Finite Group Databasebeta

← #38752069

Subgroups — #38752069

Order 25216.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#27161877C212yesmaximal, complemented
2#27161646C211yesmaximal, characteristic, complemented
3#27161257C214yesmaximal, complemented
4#27161247C214yesmaximal, complemented
5#27161096C212yesmaximal, complemented
6C2×#20263243C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C197C211yesmaximal, characteristic, complemented
8C2×#15691358C2211yescharacteristic, complemented
9#20263357—22no—
10#20263356C2214yescomplemented
11#20263352—22no—
12#20263351C2214yescomplemented
13#20263346—24no—
14C2×#15691342C2212yescomplemented
15#20263322C2212yescomplemented
16#20263313C2212yescomplemented
17#20263297—22no—
18#20263295—22no—
19#20263285C2211yescharacteristic, complemented
20#20263253C2218yescomplemented
21#20263243C2214yescomplemented
22H(5,𝔽2)×C197—24no—
23H(5,𝔽2)×C197—22no—
24H(5,𝔽2)×C197C2214yes—
25C2×#41×C197—22no—
26C2×#41×C197C2212yes—
27C22×D4×C197—22no—
28C22×D4×C197—22no—
29C22×D4×C197C2211yescharacteristic
30#15691358—22no—
31#15691358—22no—
32C2×Dic394C2311yescharacteristic, complemented
33C2×D788C2311yescharacteristic, complemented
34C2×C22⋊D197—21no—
35C2×C22⋊D197—24no—
36C2×C4×D197—21no—
37#15691348—24no—
38#15691346—24no—
39#15691345—24no—
40#15691343—24no—
41#15691342—22no—
42#15691342C2314yescomplemented
43C2×#13096802—22no—
44C4⋊Dic197—21no—
45C22⋊Dic197—21no—
46C22⋊Dic197—24no—
47C4×Dic197C2312yescomplemented
48C24×C197—22no—
49C24×C197D412yes—
50#41×C197—28no—
51#41×C197—28no—
52#41×C197—22no—
53#41×C197—22no—
54#41×C197D414yescomplemented
55#41×C197C2314yes—
56C2×Q8×C197D412yescomplemented
57C2×D4×C197—24no—
58C2×D4×C197—28no—
59C2×D4×C197—28no—
60C2×D4×C197—22no—
61C2×D4×C197—28no—
62C2×D4×C197—28no—
63C2×D4×C197—22no—
64C2×D4×C197—22no—
65C2×D4×C197D412yes—
66C2×D4×C197D412yescomplemented
67C2×D4×C197C2312yes—
68C22×C4×C197—22no—
69C22×C4×C197—22no—
70C22×C4×C197C2311yescharacteristic
71C22×D197—21no—
72Dic394—22no—
73D788—22no—
74C22⋊D197—41no—
75C22⋊D197—41no—
76C22⋊D197—84no—
77C2×Dic197—22no—
78C2×Dic197—21no—
79C4×D197—42no—
80#13096802—24no—
81C23×C197—28no—
82C23×C197—22no—
83C23×C197—22no—
84C23×C197—22no—
85C23×C197—82no—
86C23×C197—84no—
87C23×C197C2×D414yes—
88C23×C197C2×D411yescharacteristic
89Q8×C197—22no—
90Q8×C197C2×D414yes—
91D4×C197—216no—
92D4×C197—24no—
93D4×C197—24no—
94D4×C197—24no—
95D4×C197—22no—
96D4×C197—24no—
97D4×C197C2×D414yes—
98C2×C4×C197—28no—
99C2×C4×C197—22no—
100C2×C4×C197—24no—
101C2×C4×C197—24no—
102C2×C4×C197C2×D411yescharacteristic
103C2×C4×C197C2×D412yes—
104C2×C4×C197C2×D412yes—
105C2×C4×C197C2411yescharacteristic
106C22×C197—24no—
107C22×C197—24no—
108C22×C197—41no—
109C22×C197—44no—
110C22×C197—44no—
111C22×C197—44no—
112C22×C197—82no—
113C22×C197—88no—
114C22×C197#12011yescharacteristic
115C22×C197#12011yescharacteristic
116C22×C197C22×D411yescharacteristic
117C2×D197—42no—
118C2×D197—41no—
119C4×C197—24no—
120C4×C197#12012yes—
121C4×C197C22×D412yes—
122Dic197—42no—
123Dic197—41no—
124C2×C197—21no—
125C2×C197—21no—
126C2×C197—21no—
127C2×C197—44no—
128C2×C197—44no—
129C2×C197C2×#12011yescharacteristic
130D197—81no—
131C197#301911yescharacteristic, complemented, Sylow
132#3019—1971nomaximal, Sylow
133C2×H(5,𝔽2)D19711yescharacteristic, complemented
134#575—1972no—
135#535—1971no—
136#455—1974no—
137#453—1974no—
138#421—1972no—
139C2×#100—1971no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D19714yes—
143C2×#41—22no—
144C2×#41—1971no—
145C2×#41C2×D19712yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D19711yescharacteristic
149#137—3942no—
150#137—1974no—
151AGL(1,ℤ/8)—3942no—
152AGL(1,ℤ/8)—1974no—
153#135—3944no—
154C2×M16—1972no—
155#127—1972no—
156#124—1972no—
157#121—3942no—
158#120—3942no—
159#117—1971no—
160#104—1978no—
161#100—1974no—
162C24—22no—
163C24C22⋊D19712yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—3942no—
169#41—3942no—
170#41C22⋊D19714yescomplemented
171#41C22×D19714yes—
172C2×Q8—1971no—
173C2×Q8C22⋊D19712yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—3941no—
183C2×D4—3944no—
184C2×D4—1971no—
185C2×D4C22⋊D19712yes—
186C2×D4C22⋊D19712yescomplemented
187C2×D4C22×D19712yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—3941no—
191C22×C4C22×D19711yescharacteristic
192Q16—3944no—
193SD16—3944no—
194SD16—3944no—
195D8—3944no—
196M16—3942no—
197M16—1974no—
198C2×C8—3942no—
199C4⋊C4—3941no—
200C22⋊C4—3941no—
201C22⋊C4—3944no—
202C42—1972no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—3941no—
210C23C2×C22⋊D19714yes—
211C23C2×C22⋊D19711yescharacteristic
212Q8—22no—
213Q8—3942no—
214Q8C2×C22⋊D19714yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—7881no—
222D4—7881no—
223D4—15764no—
224D4—3942no—
225D4C2×C22⋊D19714yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—7882no—
231C2×C4—3942no—
232C2×C4—3941no—
233C2×C4C2×C22⋊D19711yescharacteristic
234C2×C4C2×C22⋊D19712yes—
235C2×C4C2×C22⋊D19712yes—
236C2×C4C23×D19711yescharacteristic
237C8—3944no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—7882no—
247C22—7881no—
248C22#2026329511yescharacteristic
249C22#2026329511yescharacteristic
250C22C22×C22⋊D19711yescharacteristic
251C4—24no—
252C4—7882no—
253C4—7881no—
254C4#2026329512yes—
255C4C22×C22⋊D19712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—15761no—
262C2C2×#2026329511yescharacteristic
subgroupG/H
centerC2C2×#20263295
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×H(5,𝔽2)×C197C2
socleC2×C197C2×#120
hypercenterC2×H(5,𝔽2)D197
hypocenterC197#3019
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #27161877C2semidirect
2. #27161646C2semidirect
3. #27161257C2semidirect
4. #27161247C2semidirect
5. #27161096C2semidirect
6. C2×#20263243C2semidirect
7. C2×H(5,𝔽2)×C197C2semidirect
8. C2×#15691358C22semidirect
10. #20263356C22semidirect
12. #20263351C22semidirect
14. C2×#15691342C22semidirect
15. #20263322C22semidirect
16. #20263313C22semidirect
19. #20263285C22semidirect
20. #20263253C22semidirect
21. #20263243C22semidirect
32. C2×Dic394C23semidirect
33. C2×D788C23semidirect
42. #15691342C23semidirect
47. C4×Dic197C23semidirect
54. #41×C197D4semidirect
56. C2×Q8×C197D4semidirect
66. C2×D4×C197D4semidirect
131. C197#3019semidirect
133. C2×H(5,𝔽2)D197semidirect
170. #41C22⋊D197semidirect
173. C2×Q8C22⋊D197semidirect
186. C2×D4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271618772
#271616462
#271612572
#271612472
#271610962
C2×#202632432
C2×H(5,𝔽2)×C1972
#3019197