Finite Group Databasebeta

← #38752113

Subgroups — #38752113

Order 25216.

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

normal subgroups only

This group has 772 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
1#27161915C211yesmaximal, characteristic, complemented
2#27161646C212yesmaximal, complemented
3#27161265C218yesmaximal, complemented
4#27161053C211yesmaximal, characteristic, complemented
5#27161052C212yesmaximal, complemented
6#535×C197C211yesmaximal, characteristic, complemented
7#20263374—21no—
8#20263373—22no—
9#20263372—22no—
10#20263371—21no—
11#20263371C2214yes—
12C2×#15691358—22no—
13C2×#15691358C2212yes—
14#41×D197—24no—
15#20263322—22no—
16#20263313—22no—
17#20263297—24no—
18#20263295C2218yescomplemented
19#20263285C2212yescomplemented
20#20263241C2214yescomplemented
21#20263240C2218yescomplemented
22C2×#41×C197C2211yescharacteristic
23C22×D4×C197C2211yescharacteristic, complemented
24#127×C197—21no—
25#124×C197—21no—
26#121×C197—22no—
27#120×C197C2214yescomplemented
28#117×C197C2211yescharacteristic, complemented
29#15691359—24no—
30#15691358—24no—
31#15691358—24no—
32#15691358—22no—
33#15691358—22no—
34#15691358—24no—
35#15691358—22no—
36#15691358—22no—
37#15691357—24no—
38#15691357—28no—
39C2×Dic394—22no—
40C2×Dic394D412yescomplemented
41Q8×D197—24no—
42C2×D788—22no—
43C2×D788D412yescomplemented
44C2×C22⋊D197—28no—
45C2×C22⋊D197—24no—
46C2×C22⋊D197C2314yes—
47D4×D197—24no—
48D4×D197—28no—
49C2×C4×D197—24no—
50C2×C4×D197D414yescomplemented
51C4⋊Dic197—22no—
52C22⋊Dic197—28no—
53C22⋊Dic197C2314yes—
54C4×Dic197—22no—
55C24×C197C2312yes—
56#41×C197—22no—
57#41×C197—22no—
58C2×Q8×C197D411yescharacteristic, complemented
59C2×D4×C197—24no—
60C2×D4×C197—21no—
61C2×D4×C197—21no—
62C2×D4×C197—22no—
63C2×D4×C197—22no—
64C2×D4×C197D411yescharacteristic, complemented
65C2×D4×C197C2312yes—
66C22×C4×C197D412yescomplemented
67C22×C4×C197C2311yescharacteristic
68C4⋊C4×C197—21no—
69C22⋊C4×C197—24no—
70C22⋊C4×C197C2312yes—
71C42×C197—21no—
72C22×D197—24no—
73C22×D197C2×D414yes—
74Dic394—24no—
75Dic394—22no—
76Dic394—22no—
77D788—24no—
78D788—22no—
79D788—22no—
80C22⋊D197—28no—
81C22⋊D197—24no—
82C22⋊D197—24no—
83C22⋊D197—88no—
84C2×Dic197—24no—
85C2×Dic197—24no—
86C2×Dic197C2×D414yes—
87C4×D197—28no—
88C4×D197—24no—
89C4×D197—24no—
90C23×C197—22no—
91C23×C197—24no—
92C23×C197—82no—
93C23×C197C2×D412yes—
94C23×C197C2411yescharacteristic
95Q8×C197—22no—
96D4×C197—24no—
97D4×C197—22no—
98D4×C197—41no—
99D4×C197—41no—
100D4×C197—42no—
101D4×C197—84no—
102C2×C4×C197—22no—
103C2×C4×C197—22no—
104C2×C4×C197—22no—
105C2×C4×C197C2×D412yes—
106C2×C4×C197C2×D412yes—
107C2×C4×C197C2×D412yes—
108C2×C4×C197C2×D411yescharacteristic
109C2×C4×C197C2×D411yescharacteristic
110C22×C197—22no—
111C22×C197—41no—
112C22×C197—42no—
113C22×C197—44no—
114C22×C197—48no—
115C22×C197—42no—
116C22×C197C22×D412yes—
117C22×C197C22×D411yescharacteristic
118C2×D197—24no—
119C2×D197—44no—
120C2×D197—84no—
121C4×C197—22no—
122C4×C197—42no—
123C4×C197#12011yescharacteristic
124C4×C197#12011yescharacteristic
125C4×C197#12012yes—
126Dic197—24no—
127Dic197—44no—
128C2×C197—22no—
129C2×C197—21no—
130C2×C197—42no—
131C2×C197—82no—
132C2×C197C2×#12011yescharacteristic
133D197—44no—
134C197#302711yescharacteristic, complemented, Sylow
135#3027—1971nomaximal, Sylow
136#585—1971no—
137#535—1972no—
138#535D19711yescharacteristic, complemented
139B(4,𝔽2)—1978no—
140#410—1971no—
141#410—1972no—
1422−1+4—3942no—
1432−1+4—3941no—
144H(5,𝔽2)—3942no—
145H(5,𝔽2)—3941no—
146H(5,𝔽2)—1974no—
147C2×#41—3944no—
148C2×#41—3942no—
149C2×#41—1972no—
150C2×#41C2×D19711yescharacteristic
151C22×D4C2×D19711yescharacteristic, complemented
152#127—21no—
153#127—3942no—
154#124—21no—
155#124—3942no—
156#121—22no—
157#121—3944no—
158#120—1978no—
159#120C2×D19714yescomplemented
160#117—1972no—
161#117C2×D19711yescharacteristic, complemented
162#99—1974no—
163#99—1978no—
164C24C22×D19712yes—
165#41—22no—
166#41—22no—
167#41—3944no—
168#41—3944no—
169#41—3944no—
170#41—3944no—
171#41—3948no—
172#41—3942no—
173#41—3942no—
174#41—3942no—
175#41—3944no—
176#41—3942no—
177C2×Q8—3942no—
178C2×Q8—3944no—
179C2×Q8—1972no—
180C2×Q8C22⋊D19711yescharacteristic, complemented
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—3942no—
187C2×D4—3944no—
188C2×D4—3948no—
189C2×D4—3948no—
190C2×D4—3944no—
191C2×D4—1974no—
192C2×D4—1972no—
193C2×D4C22⋊D19711yescharacteristic, complemented
194C2×D4C22×D19712yes—
195C22×C4—3944no—
196C22×C4—1974no—
197C22×C4C22⋊D19712yescomplemented
198C22×C4C22×D19711yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—3942no—
201C22⋊C4—24no—
202C22⋊C4—3948no—
203C22⋊C4—1974no—
204C22⋊C4C22×D19712yes—
205C42—21no—
206C42—3942no—
207C23—24no—
208C23—22no—
209C23—82no—
210C23—3944no—
211C23—1974no—
212C23C2×C22⋊D19712yes—
213C23C23×D19711yescharacteristic
214Q8—22no—
215Q8—3944no—
216Q8—3942no—
217Q8—3942no—
218D4—24no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—84no—
224D4—15768no—
225D4—3944no—
226D4—3948no—
227D4—3944no—
228D4—3944no—
229D4—3942no—
230D4—3942no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—3948no—
235C2×C4—3944no—
236C2×C4—3944no—
237C2×C4—3944no—
238C2×C4—3944no—
239C2×C4—1974no—
240C2×C4D4×D19712yes—
241C2×C4D4×D19712yes—
242C2×C4C2×C22⋊D19712yes—
243C2×C4C2×C22⋊D19711yescharacteristic
244C2×C4C2×C22⋊D19711yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—7884no—
252C22—15764no—
253C22—3944no—
254C22C2×D4×D19712yes—
255C22C22×C22⋊D19711yescharacteristic
256C4—22no—
257C4—42no—
258C4—7884no—
259C4—3944no—
260C4#2026329411yescharacteristic
261C4#2026329411yescharacteristic
262C4#2026329412yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—7884no—
268C2C2×#2026329411yescharacteristic
subgroupG/H
centerC2C2×#20263294
commutator, cosocleC23×C197C24
FrattiniC23C23×D197
Fitting#535×C197C2
socleC2×C197C2×#120
hypercenter#535D197
hypocenterC197#3027
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #27161915C2semidirect
2. #27161646C2semidirect
3. #27161265C2semidirect
4. #27161053C2semidirect
5. #27161052C2semidirect
6. #535×C197C2semidirect
18. #20263295C22semidirect
19. #20263285C22semidirect
20. #20263241C22semidirect
21. #20263240C22semidirect
23. C22×D4×C197C22semidirect
27. #120×C197C22semidirect
28. #117×C197C22semidirect
40. C2×Dic394D4semidirect
43. C2×D788D4semidirect
50. C2×C4×D197D4semidirect
58. C2×Q8×C197D4semidirect
64. C2×D4×C197D4semidirect
66. C22×C4×C197D4semidirect
134. C197#3027semidirect
138. #535D197semidirect
151. C22×D4C2×D197semidirect
159. #120C2×D197semidirect
161. #117C2×D197semidirect
180. C2×Q8C22⋊D197semidirect
193. C2×D4C22⋊D197semidirect
197. C22×C4C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
#271616462
#271612652
#271610532
#271610522
#535×C1972
#3027197