Finite Group Databasebeta

← #38754510

Subgroups — #38754510

Order 25216.

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

normal subgroups only

This group has 476 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#27161884C211yesmaximal, characteristic, complemented
2#27161883C212yesmaximal, complemented
3#27161880C211yesmaximal, characteristic, complemented
4#27161454C211yesmaximal, characteristic, complemented
5#27161350C211yesmaximal, characteristic, complemented
6#27161348C212yesmaximal, complemented
7#27161249C212yesmaximal, complemented
8#27161245C212yesmaximal, complemented
9#27161195C211yesmaximal, characteristic, complemented
10#27161194C211yesmaximal, characteristic, complemented
11#472×C197C211yesmaximal, characteristic, complemented
12#20263372C2212yescomplemented
13#20263371C2212yescomplemented
14#20263354—22no—
15#20263354C2212yescomplemented
16#20263354C2212yescomplemented
17#20263353—22no—
18#20263352C2212yescomplemented
19#20263350—22no—
20#20263350—22no—
21#20263348—21no—
22#20263348C2211yescharacteristic
23#20263347—22no—
24#20263346C2211yescharacteristic
25#20263345—21no—
26C2×D1576—21no—
27C2×D1576C2211yescharacteristic, complemented
28C2×#15691343C2212yescomplemented
29D8×D197—21no—
30D8×D197—22no—
31#20263334C2212yescomplemented
32#20263332C2211yescharacteristic, complemented
33#20263322C2212yescomplemented
34#20263261C2211yescharacteristic, complemented
35#20263253C2211yescharacteristic, complemented
36#20263252C2211yescharacteristic, complemented
37#20263252C2212yescomplemented
38#20263242C2212yescomplemented
39C8×Dic197C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C197C2212yescomplemented
41C2×D8×C197C2211yescharacteristic, complemented
42#131×C197C2211yescharacteristic, complemented
43#108×C197C2211yescharacteristic, complemented
44#100×C197C2212yescomplemented
45#15691359—22no—
46#15691358—22no—
47#15691358C2311yescharacteristic
48#15691358C2312yes—
49#15691357—22no—
50#15691357—22no—
51C2×D788—22no—
52C2×D788C2312yes—
53C2×C22⋊D197—42no—
54C2×C22⋊D197—42no—
55D4×D197—22no—
56D4×D197—22no—
57D4×D197—22no—
58D4×D197—42no—
59Dic788D411yescharacteristic, complemented
60#15691348D411yescharacteristic, complemented
61#15691347—22no—
62#15691347—22no—
63#15691347D412yescomplemented
64#15691346—22no—
65#15691346D411yescharacteristic, complemented
66#15691345D411yescharacteristic, complemented
67D1576—22no—
68D1576—22no—
69D1576—21no—
70D1576—21no—
71D1576D411yescharacteristic, complemented
72#15691343—22no—
73#15691343—22no—
74#15691343—22no—
75#15691343D411yescharacteristic, complemented
76#15691342C2311yescharacteristic
77#15691341—22no—
78#15691341—21no—
79C2×#13096802C2311yescharacteristic
80C8×D197—22no—
81C8×D197—21no—
82C4×Dic197C2311yescharacteristic
83#41×C197C2311yescharacteristic
84C2×D4×C197C2312yes—
85SD16×C197—22no—
86D8×C197—22no—
87D8×C197—21no—
88D8×C197—21no—
89M16×C197—21no—
90M16×C197C2311yescharacteristic
91M16×C197C2312yes—
92C2×C8×C197—21no—
93C2×C8×C197C2311yescharacteristic
94C22×D197—22no—
95C22×D197—42no—
96C22×D197—42no—
97Dic394C2×D412yes—
98Dic394C2×D411yescharacteristic
99D788—22no—
100D788—22no—
101D788—42no—
102D788C2×D412yes—
103D788C2×D411yescharacteristic
104C22⋊D197—22no—
105C22⋊D197—21no—
106C22⋊D197—42no—
107C22⋊D197—42no—
108C22⋊D197—42no—
109C22⋊D197—84no—
110C2×Dic197—21no—
111C2×Dic197—42no—
112C4×D197—21no—
113C4×D197—22no—
114C4×D197—42no—
115#13096802—21no—
116#13096802C2×D411yescharacteristic
117#13096802C2×D411yescharacteristic
118C23×C197—22no—
119Q8×C197C2×D411yescharacteristic
120D4×C197—21no—
121D4×C197—22no—
122D4×C197—22no—
123D4×C197C2×D411yescharacteristic
124C2×C4×C197—21no—
125C2×C4×C197C2411yescharacteristic
126C8×C197—21no—
127C8×C197—22no—
128C8×C197C2×D411yescharacteristic
129C8×C197C2×D411yescharacteristic
130C22×C197—21no—
131C22×C197—42no—
132C22×C197—82no—
133C22×C197H(5,𝔽2)11yescharacteristic
134C2×D197—21no—
135C2×D197—22no—
136C2×D197—42no—
137C2×D197—82no—
138C2×D197—162no—
139C2×D197—162no—
140C4×C197—21no—
141C4×C197C22×D411yescharacteristic
142C4×C197C22×D411yescharacteristic
143Dic197—22no—
144Dic197—22no—
145Dic197—21no—
146C2×C197—21no—
147C2×C197—41no—
148C2×C197—82no—
149C2×C197D4211yescharacteristic
150D197—41no—
151D197—42no—
152D197—82no—
153C197#329211yescharacteristic, complemented, Sylow
154#3292—1971nomaximal, Sylow
155#576—1972no—
156#576—1971no—
157#576—1971no—
158#493—1971no—
159#472—1971no—
160#472—1972no—
161#472D19711yescharacteristic, complemented
162#453—1972no—
163#453—1972no—
164#443—1971no—
165#443—1971no—
166H(5,𝔽2)—1972no—
167H(5,𝔽2)—1972no—
168AGL(1,ℤ/8)—3942no—
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—3941no—
177#135—3941no—
178#135—3942no—
179#135—1971no—
180#135—1971no—
181C2×D8—3941no—
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#108C2×D19711yescharacteristic, complemented
193#104—1971no—
194#104—1972no—
195#104—1971no—
196#100—1972no—
197#100C2×D19712yescomplemented
198C4×C8—1971no—
199#41—3942no—
200#41—3942no—
201#41—3942no—
202#41—3942no—
203#41—1971no—
204#41—1972no—
205#41C22×D19711yescharacteristic
206C2×D4—7882no—
207C2×D4—7882no—
208C2×D4—7882no—
209C2×D4—3942no—
210C2×D4—3942no—
211C2×D4—3942no—
212C2×D4—3942no—
213C2×D4—1972no—
214C2×D4C22×D19712yes—
215Q16—1971no—
216Q16—1971no—
217SD16—22no—
218SD16—3942no—
219SD16—3942no—
220SD16—3942no—
221SD16—1971no—
222SD16—1971no—
223SD16—1972no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3942no—
228D8—3942no—
229D8—3942no—
230D8—3942no—
231D8—3942no—
232D8—3941no—
233D8—3941no—
234D8—1971no—
235D8—1971no—
236M16—21no—
237M16—3942no—
238M16—3941no—
239M16—1971no—
240M16C22×D19712yes—
241M16C22×D19711yescharacteristic
242C2×C8—21no—
243C2×C8—3942no—
244C2×C8—3941no—
245C2×C8—1971no—
246C2×C8C22×D19711yescharacteristic
247C42—1971no—
248C23—22no—
249C23—7882no—
250C23—7882no—
251C23—3942no—
252Q8—1971no—
253Q8—1972no—
254Q8D4×D19711yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—7882no—
259D4—7882no—
260D4—7882no—
261D4—7882no—
262D4—15764no—
263D4—3942no—
264D4—3941no—
265D4—3942no—
266D4—3942no—
267D4—1971no—
268D4—1972no—
269D4D4×D19711yescharacteristic
270C2×C4—21no—
271C2×C4—7882no—
272C2×C4—7882no—
273C2×C4—3941no—
274C2×C4—3942no—
275C2×C4—3941no—
276C2×C4C23×D19711yescharacteristic
277C8—21no—
278C8—22no—
279C8—3941no—
280C8—1971no—
281C8—1971no—
282C8D4×D19711yescharacteristic
283C8D4×D19711yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—7882no—
288C22—15762no—
289C22—31522no—
290C22—31522no—
291C22—3942no—
292C22—3941no—
293C22#2026337211yescharacteristic
294C4—21no—
295C4—3942no—
296C4—3942no—
297C4—3941no—
298C4C2×D4×D19711yescharacteristic
299C4C2×D4×D19711yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—7882no—
304C2—7881no—
305C2—15762no—
306C2#2716171311yescharacteristic
subgroupG/H
centerC2#27161713
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#472×C197C2
socleC2×C197D42
hypercenter#472D197
hypocenterC197#3292
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #27161884C2semidirect
2. #27161883C2semidirect
3. #27161880C2semidirect
4. #27161454C2semidirect
5. #27161350C2semidirect
6. #27161348C2semidirect
7. #27161249C2semidirect
8. #27161245C2semidirect
9. #27161195C2semidirect
10. #27161194C2semidirect
11. #472×C197C2semidirect
12. #20263372C22semidirect
13. #20263371C22semidirect
15. #20263354C22semidirect
16. #20263354C22semidirect
18. #20263352C22semidirect
27. C2×D1576C22semidirect
28. C2×#15691343C22semidirect
31. #20263334C22semidirect
32. #20263332C22semidirect
33. #20263322C22semidirect
34. #20263261C22semidirect
35. #20263253C22semidirect
36. #20263252C22semidirect
37. #20263252C22semidirect
38. #20263242C22semidirect
39. C8×Dic197C22semidirect
40. AGL(1,ℤ/8)×C197C22semidirect
41. C2×D8×C197C22semidirect
42. #131×C197C22semidirect
43. #108×C197C22semidirect
44. #100×C197C22semidirect
59. Dic788D4semidirect
60. #15691348D4semidirect
63. #15691347D4semidirect
65. #15691346D4semidirect
66. #15691345D4semidirect
71. D1576D4semidirect
75. #15691343D4semidirect
153. C197#3292semidirect
161. #472D197semidirect
175. AGL(1,ℤ/8)C2×D197semidirect
186. C2×D8C2×D197semidirect
189. #131C2×D197semidirect
192. #108C2×D197semidirect
197. #100C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618842
#271618832
#271618802
#271614542
#271613502
#271613482
#271612492
#271612452
#271611952
#271611942
#472×C1972
#3292197