Databasebeta

← #24917378

Subgroups — #24917378

Order 10624.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#19128212C211yesmaximal, characteristic
2C2×#14581751C211yesmaximal, characteristic, complemented
3#19128043C212yesmaximal, complemented
4#19128037C211yesmaximal, characteristic, complemented
5#19128032C211yesmaximal, characteristic, complemented
6#19128026C212yesmaximal, complemented
7#19128020C211yesmaximal, characteristic, complemented
8#19128015C211yesmaximal, characteristic, complemented
9#19127739C212yesmaximal, complemented
10#19127698C212yesmaximal, complemented
11C2×C22⋊C8×C83C211yesmaximal, characteristic, complemented
12C2×#13050590C2211yescharacteristic
13#14581775—21no—
14#14581771C2211yescharacteristic
15#14581771C2211yescharacteristic
16#14581766C2211yescharacteristic
17#14581766C2211yescharacteristic
18#14581761—21no—
19C4×C22⋊D83—21no—
20C4×C22⋊D83—21no—
21#14581751—21no—
22#14581751—21no—
23#14581751C2212yes—
24#14581751C2212yes—
25#14581751C2211yescharacteristic
26C2×C4⋊Dic83C2211yescharacteristic
27C2×C4⋊Dic83C2211yescharacteristic
28C2×C22⋊Dic83—21no—
29C2×C4×Dic83—21no—
30#14581726C2212yescomplemented
31#14581726C2212yescomplemented
32#14581724C2212yescomplemented
33#14581724C2212yescomplemented
34#14581717C2212yescomplemented
35#14581717C2212yescomplemented
36#14581714C2212yescomplemented
37#14581714C2212yescomplemented
38C23×C4×C83C2211yescharacteristic
39C22×C8×C83C2212yes—
40C22⋊C8×C83C2211yescharacteristic
41C22⋊C8×C83C2211yescharacteristic
42C22⋊C8×C83C2212yes—
43#13050590—24no—
44C2×Dic166C2311yescharacteristic
45C2×D332C2311yescharacteristic
46C2×C22⋊D83—21no—
47C2×C22⋊D83—21no—
48C2×C22⋊D83—21no—
49C22×Dic83—21no—
50C22×Dic83—21no—
51C2×C4×D83—21no—
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83C2311yescharacteristic
55C4⋊Dic83C2311yescharacteristic
56C4⋊Dic83C2312yes—
57C4⋊Dic83C2312yes—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60C22⋊Dic83—21no—
61C22⋊Dic83—21no—
62C22⋊Dic83—21no—
63C22⋊Dic83—22no—
64C22⋊Dic83—22no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67C4×Dic83—22no—
68C4×Dic83—22no—
69C4×Dic83—21no—
70C24×C83D411yescharacteristic
71C22×C4×C83—24no—
72C22×C4×C83D411yescharacteristic
73C22×C4×C83D411yescharacteristic
74C22×C4×C83D411yescharacteristic
75C22×C4×C83C2311yescharacteristic
76C22×C4×C83C2311yescharacteristic
77C22×C4×C83C2311yescharacteristic
78C2×C8×C83—24no—
79C2×C8×C83C2312yes—
80C2×C8×C83C2312yes—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—41no—
84C23×C83C2×D411yescharacteristic
85C23×C83C2×D411yescharacteristic
86C23×C83C2×D411yescharacteristic
87C22×D83—21no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83#4114yes—
97C2×C4×C83C2411yescharacteristic
98C22⋊D83—42no—
99C22⋊D83—42no—
100C22⋊D83—42no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—21no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—42no—
107C2×Dic83—42no—
108D332—22no—
109C4×D83—42no—
110Dic166—22no—
111C8×C83—22no—
112C8×C83—22no—
113C22×C83—21no—
114C22×C83—21no—
115C22×C83—21no—
116C22×C83—21no—
117C22×C83—21no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83—44no—
121C22×C83#12312yes—
122C22×C83#13512yes—
123C22×C83C22×D411yescharacteristic
124C2×D83—41no—
125C2×D83—41no—
126C2×D83—41no—
127C4×C83—21no—
128C4×C83—21no—
129C4×C83#12312yes—
130C4×C83C2×#4112yes—
131Dic83—41no—
132Dic83—42no—
133Dic83—41no—
134Dic83—42no—
135Dic83—41no—
136C2×C83—22no—
137C2×C83—22no—
138C2×C83—41no—
139C2×C83C2×#12311yescharacteristic
140C2×C83C2×#13511yescharacteristic
141C2×C83#57511yescharacteristic
142D83—81no—
143#3091—831nomaximal, Sylow
144C83#309111yescharacteristic, complemented, Sylow
145#525—831no—
146C2×#117—831no—
147#485—832no—
148#484—831no—
149#483—831no—
150#482—832no—
151#481—831no—
152#480—831no—
153#427—832no—
154#416—832no—
155C2×C22⋊C8D8311yescharacteristic, complemented
156C2×#41—831no—
157C23×C4C2×D8311yescharacteristic
158C22×C8C2×D8312yes—
159#123—1661no—
160C22⋊Q8—831no—
161C22⋊Q8—831no—
162#121—831no—
163#121—831no—
164#120—1661no—
165C4×D4—1661no—
166C4×D4—1661no—
167#117—832no—
168#117—832no—
169#117—831no—
170#117—1661no—
171#117—1661no—
172C2×C4⋊C4—831no—
173C2×C4⋊C4—831no—
174C2×C22⋊C4—1661no—
175C2×C42—1661no—
176#107—832no—
177#107—832no—
178#106—832no—
179#106—832no—
180#103—832no—
181#103—832no—
182#102—832no—
183#102—832no—
184C22⋊C8C2×D8311yescharacteristic
185C22⋊C8C2×D8311yescharacteristic
186C22⋊C8C2×D8312yes—
187C24D33211yescharacteristic
188#41—1664no—
189C2×Q8—831no—
190C2×D4—831no—
191C2×D4—1661no—
192C2×D4—1661no—
193C2×D4—1661no—
194C22×C4—24no—
195C22×C4—1661no—
196C22×C4—1661no—
197C22×C4—1661no—
198C22×C4D33211yescharacteristic
199C22×C4D33211yescharacteristic
200C22×C4D33211yescharacteristic
201C22×C4C22×D8311yescharacteristic
202C22×C4C22×D8311yescharacteristic
203C22×C4C22×D8311yescharacteristic
204C2×C8—24no—
205C2×C8C22×D8312yes—
206C2×C8C22×D8312yes—
207C4⋊C4—832no—
208C4⋊C4—832no—
209C4⋊C4—831no—
210C4⋊C4—831no—
211C4⋊C4—1661no—
212C4⋊C4—1661no—
213C4⋊C4—1661no—
214C4⋊C4—1661no—
215C22⋊C4—1662no—
216C22⋊C4—1662no—
217C22⋊C4—1661no—
218C22⋊C4—1661no—
219C22⋊C4—1661no—
220C22⋊C4—1661no—
221C22⋊C4—1661no—
222C42—1662no—
223C42—1662no—
224C42—1661no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—1661no—
229C23C2×D33211yescharacteristic
230C23C2×D33211yescharacteristic
231C23C2×D33211yescharacteristic
232Q8—1662no—
233D4—3322no—
234D4—3322no—
235D4—3322no—
236D4—1662no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—3322no—
243C2×C4—3322no—
244C2×C4—3322no—
245C2×C4—1662no—
246C2×C4—1662no—
247C2×C4—1661no—
248C2×C4—1661no—
249C2×C4—1661no—
250C2×C4C2×D33211yescharacteristic
251C2×C4C2×D33211yescharacteristic
252C2×C4C2×D33211yescharacteristic
253C2×C4#1305058914yes—
254C2×C4C23×D8311yescharacteristic
255C8—22no—
256C8—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—3321no—
266C22—3321no—
267C22—3321no—
268C22#1458177612yes—
269C22#1458181412yes—
270C22C22×D33211yescharacteristic
271C4—21no—
272C4—21no—
273C4—3321no—
274C4—3322no—
275C4—3321no—
276C4—3322no—
277C4—3321no—
278C4#1458177612yes—
279C4C2×#1305058912yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—6641no—
284C2C2×#1458177611yescharacteristic
285C2C2×#1458181411yescharacteristic
286C2#1912850111yescharacteristic
subgroupG/H
centerC2×C4C2×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×C22⋊C8×C83C2
socleC22×C83C22×D4
prime radical#19128212C2
hypercenterC2×C22⋊C8D83
hypocenterC83#3091
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#14581751C2semidirect
3. #19128043C2semidirect
4. #19128037C2semidirect
5. #19128032C2semidirect
6. #19128026C2semidirect
7. #19128020C2semidirect
8. #19128015C2semidirect
9. #19127739C2semidirect
10. #19127698C2semidirect
11. C2×C22⋊C8×C83C2semidirect
30. #14581726C22semidirect
31. #14581726C22semidirect
32. #14581724C22semidirect
33. #14581724C22semidirect
34. #14581717C22semidirect
35. #14581717C22semidirect
36. #14581714C22semidirect
37. #14581714C22semidirect
144. C83#3091semidirect
155. C2×C22⋊C8D83semidirect

Maximal subgroups

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

subgroupindex
#191282122
C2×#145817512
#191280432
#191280372
#191280322
#191280262
#191280202
#191280152
#191277392
#191276982
C2×C22⋊C8×C832
#309183