Finite Group Databasebeta

← #24915922

Subgroups — #24915922

Order 10624.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#14581838C211yesmaximal, characteristic
2#19128175C212yesmaximal, complemented
3#19127675C211yesmaximal, characteristic, complemented
4#19127674C212yesmaximal, complemented
5#99×D83C218yesmaximal, complemented
6#410×C83C211yesmaximal, characteristic, complemented
7#14581838—24no—
8#14581838—24no—
9C2×#13050591C412yescomplemented
10C2×#13050590C412yescomplemented
11C2×#13050590C2211yescharacteristic
12C22×D332C412yescomplemented
13C22×D332C2211yescharacteristic
14C2×D4×D83C412yescomplemented
15C2×D4×D83C2214yes—
16C4×D332—24no—
17#14581754—24no—
18#14581751C2212yescomplemented
19C22⋊C4×D83C2218yescomplemented
20#14581707C2214yescomplemented
21#14581706C2218yescomplemented
22C2×#41×C83C2211yescharacteristic
23#117×C83C2212yescomplemented
24#99×C83C2214yescomplemented
25C23×D83C2×C414yescomplemented
26C23×D83C2312yes—
27#13050591—28no—
28#13050590—22no—
29#13050590—22no—
30#13050590—24no—
31#13050590—24no—
32C2×Dic166C2×C411yescharacteristic, complemented
33C2×D332—24no—
34C2×D332—24no—
35C2×D332C2×C412yes—
36C2×D332C2×C412yes—
37C2×D332C2×C414yescomplemented
38C2×D332C2×C411yescharacteristic, complemented
39C2×D332D414yes—
40C2×D332D414yes—
41C2×C22⋊D83C2×C414yescomplemented
42C2×C22⋊D83C2312yes—
43D4×D83—216no—
44D4×D83—28no—
45C2×C4×D83—28no—
46C2×C4×D83C2×C412yescomplemented
47C2×C4×D83C2×C414yescomplemented
48C4⋊Dic83—22no—
49C22⋊Dic83C2314yes—
50(C2×D83)⋊C4—28no—
51C4×Dic83—22no—
52#41×C83—22no—
53#41×C83—22no—
54C2×Q8×C83C2×C411yescharacteristic, complemented
55C2×D4×C83C2×C411yescharacteristic, complemented
56C2×D4×C83C2312yes—
57C22×C4×C83C2×C412yescomplemented
58C22×C4×C83C2311yescharacteristic
59C4⋊C4×C83—22no—
60C22⋊C4×C83C2314yes—
61C42×C83—22no—
62C23×C83C22×C412yes—
63C23×C83C2411yescharacteristic
64C22×D83—28no—
65C22×D83—82no—
66C22×D83—84no—
67C22×D83C22×C412yes—
68C22×D83C22×C414yes—
69C22×D83C2×D418yes—
70Q8×C83—22no—
71D4×C83—24no—
72D4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—24no—
76C2×C4×C83C22×C412yes—
77C2×C4×C83C22×C411yescharacteristic
78C2×C4×C83C22×C411yescharacteristic
79C2×C4×C83C2×D412yes—
80C2×C4×C83C2×D412yes—
81C22⋊D83—24no—
82C22⋊D83—28no—
83C2×Dic83—24no—
84C2×Dic83C22×C412yes—
85D332—28no—
86D332—21no—
87D332—21no—
88D332C22⋊C418yes—
89D332C22⋊C414yes—
90D332C22⋊C414yes—
91C4×D83—28no—
92C4×D83—24no—
93C4×D83—88no—
94Dic166—21no—
95Dic166—21no—
96C22×C83—22no—
97C22×C83—41no—
98C22×C83—42no—
99C22×C83C23×C411yescharacteristic
100C22×C83C22×D412yes—
101C2×D83—22no—
102C2×D83—48no—
103C2×D83—44no—
104C2×D83—42no—
105C2×D83—88no—
106C2×D83C2×C22⋊C418yes—
107C4×C83—22no—
108C4×C83—44no—
109C4×C83C2×C22⋊C411yescharacteristic
110C4×C83C2×C22⋊C411yescharacteristic
111C4×C83C2×C22⋊C412yes—
112Dic83—22no—
113Dic83—44no—
114C2×C83—22no—
115C2×C83—21no—
116C2×C83—42no—
117C2×C83C22×C22⋊C411yescharacteristic
118D83—28no—
119D83—42no—
120#2883—831nomaximal, Sylow
121C83#288311yescharacteristic, complemented, Sylow
122C2×H(5,𝔽2)—831no—
123#518—832no—
124#410—831no—
125#410—832no—
126#410D8311yescharacteristic, complemented
127C2×#99—838no—
128H(5,𝔽2)—1664no—
129H(5,𝔽2)—1664no—
130C2×#41—832no—
131C2×#41—832no—
132C2×#41—831no—
133C2×#41C2×D8311yescharacteristic
134C22×D4—832no—
135C22×D4—834no—
136C22×D4—832no—
137C22×D4—831no—
138C4×D4—1664no—
139C4×D4—1664no—
140#117—832no—
141#117C2×D8312yescomplemented
142C2×C22⋊C4—838no—
143#99—834no—
144#99—838no—
145#99C2×D8314yescomplemented
146C24—832no—
147C24—834no—
148#41—22no—
149#41—22no—
150#41—1664no—
151#41—1664no—
152#41—1668no—
153#41—1662no—
154#41—1662no—
155C2×Q8—831no—
156C2×Q8C4×D8311yescharacteristic, complemented
157C2×D4—834no—
158C2×D4—832no—
159C2×D4—832no—
160C2×D4—832no—
161C2×D4—834no—
162C2×D4—831no—
163C2×D4—834no—
164C2×D4—834no—
165C2×D4—1664no—
166C2×D4—1664no—
167C2×D4—1668no—
168C2×D4—16616no—
169C2×D4C4×D8311yescharacteristic, complemented
170C2×D4C22×D8312yes—
171C22×C4—834no—
172C22×C4—832no—
173C22×C4—1668no—
174C22×C4C4×D8312yescomplemented
175C22×C4C22×D8311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—1662no—
178C22⋊C4—834no—
179C22⋊C4—1668no—
180C22⋊C4C22×D8314yes—
181C42—22no—
182C42—1662no—
183C23—834no—
184C23—838no—
185C23—832no—
186C23—6642no—
187C23—6644no—
188C23—1668no—
189C23C2×C4×D8312yes—
190C23C23×D8311yescharacteristic
191Q8—22no—
192Q8—1661no—
193Q8—1661no—
194D4—24no—
195D4—22no—
196D4—834no—
197D4—834no—
198D4—838no—
199D4—1668no—
200D4—1668no—
201D4—1664no—
202D4—1661no—
203D4—1661no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—832no—
208C2×C4—6648no—
209C2×C4—1668no—
210C2×C4—1664no—
211C2×C4—1664no—
212C2×C4C2×C4×D8312yes—
213C2×C4C2×C4×D8311yescharacteristic
214C2×C4C2×C4×D8311yescharacteristic
215C2×C4D4×D8312yes—
216C2×C4D4×D8312yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—3322no—
221C22—3328no—
222C22—3324no—
223C22—838no—
224C22—6648no—
225C22—1662no—
226C22C22×C4×D8311yescharacteristic
227C22C2×D4×D8312yes—
228C4—22no—
229C4—44no—
230C4—3324no—
231C4—1662no—
232C4C22⋊C4×D8311yescharacteristic
233C4C22⋊C4×D8311yescharacteristic
234C4C22⋊C4×D8312yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—3322no—
239C2—1668no—
240C2C2×C22⋊C4×D8311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D83
commutatorC22×C83C23×C4
FrattiniC23C23×D83
Fitting#410×C83C2
socleC2×C83C22×C22⋊C4
prime radicalC2×#14581838C2
cosocleC23×C83C24
hypercenter#410D83
hypocenterC83#2883
LCSG ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #19128175C2semidirect
3. #19127675C2semidirect
4. #19127674C2semidirect
5. #99×D83C2semidirect
6. #410×C83C2semidirect
9. C2×#13050591C4semidirect
10. C2×#13050590C4semidirect
12. C22×D332C4semidirect
14. C2×D4×D83C4semidirect
18. #14581751C22semidirect
19. C22⋊C4×D83C22semidirect
20. #14581707C22semidirect
21. #14581706C22semidirect
23. #117×C83C22semidirect
24. #99×C83C22semidirect
25. C23×D83C2×C4semidirect
32. C2×Dic166C2×C4semidirect
37. C2×D332C2×C4semidirect
38. C2×D332C2×C4semidirect
41. C2×C22⋊D83C2×C4semidirect
46. C2×C4×D83C2×C4semidirect
47. C2×C4×D83C2×C4semidirect
54. C2×Q8×C83C2×C4semidirect
55. C2×D4×C83C2×C4semidirect
57. C22×C4×C83C2×C4semidirect
121. C83#2883semidirect
126. #410D83semidirect
141. #117C2×D83semidirect
145. #99C2×D83semidirect
156. C2×Q8C4×D83semidirect
169. C2×D4C4×D83semidirect
174. C22×C4C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818382
#191281752
#191276752
#191276742
#99×D832
#410×C832
#288383