Finite Group Databasebeta

← #38754478

Subgroups — #38754478

Order 25216.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#27161885C212yesmaximal, complemented
2#27161880C212yesmaximal, complemented
3#27161346C212yesmaximal, complemented
4#27161339C212yesmaximal, complemented
5#27161256C211yesmaximal, characteristic, complemented
6#27161251C211yesmaximal, characteristic, complemented
7#27161246C212yesmaximal, complemented
8#27161202C212yesmaximal, complemented
9#496×C197C211yesmaximal, characteristic, complemented
10#20263374C2211yescharacteristic, complemented
11#20263371C2211yescharacteristic
12#20263371C2212yescomplemented
13#20263357C2212yescomplemented
14#20263355—22no—
15#20263353—22no—
16#20263352C2212yes—
17#20263352C2212yescomplemented
18#20263352C2212yescomplemented
19#20263350—22no—
20#20263350—22no—
21#20263348C2212yes—
22#20263347—22no—
23#20263345—22no—
24SD16×D197—22no—
25D8×D197—22no—
26#20263332C2212yescomplemented
27#20263332C2212yescomplemented
28#20263262C2212yescomplemented
29#20263254C2212yescomplemented
30#20263254C2212yescomplemented
31#20263245C2211yescharacteristic, complemented
32#20263243C2211yescharacteristic, complemented
33#20263243C2212yescomplemented
34C2×SD16×C197C2212yescomplemented
35C2×D8×C197C2212yescomplemented
36#127×C197C2211yescharacteristic, complemented
37#124×C197C2211yescharacteristic, complemented
38#97×C197C2211yescharacteristic, complemented
39#15691359—22no—
40#15691358—41no—
41#15691358C2312yes—
42#15691358C2312yes—
43#15691357—22no—
44#15691357—22no—
45#15691357—22no—
46Q8×D197—22no—
47C2×C22⋊D197—41no—
48C2×C22⋊D197—42no—
49D4×D197—22no—
50D4×D197—22no—
51D4×D197—22no—
52Dic788D412yescomplemented
53#15691348—22no—
54#15691347D412yescomplemented
55#15691347D412yescomplemented
56#15691346—22no—
57#15691346—22no—
58#15691346—22no—
59#15691345—22no—
60D1576D412yescomplemented
61#15691343—22no—
62#15691343—22no—
63#15691343—22no—
64#15691342C2312yes—
65#15691342C2312yes—
66#15691341—22no—
67#15691341—22no—
68C8×D197—22no—
69C8×D197—22no—
70C2×Q8×C197C2311yescharacteristic
71C2×D4×C197—41no—
72C2×D4×C197C2311yescharacteristic
73C2×D4×C197C2312yes—
74SD16×C197—22no—
75SD16×C197—22no—
76D8×C197—22no—
77D8×C197—22no—
78C2×C8×C197C2312yes—
79C22⋊C4×C197—41no—
80C42×C197C2311yescharacteristic
81C22×D197—42no—
82C22×D197—41no—
83Dic394—41no—
84Dic394C2×D412yes—
85Dic394C2×D412yes—
86D788—41no—
87D788C2×D412yes—
88D788C2×D412yes—
89C22⋊D197—22no—
90C22⋊D197—22no—
91C22⋊D197—42no—
92C22⋊D197—42no—
93C22⋊D197—42no—
94C2×Dic197—42no—
95C2×Dic197—41no—
96C4×D197—22no—
97C4×D197—22no—
98C4×D197—42no—
99#13096802—22no—
100#13096802—22no—
101C23×C197—21no—
102C23×C197—22no—
103Q8×C197—22no—
104D4×C197—22no—
105D4×C197—22no—
106D4×C197—22no—
107D4×C197—82no—
108C2×C4×C197—21no—
109C2×C4×C197—21no—
110C2×C4×C197C2411yescharacteristic
111C8×C197C2×D412yes—
112C8×C197C2×D412yes—
113C22×C197—41no—
114C22×C197—42no—
115C22×C197—82no—
116C22×C197—81no—
117C22×C197H(5,𝔽2)11yescharacteristic
118C2×D197—22no—
119C2×D197—22no—
120C2×D197—162no—
121C2×D197—161no—
122C4×C197—41no—
123C4×C197—41no—
124C4×C197C22×D412yes—
125Dic197—22no—
126Dic197—22no—
127C2×C197—21no—
128C2×C197—82no—
129C2×C197—81no—
130C2×C197D4211yescharacteristic
131D197—42no—
132D197—42no—
133C197#328811yescharacteristic, complemented, Sylow
134#3288—1971nomaximal, Sylow
135#577—1972no—
136#576—1972no—
137#496D19711yescharacteristic, complemented
138#472—1972no—
139#471—1972no—
140#455—1971no—
141#454—1971no—
142#453—1972no—
143#444—1972no—
1442−1+4—1971no—
145H(5,𝔽2)—1971no—
146H(5,𝔽2)—1972no—
147#137—3942no—
148#137—1972no—
149AGL(1,ℤ/8)—3942no—
150AGL(1,ℤ/8)—3942no—
151AGL(1,ℤ/8)—3942no—
152AGL(1,ℤ/8)—1972no—
153AGL(1,ℤ/8)—1972no—
154AGL(1,ℤ/8)—1972no—
155#135—3942no—
156#135—3942no—
157#135—1972no—
158C2×SD16—3942no—
159C2×SD16C2×D19712yescomplemented
160C2×D8—3942no—
161C2×D8C2×D19712yescomplemented
162#131—1972no—
163#131—1972no—
164#127C2×D19711yescharacteristic, complemented
165#124C2×D19711yescharacteristic, complemented
166#108—1972no—
167#104—1972no—
168#104—1972no—
169#101—1971no—
170#100—1972no—
171#100—1971no—
172#97C2×D19711yescharacteristic, complemented
173#41—7881no—
174#41—3942no—
175#41—3942no—
176#41—3942no—
177#41—3942no—
178#41—1972no—
179#41—1972no—
180C2×Q8—3942no—
181C2×Q8C22×D19711yescharacteristic
182C2×D4—41no—
183C2×D4—7881no—
184C2×D4—7882no—
185C2×D4—3942no—
186C2×D4—3942no—
187C2×D4—3942no—
188C2×D4C22×D19711yescharacteristic
189C2×D4C22×D19712yes—
190Q16—3942no—
191Q16—1972no—
192SD16—22no—
193SD16—22no—
194SD16—3942no—
195SD16—3942no—
196SD16—3942no—
197SD16—3942no—
198SD16—1972no—
199SD16—1972no—
200D8—22no—
201D8—22no—
202D8—3942no—
203D8—3942no—
204D8—3942no—
205D8—1972no—
206M16—3942no—
207M16—3942no—
208M16—1972no—
209M16—1972no—
210C2×C8—3942no—
211C2×C8—3942no—
212C2×C8C22×D19712yes—
213C22⋊C4—41no—
214C42C22×D19711yescharacteristic
215C23—21no—
216C23—22no—
217C23—7882no—
218C23—7881no—
219Q8—22no—
220Q8—7881no—
221Q8—1972no—
222Q8—1972no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—7881no—
228D4—7882no—
229D4—7882no—
230D4—7882no—
231D4—3942no—
232D4—3942no—
233D4—1972no—
234D4—1972no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—7882no—
238C2×C4—7881no—
239C2×C4—7882no—
240C2×C4—3942no—
241C2×C4—3942no—
242C2×C4C23×D19711yescharacteristic
243C8—3942no—
244C8—3942no—
245C8D4×D19712yes—
246C8D4×D19712yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—31521no—
252C22—31522no—
253C22—3942no—
254C22—3942no—
255C22#2026337111yescharacteristic
256C4—41no—
257C4—41no—
258C4—3942no—
259C4—3942no—
260C4C2×D4×D19712yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—7882no—
265C2—7882no—
266C2#2716171411yescharacteristic
subgroupG/H
centerC2#27161714
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#496×C197C2
socleC2×C197D42
hypercenter#496D197
hypocenterC197#3288
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #27161885C2semidirect
2. #27161880C2semidirect
3. #27161346C2semidirect
4. #27161339C2semidirect
5. #27161256C2semidirect
6. #27161251C2semidirect
7. #27161246C2semidirect
8. #27161202C2semidirect
9. #496×C197C2semidirect
10. #20263374C22semidirect
12. #20263371C22semidirect
13. #20263357C22semidirect
17. #20263352C22semidirect
18. #20263352C22semidirect
26. #20263332C22semidirect
27. #20263332C22semidirect
28. #20263262C22semidirect
29. #20263254C22semidirect
30. #20263254C22semidirect
31. #20263245C22semidirect
32. #20263243C22semidirect
33. #20263243C22semidirect
34. C2×SD16×C197C22semidirect
35. C2×D8×C197C22semidirect
36. #127×C197C22semidirect
37. #124×C197C22semidirect
38. #97×C197C22semidirect
52. Dic788D4semidirect
54. #15691347D4semidirect
55. #15691347D4semidirect
60. D1576D4semidirect
133. C197#3288semidirect
137. #496D197semidirect
159. C2×SD16C2×D197semidirect
161. C2×D8C2×D197semidirect
164. #127C2×D197semidirect
165. #124C2×D197semidirect
172. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618852
#271618802
#271613462
#271613392
#271612562
#271612512
#271612462
#271612022
#496×C1972
#3288197