Finite Group Databasebeta

← #38752083

Subgroups — #38752083

Order 25216.

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

normal subgroups only

This group has 736 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
2C2×#20263352C212yesmaximal, complemented
3#27161251C214yesmaximal, complemented
4#27161246C214yesmaximal, complemented
5#27161097C212yesmaximal, complemented
6#27161063C211yesmaximal, characteristic, complemented
7#535×C197C211yesmaximal, characteristic, complemented
8#20263374C2212yescomplemented
9#20263373—22no—
10#20263372—22no—
11#20263371—22no—
12#20263371C2212yescomplemented
13C2×#15691358—22no—
14C2×#15691358C2212yes—
15#41×D197—24no—
16#20263352—24no—
17#20263352C2218yescomplemented
18C2×#15691346—22no—
19C2×#15691343—22no—
20C2×#15691342C2212yescomplemented
21#20263254C2218yescomplemented
22#20263245C2212yescomplemented
23#20263243C2212yescomplemented
24C2×#41×C197C2211yescharacteristic
25C22×D4×C197C2211yescharacteristic, complemented
26#127×C197C2212yescomplemented
27#124×C197C2212yescomplemented
28#121×C197—22no—
29#120×C197—22no—
30#117×C197C2211yescharacteristic, complemented
31#15691359—24no—
32#15691358—24no—
33#15691358—24no—
34#15691358—22no—
35#15691358—24no—
36#15691358—22no—
37#15691358D414yescomplemented
38#15691358C2314yes—
39#15691357—28no—
40#15691357—24no—
41C2×Dic394—22no—
42C2×Dic394D412yescomplemented
43Q8×D197—24no—
44C2×D788—22no—
45C2×D788D412yescomplemented
46C2×C22⋊D197—24no—
47C2×C22⋊D197—22no—
48D4×D197—28no—
49D4×D197—24no—
50C2×C4×D197—24no—
51C2×C4×D197—22no—
52#15691346—28no—
53#15691343—28no—
54#15691342—22no—
55#15691342C2314yes—
56C2×#13096802—22no—
57C24×C197—21no—
58#41×C197—22no—
59#41×C197—22no—
60C2×Q8×C197C2311yescharacteristic
61C2×D4×C197—24no—
62C2×D4×C197—22no—
63C2×D4×C197—22no—
64C2×D4×C197—21no—
65C2×D4×C197D412yes—
66C2×D4×C197C2312yes—
67C2×D4×C197C2311yescharacteristic
68C22×C4×C197D412yes—
69C22×C4×C197C2311yescharacteristic
70C4⋊C4×C197—21no—
71C22⋊C4×C197—21no—
72C22⋊C4×C197—24no—
73C42×C197C2312yes—
74C22×D197—24no—
75C22×D197—22no—
76Dic394—24no—
77Dic394—22no—
78Dic394C2×D414yes—
79D788—24no—
80D788—22no—
81D788C2×D414yes—
82C22⋊D197—28no—
83C22⋊D197—24no—
84C22⋊D197—24no—
85C2×Dic197—24no—
86C2×Dic197—22no—
87C4×D197—28no—
88C4×D197—24no—
89C4×D197—24no—
90#13096802—24no—
91C23×C197—22no—
92C23×C197—42no—
93C23×C197—82no—
94C23×C197C2×D412yes—
95C23×C197C2×D411yescharacteristic
96Q8×C197—22no—
97D4×C197—24no—
98D4×C197—22no—
99D4×C197—24no—
100D4×C197—41no—
101D4×C197—41no—
102D4×C197—84no—
103C2×C4×C197—22no—
104C2×C4×C197—22no—
105C2×C4×C197—22no—
106C2×C4×C197C2×D412yes—
107C2×C4×C197C2×D412yes—
108C2×C4×C197C2×D412yes—
109C2×C4×C197C2×D411yescharacteristic
110C2×C4×C197C2411yescharacteristic
111C22×C197—22no—
112C22×C197—41no—
113C22×C197—42no—
114C22×C197—44no—
115C22×C197—42no—
116C22×C197—84no—
117C22×C197#12012yes—
118C22×C197C22×D411yescharacteristic
119C2×D197—24no—
120C2×D197—82no—
121C2×D197—84no—
122C4×C197—22no—
123C4×C197—42no—
124C4×C197#12011yescharacteristic
125C4×C197#12011yescharacteristic
126C4×C197C22×D412yes—
127Dic197—24no—
128C2×C197—21no—
129C2×C197—22no—
130C2×C197—42no—
131C2×C197—82no—
132C2×C197C2×#12011yescharacteristic
133D197—44no—
134C197#302111yescharacteristic, complemented, Sylow
135#3021—1971nomaximal, Sylow
136#585—1971no—
137C2×AGL(1,ℤ/8)—1972no—
138#535D19711yescharacteristic, complemented
139#454—1974no—
140#453—1974no—
141#421—1972no—
142#413—1971no—
1432−1+4—3942no—
1442−1+4—1972no—
145H(5,𝔽2)—3942no—
146H(5,𝔽2)—3942no—
147H(5,𝔽2)—1972no—
148C2×#41—3944no—
149C2×#41—3942no—
150C2×#41—1972no—
151C2×#41C2×D19711yescharacteristic
152C22×D4C2×D19711yescharacteristic, complemented
153AGL(1,ℤ/8)—3944no—
154AGL(1,ℤ/8)—1978no—
155C2×SD16—3942no—
156C2×D8—3942no—
157C2×M16—1972no—
158#127C2×D19712yescomplemented
159#124C2×D19712yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D19711yescharacteristic, complemented
163#104—1978no—
164#101—1972no—
165#100—1972no—
166C24—21no—
167#41—22no—
168#41—22no—
169#41—3944no—
170#41—3944no—
171#41—3948no—
172#41—3944no—
173#41—3944no—
174#41—3942no—
175#41—3942no—
176#41—3944no—
177#41—1974no—
178#41—1974no—
179C2×Q8—3942no—
180C2×Q8—3944no—
181C2×Q8—1972no—
182C2×Q8C22×D19711yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—3942no—
188C2×D4—3948no—
189C2×D4—3944no—
190C2×D4—3944no—
191C2×D4—3942no—
192C2×D4—1972no—
193C2×D4C22⋊D19712yes—
194C2×D4C22×D19712yes—
195C2×D4C22×D19711yescharacteristic
196C22×C4—3944no—
197C22×C4—3942no—
198C22×C4C22⋊D19712yes—
199C22×C4C22×D19711yescharacteristic
200SD16—3948no—
201D8—3948no—
202M16—3942no—
203M16—1974no—
204C2×C8—3942no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D19712yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—3944no—
213C23—3942no—
214C23C2×C22⋊D19712yes—
215C23C2×C22⋊D19711yescharacteristic
216Q8—22no—
217Q8—3944no—
218Q8—3942no—
219Q8—1974no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—3944no—
227D4—3948no—
228D4—3944no—
229D4—3944no—
230D4—3942no—
231D4—1974no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—3948no—
236C2×C4—3944no—
237C2×C4—3942no—
238C2×C4—3944no—
239C2×C4—3944no—
240C2×C4D4×D19712yes—
241C2×C4D4×D19712yes—
242C2×C4C2×C22⋊D19712yes—
243C2×C4C2×C22⋊D19711yescharacteristic
244C2×C4C23×D19711yescharacteristic
245C8—3944no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—15762no—
253C22—15764no—
254C22—3944no—
255C22#2026329412yes—
256C22C22×C22⋊D19711yescharacteristic
257C4—22no—
258C4—42no—
259C4—3944no—
260C4#2026329411yescharacteristic
261C4#2026329411yescharacteristic
262C4C2×D4×D19712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—7884no—
268C2C2×#2026329411yescharacteristic
subgroupG/H
centerC2C2×#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#535×C197C2
socleC2×C197C2×#120
hypercenter#535D197
hypocenterC197#3021
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. #27161915C2semidirect
2. C2×#20263352C2semidirect
3. #27161251C2semidirect
4. #27161246C2semidirect
5. #27161097C2semidirect
6. #27161063C2semidirect
7. #535×C197C2semidirect
8. #20263374C22semidirect
12. #20263371C22semidirect
17. #20263352C22semidirect
20. C2×#15691342C22semidirect
21. #20263254C22semidirect
22. #20263245C22semidirect
23. #20263243C22semidirect
25. C22×D4×C197C22semidirect
26. #127×C197C22semidirect
27. #124×C197C22semidirect
30. #117×C197C22semidirect
37. #15691358D4semidirect
42. C2×Dic394D4semidirect
45. C2×D788D4semidirect
134. C197#3021semidirect
138. #535D197semidirect
152. C22×D4C2×D197semidirect
158. #127C2×D197semidirect
159. #124C2×D197semidirect
162. #117C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
C2×#202633522
#271612512
#271612462
#271610972
#271610632
#535×C1972
#3021197