Finite Group Databasebeta

← #38751171

Subgroups — #38751171

Order 25216.

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×#20263372C211yesmaximal, characteristic
2#27161553C212yesmaximal, complemented
3#27161053C211yesmaximal, characteristic, complemented
4#27161052C212yesmaximal, complemented
5#99×D197C218yesmaximal, complemented
6#410×C197C211yesmaximal, characteristic, complemented
7#20263372—24no—
8#20263372—24no—
9C2×#15691359C412yescomplemented
10C2×#15691358C412yescomplemented
11C2×#15691358C2211yescharacteristic
12C22×D788C412yescomplemented
13C22×D788C2211yescharacteristic
14C2×D4×D197C412yescomplemented
15C2×D4×D197C2214yes—
16C4×D788—24no—
17#20263288—24no—
18#20263285C2212yescomplemented
19C22⋊C4×D197C2218yescomplemented
20#20263241C2214yescomplemented
21#20263240C2218yescomplemented
22C2×#41×C197C2211yescharacteristic
23#117×C197C2212yescomplemented
24#99×C197C2214yescomplemented
25C23×D197C2×C414yescomplemented
26C23×D197C2312yes—
27#15691359—28no—
28#15691358—24no—
29#15691358—24no—
30#15691358—22no—
31#15691358—22no—
32C2×Dic394C2×C411yescharacteristic, complemented
33C2×D788—24no—
34C2×D788—24no—
35C2×D788C2×C412yes—
36C2×D788C2×C412yes—
37C2×D788C2×C414yescomplemented
38C2×D788C2×C411yescharacteristic, complemented
39C2×D788D414yes—
40C2×D788D414yes—
41C2×C22⋊D197C2×C414yescomplemented
42C2×C22⋊D197C2312yes—
43D4×D197—216no—
44D4×D197—28no—
45C2×C4×D197—28no—
46C2×C4×D197C2×C412yescomplemented
47C2×C4×D197C2×C414yescomplemented
48C4⋊Dic197—22no—
49C22⋊Dic197C2314yes—
50#15691335—28no—
51C4×Dic197—22no—
52#41×C197—22no—
53#41×C197—22no—
54C2×Q8×C197C2×C411yescharacteristic, complemented
55C2×D4×C197C2×C411yescharacteristic, complemented
56C2×D4×C197C2312yes—
57C22×C4×C197C2×C412yescomplemented
58C22×C4×C197C2311yescharacteristic
59C4⋊C4×C197—22no—
60C22⋊C4×C197C2314yes—
61C42×C197—22no—
62C22×D197—28no—
63C22×D197—82no—
64C22×D197—84no—
65C22×D197C22×C412yes—
66C22×D197C22×C414yes—
67C22×D197C2×D418yes—
68Dic394—21no—
69Dic394—21no—
70D788—28no—
71D788—21no—
72D788—21no—
73D788C22⋊C418yes—
74D788C22⋊C414yes—
75D788C22⋊C414yes—
76C22⋊D197—24no—
77C22⋊D197—28no—
78C2×Dic197—24no—
79C2×Dic197C22×C412yes—
80C4×D197—28no—
81C4×D197—24no—
82C4×D197—88no—
83C23×C197C22×C412yes—
84C23×C197C2411yescharacteristic
85Q8×C197—22no—
86D4×C197—24no—
87D4×C197—22no—
88C2×C4×C197—22no—
89C2×C4×C197—22no—
90C2×C4×C197—24no—
91C2×C4×C197C22×C412yes—
92C2×C4×C197C22×C411yescharacteristic
93C2×C4×C197C22×C411yescharacteristic
94C2×C4×C197C2×D412yes—
95C2×C4×C197C2×D412yes—
96C22×C197—22no—
97C22×C197—41no—
98C22×C197—42no—
99C22×C197C23×C411yescharacteristic
100C22×C197C22×D412yes—
101C2×D197—22no—
102C2×D197—48no—
103C2×D197—44no—
104C2×D197—42no—
105C2×D197—88no—
106C2×D197C2×C22⋊C418yes—
107C4×C197—22no—
108C4×C197—44no—
109C4×C197C2×C22⋊C411yescharacteristic
110C4×C197C2×C22⋊C411yescharacteristic
111C4×C197C2×C22⋊C412yes—
112Dic197—22no—
113Dic197—44no—
114C2×C197—22no—
115C2×C197—21no—
116C2×C197—42no—
117C2×C197C22×C22⋊C411yescharacteristic
118D197—28no—
119D197—42no—
120C197#288311yescharacteristic, complemented, Sylow
121#2883—1971nomaximal, Sylow
122C2×H(5,𝔽2)—1971no—
123#518—1972no—
124#410—1971no—
125#410—1972no—
126#410D19711yescharacteristic, complemented
127C2×#99—1978no—
128H(5,𝔽2)—3944no—
129H(5,𝔽2)—3944no—
130C2×#41—1972no—
131C2×#41—1972no—
132C2×#41—1971no—
133C2×#41C2×D19711yescharacteristic
134C22×D4—1972no—
135C22×D4—1974no—
136C22×D4—1972no—
137C22×D4—1971no—
138C4×D4—3944no—
139C4×D4—3944no—
140#117—1972no—
141#117C2×D19712yescomplemented
142C2×C22⋊C4—1978no—
143#99—1974no—
144#99—1978no—
145#99C2×D19714yescomplemented
146C24—1974no—
147C24—1972no—
148#41—22no—
149#41—22no—
150#41—3944no—
151#41—3944no—
152#41—3948no—
153#41—3942no—
154#41—3942no—
155C2×Q8—1971no—
156C2×Q8C4×D19711yescharacteristic, complemented
157C2×D4—3944no—
158C2×D4—3944no—
159C2×D4—39416no—
160C2×D4—3948no—
161C2×D4—1974no—
162C2×D4—1972no—
163C2×D4—1974no—
164C2×D4—1971no—
165C2×D4—1972no—
166C2×D4—1972no—
167C2×D4—1974no—
168C2×D4—1974no—
169C2×D4C4×D19711yescharacteristic, complemented
170C2×D4C22×D19712yes—
171C22×C4—3948no—
172C22×C4—1974no—
173C22×C4—1972no—
174C22×C4C4×D19712yescomplemented
175C22×C4C22×D19711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—3942no—
178C22⋊C4—3948no—
179C22⋊C4—1974no—
180C22⋊C4C22×D19714yes—
181C42—22no—
182C42—3942no—
183C23—15762no—
184C23—15764no—
185C23—3948no—
186C23—1972no—
187C23—1974no—
188C23—1978no—
189C23C2×C4×D19712yes—
190C23C23×D19711yescharacteristic
191Q8—22no—
192Q8—3941no—
193Q8—3941no—
194D4—24no—
195D4—22no—
196D4—3948no—
197D4—3948no—
198D4—3944no—
199D4—3941no—
200D4—3941no—
201D4—1974no—
202D4—1974no—
203D4—1978no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—15768no—
208C2×C4—3948no—
209C2×C4—3944no—
210C2×C4—3944no—
211C2×C4—1972no—
212C2×C4C2×C4×D19712yes—
213C2×C4C2×C4×D19711yescharacteristic
214C2×C4C2×C4×D19711yescharacteristic
215C2×C4D4×D19712yes—
216C2×C4D4×D19712yes—
217C22—22no—
218C22—41no—
219C22—42no—
220C22—7882no—
221C22—7888no—
222C22—7884no—
223C22—15768no—
224C22—3942no—
225C22—1978no—
226C22C22×C4×D19711yescharacteristic
227C22C2×D4×D19712yes—
228C4—22no—
229C4—44no—
230C4—7884no—
231C4—3942no—
232C4C22⋊C4×D19711yescharacteristic
233C4C22⋊C4×D19711yescharacteristic
234C4C22⋊C4×D19712yes—
235C2—22no—
236C2—21no—
237C2—42no—
238C2—7882no—
239C2—3948no—
240C2C2×C22⋊C4×D19711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D197
commutatorC22×C197C23×C4
FrattiniC23C23×D197
Fitting#410×C197C2
socleC2×C197C22×C22⋊C4
prime radicalC2×#20263372C2
cosocleC23×C197C24
hypercenter#410D197
hypocenterC197#2883
LCSG ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27161553C2semidirect
3. #27161053C2semidirect
4. #27161052C2semidirect
5. #99×D197C2semidirect
6. #410×C197C2semidirect
9. C2×#15691359C4semidirect
10. C2×#15691358C4semidirect
12. C22×D788C4semidirect
14. C2×D4×D197C4semidirect
18. #20263285C22semidirect
19. C22⋊C4×D197C22semidirect
20. #20263241C22semidirect
21. #20263240C22semidirect
23. #117×C197C22semidirect
24. #99×C197C22semidirect
25. C23×D197C2×C4semidirect
32. C2×Dic394C2×C4semidirect
37. C2×D788C2×C4semidirect
38. C2×D788C2×C4semidirect
41. C2×C22⋊D197C2×C4semidirect
46. C2×C4×D197C2×C4semidirect
47. C2×C4×D197C2×C4semidirect
54. C2×Q8×C197C2×C4semidirect
55. C2×D4×C197C2×C4semidirect
57. C22×C4×C197C2×C4semidirect
120. C197#2883semidirect
126. #410D197semidirect
141. #117C2×D197semidirect
145. #99C2×D197semidirect
156. C2×Q8C4×D197semidirect
169. C2×D4C4×D197semidirect
174. C22×C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633722
#271615532
#271610532
#271610522
#99×D1972
#410×C1972
#2883197