Finite Group Databasebeta

← #38752068

Subgroups — #38752068

Order 25216.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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×#20263371C211yesmaximal, characteristic, complemented
2#27161877C212yesmaximal, complemented
3#27161256C214yesmaximal, complemented
4#27161246C214yesmaximal, complemented
5#27161097C212yesmaximal, complemented
6C2×#20263243C211yesmaximal, characteristic, complemented
7#535×C197C211yesmaximal, characteristic, complemented
8#20263371—24no—
9#20263371—22no—
10#20263371C2214yes—
11C2×#15691358C2212yes—
12C2×#15691357—22no—
13C22×C22⋊D197—22no—
14C2×D4×D197—22no—
15#20263357C2214yescomplemented
16#20263356—22no—
17#20263352C2214yescomplemented
18#20263351—22no—
19#20263346—24no—
20C2×#15691342C2212yescomplemented
21#20263254C2218yescomplemented
22#20263243C2214yescomplemented
23C2×#41×C197C2211yescharacteristic, complemented
24C22×D4×C197C2211yescharacteristic
25#127×C197C2212yescomplemented
26#124×C197C2212yescomplemented
27#121×C197—22no—
28#120×C197—22no—
29#117×C197C2211yescharacteristic, complemented
30C23×D197—22no—
31#15691358—22no—
32#15691358—22no—
33#15691358D414yescomplemented
34#15691358C2314yes—
35#15691357—28no—
36#15691357—28no—
37C2×Dic394D412yescomplemented
38C2×D788D412yescomplemented
39C2×C22⋊D197—24no—
40C2×C22⋊D197—28no—
41C2×C22⋊D197—22no—
42C2×C22⋊D197—28no—
43D4×D197—28no—
44D4×D197—28no—
45C22×Dic197—22no—
46C2×C4×D197—22no—
47#15691348—24no—
48#15691346—24no—
49#15691345—24no—
50#15691343—24no—
51#15691342—22no—
52#15691342C2314yescomplemented
53C2×#13096802—22no—
54C24×C197D412yes—
55#41×C197—22no—
56#41×C197—22no—
57C2×Q8×C197C2311yescharacteristic, complemented
58C2×D4×C197—21no—
59C2×D4×C197—24no—
60C2×D4×C197—22no—
61C2×D4×C197—22no—
62C2×D4×C197D412yes—
63C2×D4×C197C2312yes—
64C2×D4×C197C2311yescharacteristic, complemented
65C22×C4×C197—21no—
66C22×C4×C197C2311yescharacteristic
67C4⋊C4×C197—21no—
68C22⋊C4×C197—21no—
69C22⋊C4×C197—24no—
70C42×C197C2312yescomplemented
71C22×D197—22no—
72C22×D197—28no—
73C22×D197—84no—
74Dic394—22no—
75Dic394C2×D414yes—
76D788—22no—
77D788C2×D414yes—
78C22⋊D197—216no—
79C22⋊D197—24no—
80C22⋊D197—24no—
81C2×Dic197—28no—
82C2×Dic197—22no—
83C4×D197—24no—
84C4×D197—24no—
85#13096802—24no—
86C23×C197—22no—
87C23×C197—22no—
88C23×C197—21no—
89C23×C197—82no—
90C23×C197C2×D414yes—
91C23×C197C2×D411yescharacteristic
92Q8×C197—22no—
93D4×C197—24no—
94D4×C197—24no—
95D4×C197—22no—
96D4×C197—41no—
97D4×C197—41no—
98D4×C197—84no—
99C2×C4×C197—22no—
100C2×C4×C197—21no—
101C2×C4×C197—42no—
102C2×C4×C197C2×D412yes—
103C2×C4×C197C2×D412yes—
104C2×C4×C197C2×D411yescharacteristic
105C2×C4×C197C2411yescharacteristic
106C22×C197—24no—
107C22×C197—41no—
108C22×C197—44no—
109C22×C197—44no—
110C22×C197—42no—
111C22×C197—41no—
112C22×C197—82no—
113C22×C197#12011yescharacteristic
114C22×C197#12011yescharacteristic
115C22×C197C22×D411yescharacteristic
116C2×D197—24no—
117C2×D197—44no—
118C2×D197—88no—
119C4×C197—42no—
120C4×C197—41no—
121C4×C197#12012yes—
122C4×C197C22×D412yes—
123Dic197—24no—
124C2×C197—21no—
125C2×C197—21no—
126C2×C197—21no—
127C2×C197—44no—
128C2×C197—81no—
129C2×C197C2×#12011yescharacteristic
130D197—44no—
131C197#301911yescharacteristic, complemented, Sylow
132#3019—1971nomaximal, Sylow
133C2×H(5,𝔽2)—1971no—
134#575—1972no—
135#535D19711yescharacteristic, complemented
136#455—1974no—
137#453—1974no—
138#421—1972no—
139C2×#100—1971no—
140H(5,𝔽2)—3944no—
141H(5,𝔽2)—3942no—
142H(5,𝔽2)—1974no—
143C2×#41—3942no—
144C2×#41—1972no—
145C2×#41C2×D19711yescharacteristic, complemented
146C22×D4—3942no—
147C22×D4—3942no—
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#127C2×D19712yescomplemented
156#124C2×D19712yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D19711yescharacteristic, complemented
160#104—1978no—
161#100—1974no—
162C24—3942no—
163C24C22⋊D19712yes—
164#41—22no—
165#41—22no—
166#41—3948no—
167#41—3948no—
168#41—3942no—
169#41—3942no—
170#41—1974no—
171#41—1974no—
172C2×Q8—1972no—
173C2×Q8C22×D19711yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—3948no—
179C2×D4—3948no—
180C2×D4—3944no—
181C2×D4—3948no—
182C2×D4—3942no—
183C2×D4—3948no—
184C2×D4—1972no—
185C2×D4C22⋊D19712yes—
186C2×D4C22×D19712yes—
187C2×D4C22×D19711yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—3942no—
190C22×C4—3942no—
191C22×C4C22×D19711yescharacteristic
192Q16—3944no—
193SD16—3944no—
194SD16—3944no—
195D8—3944no—
196M16—3942no—
197M16—1974no—
198C2×C8—3942no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D19712yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—15764no—
208C23—3948no—
209C23—3942no—
210C23C2×C22⋊D19714yes—
211C23C2×C22⋊D19711yescharacteristic
212Q8—22no—
213Q8—3942no—
214Q8—1974no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—39416no—
222D4—3944no—
223D4—3944no—
224D4—3942no—
225D4—1974no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—3948no—
230C2×C4—3942no—
231C2×C4—3944no—
232C2×C4—3944no—
233C2×C4D4×D19712yes—
234C2×C4D4×D19712yes—
235C2×C4C2×C22⋊D19711yescharacteristic
236C2×C4C23×D19711yescharacteristic
237C8—3944no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—7884no—
246C22—15768no—
247C22—3944no—
248C22#2026329411yescharacteristic
249C22#2026329411yescharacteristic
250C22C22×C22⋊D19711yescharacteristic
251C4—42no—
252C4—41no—
253C4—3944no—
254C4#2026329412yes—
255C4C2×D4×D19712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—7884no—
262C2C2×#2026329411yescharacteristic
subgroupG/H
centerC2C2×#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#535×C197C2
socleC2×C197C2×#120
hypercenter#535D197
hypocenterC197#3019
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20263371C2semidirect
2. #27161877C2semidirect
3. #27161256C2semidirect
4. #27161246C2semidirect
5. #27161097C2semidirect
6. C2×#20263243C2semidirect
7. #535×C197C2semidirect
15. #20263357C22semidirect
17. #20263352C22semidirect
20. C2×#15691342C22semidirect
21. #20263254C22semidirect
22. #20263243C22semidirect
23. C2×#41×C197C22semidirect
25. #127×C197C22semidirect
26. #124×C197C22semidirect
29. #117×C197C22semidirect
33. #15691358D4semidirect
37. C2×Dic394D4semidirect
38. C2×D788D4semidirect
52. #15691342C23semidirect
57. C2×Q8×C197C23semidirect
64. C2×D4×C197C23semidirect
70. C42×C197C23semidirect
131. C197#3019semidirect
135. #535D197semidirect
145. C2×#41C2×D197semidirect
155. #127C2×D197semidirect
156. #124C2×D197semidirect
159. #117C2×D197semidirect
173. C2×Q8C22×D197semidirect
187. C2×D4C22×D197semidirect
202. C42C22×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633712
#271618772
#271612562
#271612462
#271610972
C2×#202632432
#535×C1972
#3019197