Finite Group Databasebeta

← #38754509

Subgroups — #38754509

Order 25216.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#27161882C212yesmaximal, complemented
2#27161881C211yesmaximal, characteristic, complemented
3#27161454C211yesmaximal, characteristic, complemented
4#27161349C211yesmaximal, characteristic, complemented
5#27161347C212yesmaximal, complemented
6#27161346C211yesmaximal, characteristic, complemented
7#27161247C212yesmaximal, complemented
8#27161245C212yesmaximal, complemented
9#27161196C211yesmaximal, characteristic, complemented
10#27161194C211yesmaximal, characteristic, complemented
11#576×C197C211yesmaximal, characteristic, complemented
12#20263372C2212yescomplemented
13#20263353—22no—
14#20263352—22no—
15#20263352C2212yescomplemented
16#20263351—22no—
17#20263351C2212yescomplemented
18#20263351C2212yescomplemented
19#20263349—21no—
20#20263346—22no—
21#20263346C2211yescharacteristic
22C2×D1576C2211yescharacteristic, complemented
23C2×#15691343—22no—
24C2×#15691343C2212yescomplemented
25D8×D197—21no—
26#20263333C2212yescomplemented
27#20263332C2211yescharacteristic, complemented
28#20263322C2212yescomplemented
29#20263262C2211yescharacteristic, complemented
30#20263261C2211yescharacteristic, complemented
31#20263253C2211yescharacteristic, complemented
32#20263253C2212yescomplemented
33#20263252C2211yescharacteristic, complemented
34#20263243C2212yescomplemented
35#20263242C2212yescomplemented
36C8×Dic197C2211yescharacteristic, complemented
37H(5,𝔽2)×C197C2212yescomplemented
38AGL(1,ℤ/8)×C197—22no—
39AGL(1,ℤ/8)×C197C2212yescomplemented
40#135×C197—21no—
41#135×C197C2211yescharacteristic
42C2×D8×C197—21no—
43C2×D8×C197C2211yescharacteristic, complemented
44#131×C197C2211yescharacteristic, complemented
45#15691359—22no—
46#15691358—22no—
47#15691358C2311yescharacteristic
48C2×D788—22no—
49C2×D788C2312yes—
50C2×C22⋊D197—42no—
51D4×D197—22no—
52D4×D197—42no—
53#15691348D411yescharacteristic, complemented
54#15691346—22no—
55#15691346D411yescharacteristic, complemented
56#15691345—22no—
57#15691345—22no—
58#15691345D411yescharacteristic, complemented
59D1576—21no—
60D1576—21no—
61#15691343—22no—
62#15691343—22no—
63#15691343—22no—
64#15691343—22no—
65#15691343—22no—
66#15691343D411yescharacteristic, complemented
67#15691342—22no—
68#15691342C2312yes—
69#15691342C2311yescharacteristic
70#15691341—21no—
71C2×#13096802—22no—
72C2×#13096802C2311yescharacteristic
73C8×D197—21no—
74C4×Dic197C2311yescharacteristic
75#41×C197—22no—
76#41×C197—22no—
77#41×C197C2311yescharacteristic
78#41×C197C2312yes—
79C2×D4×C197—22no—
80C2×D4×C197—22no—
81C2×D4×C197—42no—
82C2×D4×C197C2312yes—
83Q16×C197D411yescharacteristic, complemented
84SD16×C197—22no—
85SD16×C197D412yescomplemented
86D8×C197—22no—
87D8×C197—21no—
88D8×C197—21no—
89D8×C197D411yescharacteristic, complemented
90M16×C197—21no—
91M16×C197C2311yescharacteristic
92C2×C8×C197—21no—
93C2×C8×C197C2311yescharacteristic
94C22×D197—22no—
95C22×D197—42no—
96Dic394C2×D411yescharacteristic
97D788—22no—
98D788—22no—
99D788—42no—
100D788C2×D411yescharacteristic
101C22⋊D197—21no—
102C22⋊D197—42no—
103C22⋊D197—84no—
104C2×Dic197—21no—
105C4×D197—21no—
106C4×D197—42no—
107#13096802—21no—
108#13096802—22no—
109#13096802C2×D411yescharacteristic
110#13096802C2×D411yescharacteristic
111C23×C197—22no—
112C23×C197—42no—
113Q8×C197C2×D411yescharacteristic
114Q8×C197C2×D412yes—
115D4×C197—21no—
116D4×C197—22no—
117D4×C197—22no—
118D4×C197—22no—
119D4×C197—42no—
120D4×C197—42no—
121D4×C197C2×D411yescharacteristic
122D4×C197C2×D412yes—
123C2×C4×C197—21no—
124C2×C4×C197—22no—
125C2×C4×C197—42no—
126C2×C4×C197C2411yescharacteristic
127C8×C197—21no—
128C8×C197C2×D411yescharacteristic
129C8×C197C2×D411yescharacteristic
130C22×C197—21no—
131C22×C197—22no—
132C22×C197—42no—
133C22×C197—82no—
134C22×C197—162no—
135C22×C197H(5,𝔽2)11yescharacteristic
136C2×D197—21no—
137C2×D197—42no—
138C2×D197—82no—
139C2×D197—162no—
140C4×C197—21no—
141C4×C197—22no—
142C4×C197C22×D411yescharacteristic
143C4×C197C22×D411yescharacteristic
144Dic197—22no—
145Dic197—21no—
146C2×C197—21no—
147C2×C197—42no—
148C2×C197—41no—
149C2×C197—82no—
150C2×C197D4211yescharacteristic
151D197—41no—
152D197—82no—
153C197#329211yescharacteristic, complemented, Sylow
154#3292—1971nomaximal, Sylow
155#576—1972no—
156#576—1971no—
157#576D19711yescharacteristic, complemented
158#493—1971no—
159#472—1972no—
160#472—1971no—
161#472—1971no—
162#453—1972no—
163#453—1972no—
164#443—1971no—
165#443—1971no—
166H(5,𝔽2)—1972no—
167H(5,𝔽2)C2×D19712yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—3942no—
170AGL(1,ℤ/8)—3942no—
171AGL(1,ℤ/8)—3942no—
172AGL(1,ℤ/8)—1972no—
173AGL(1,ℤ/8)—1972no—
174AGL(1,ℤ/8)—1972no—
175AGL(1,ℤ/8)C2×D19712yescomplemented
176#135—21no—
177#135—3942no—
178#135—3941no—
179#135—1971no—
180#135C2×D19711yescharacteristic
181C2×D8—21no—
182C2×D8—3942no—
183C2×D8—3941no—
184C2×D8—1972no—
185C2×D8—1971no—
186C2×D8C2×D19711yescharacteristic, complemented
187#131—1972no—
188#131—1971no—
189#131C2×D19711yescharacteristic, complemented
190#127—1972no—
191#108—1971no—
192#108—1971no—
193#104—1971no—
194#104—1972no—
195#104—1971no—
196#100—1972no—
197#100—1972no—
198C4×C8—1971no—
199#41—22no—
200#41—22no—
201#41—3942no—
202#41—3942no—
203#41—1971no—
204#41C22×D19711yescharacteristic
205#41C22×D19712yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—7882no—
210C2×D4—7882no—
211C2×D4—3942no—
212C2×D4—3942no—
213C2×D4—1972no—
214C2×D4C22×D19712yes—
215Q16—1971no—
216Q16C22⋊D19711yescharacteristic, complemented
217SD16—22no—
218SD16—3942no—
219SD16—3942no—
220SD16—3942no—
221SD16—1971no—
222SD16—1971no—
223SD16C22⋊D19712yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3942no—
228D8—3942no—
229D8—3942no—
230D8—3942no—
231D8—3942no—
232D8—3941no—
233D8—3941no—
234D8—1971no—
235D8C22⋊D19711yescharacteristic, complemented
236M16—21no—
237M16—3942no—
238M16—3941no—
239M16—1972no—
240M16—1971no—
241M16C22×D19711yescharacteristic
242C2×C8—21no—
243C2×C8—3942no—
244C2×C8—3941no—
245C2×C8—1971no—
246C2×C8C22×D19711yescharacteristic
247C42—1971no—
248C23—22no—
249C23—42no—
250C23—7882no—
251C23—3942no—
252Q8—1971no—
253Q8D4×D19711yescharacteristic
254Q8C2×C22⋊D19712yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—7882no—
262D4—7882no—
263D4—15764no—
264D4—3941no—
265D4—3942no—
266D4—3942no—
267D4—1971no—
268D4D4×D19711yescharacteristic
269D4C2×C22⋊D19712yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—7882no—
274C2×C4—3941no—
275C2×C4—3941no—
276C2×C4C23×D19711yescharacteristic
277C8—21no—
278C8—3942no—
279C8—3941no—
280C8—1971no—
281C8—1971no—
282C8C2×C22⋊D19711yescharacteristic
283C8C2×C22⋊D19711yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—7882no—
290C22—15762no—
291C22—31522no—
292C22—3941no—
293C22#2026337111yescharacteristic
294C4—21no—
295C4—22no—
296C4—3942no—
297C4—3941no—
298C4C2×D4×D19711yescharacteristic
299C4C22×C22⋊D19711yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—7881no—
305C2—15762no—
306C2D4×C22⋊D19711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#576×C197C2
socleC2×C197D42
hypercenter#576D197
hypocenterC197#3292
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #27161882C2semidirect
2. #27161881C2semidirect
3. #27161454C2semidirect
4. #27161349C2semidirect
5. #27161347C2semidirect
6. #27161346C2semidirect
7. #27161247C2semidirect
8. #27161245C2semidirect
9. #27161196C2semidirect
10. #27161194C2semidirect
11. #576×C197C2semidirect
12. #20263372C22semidirect
15. #20263352C22semidirect
17. #20263351C22semidirect
18. #20263351C22semidirect
22. C2×D1576C22semidirect
24. C2×#15691343C22semidirect
26. #20263333C22semidirect
27. #20263332C22semidirect
28. #20263322C22semidirect
29. #20263262C22semidirect
30. #20263261C22semidirect
31. #20263253C22semidirect
32. #20263253C22semidirect
33. #20263252C22semidirect
34. #20263243C22semidirect
35. #20263242C22semidirect
36. C8×Dic197C22semidirect
37. H(5,𝔽2)×C197C22semidirect
39. AGL(1,ℤ/8)×C197C22semidirect
43. C2×D8×C197C22semidirect
44. #131×C197C22semidirect
53. #15691348D4semidirect
55. #15691346D4semidirect
58. #15691345D4semidirect
66. #15691343D4semidirect
83. Q16×C197D4semidirect
85. SD16×C197D4semidirect
89. D8×C197D4semidirect
153. C197#3292semidirect
157. #576D197semidirect
167. H(5,𝔽2)C2×D197semidirect
175. AGL(1,ℤ/8)C2×D197semidirect
186. C2×D8C2×D197semidirect
189. #131C2×D197semidirect
216. Q16C22⋊D197semidirect
223. SD16C22⋊D197semidirect
235. D8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271618822
#271618812
#271614542
#271613492
#271613472
#271613462
#271612472
#271612452
#271611962
#271611942
#576×C1972
#3292197