Finite Group Databasebeta

← #27182930

Subgroups — #27182930

Order 12928.

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#20265982C211yesmaximal, characteristic, complemented
2#20265981C212yesmaximal, complemented
3#20265978C211yesmaximal, characteristic, complemented
4#20265552C211yesmaximal, characteristic, complemented
5#20265448C211yesmaximal, characteristic, complemented
6#20265446C212yesmaximal, complemented
7#20265347C212yesmaximal, complemented
8#20265343C212yesmaximal, complemented
9#20265293C211yesmaximal, characteristic, complemented
10#20265292C211yesmaximal, characteristic, complemented
11#472×C101C211yesmaximal, characteristic, complemented
12#15692301C2212yescomplemented
13#15692300C2212yescomplemented
14#15692283—22no—
15#15692283C2212yescomplemented
16#15692283C2212yescomplemented
17#15692282—22no—
18#15692281C2212yescomplemented
19#15692279—22no—
20#15692279—22no—
21#15692277—21no—
22#15692277C2211yescharacteristic
23#15692276—22no—
24#15692275C2211yescharacteristic
25#15692274—21no—
26C2×D808—21no—
27C2×D808C2211yescharacteristic, complemented
28C2×#13108059C2212yescomplemented
29D8×D101—21no—
30D8×D101—22no—
31#15692263C2212yescomplemented
32#15692261C2211yescharacteristic, complemented
33#15692251C2212yescomplemented
34#15692190C2211yescharacteristic, complemented
35#15692182C2211yescharacteristic, complemented
36#15692181C2211yescharacteristic, complemented
37#15692181C2212yescomplemented
38#15692171C2212yescomplemented
39C8×Dic101C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C101C2212yescomplemented
41C2×D8×C101C2211yescharacteristic, complemented
42#131×C101C2211yescharacteristic, complemented
43#108×C101C2211yescharacteristic, complemented
44#100×C101C2212yescomplemented
45#13108075—22no—
46#13108074—22no—
47#13108074C2311yescharacteristic
48#13108074C2312yes—
49#13108073—22no—
50#13108073—22no—
51C2×D404—22no—
52C2×D404C2312yes—
53C2×C22⋊D101—42no—
54C2×C22⋊D101—42no—
55D4×D101—22no—
56D4×D101—22no—
57D4×D101—22no—
58D4×D101—42no—
59Dic404D411yescharacteristic, complemented
60#13108064D411yescharacteristic, complemented
61#13108063—22no—
62#13108063—22no—
63#13108063D412yescomplemented
64#13108062—22no—
65#13108062D411yescharacteristic, complemented
66#13108061D411yescharacteristic, complemented
67D808—22no—
68D808—22no—
69D808—21no—
70D808—21no—
71D808D411yescharacteristic, complemented
72#13108059—22no—
73#13108059—22no—
74#13108059—22no—
75#13108059D411yescharacteristic, complemented
76#13108058C2311yescharacteristic
77#13108057—21no—
78#13108057—22no—
79C2×#11717579C2311yescharacteristic
80C8×D101—21no—
81C8×D101—22no—
82C4×Dic101C2311yescharacteristic
83#41×C101C2311yescharacteristic
84C2×D4×C101C2312yes—
85SD16×C101—22no—
86D8×C101—22no—
87D8×C101—21no—
88D8×C101—21no—
89M16×C101—21no—
90M16×C101C2311yescharacteristic
91M16×C101C2312yes—
92C2×C8×C101—21no—
93C2×C8×C101C2311yescharacteristic
94C23×C101—22no—
95C22×D101—22no—
96C22×D101—42no—
97C22×D101—42no—
98Q8×C101C2×D411yescharacteristic
99D4×C101—21no—
100D4×C101—22no—
101D4×C101—22no—
102D4×C101C2×D411yescharacteristic
103C2×C4×C101—21no—
104C2×C4×C101C2411yescharacteristic
105C22⋊D101—22no—
106C22⋊D101—21no—
107C22⋊D101—42no—
108C22⋊D101—42no—
109C22⋊D101—42no—
110C22⋊D101—84no—
111C2×Dic101—21no—
112C2×Dic101—42no—
113D404—22no—
114D404—22no—
115D404—42no—
116D404C2×D412yes—
117D404C2×D411yescharacteristic
118C4×D101—21no—
119C4×D101—22no—
120C4×D101—42no—
121Dic202C2×D412yes—
122Dic202C2×D411yescharacteristic
123C8×C101—21no—
124C8×C101—22no—
125C8×C101C2×D411yescharacteristic
126C8×C101C2×D411yescharacteristic
127#11717579—21no—
128#11717579C2×D411yescharacteristic
129#11717579C2×D411yescharacteristic
130C22×C101—21no—
131C22×C101—42no—
132C22×C101—82no—
133C22×C101H(5,𝔽2)11yescharacteristic
134C2×D101—22no—
135C2×D101—21no—
136C2×D101—42no—
137C2×D101—82no—
138C2×D101—162no—
139C2×D101—162no—
140C4×C101—21no—
141C4×C101C22×D411yescharacteristic
142C4×C101C22×D411yescharacteristic
143Dic101—22no—
144Dic101—22no—
145Dic101—21no—
146C2×C101—21no—
147C2×C101—41no—
148C2×C101—82no—
149C2×C101D4211yescharacteristic
150D101—42no—
151D101—41no—
152D101—82no—
153#3292—1011nomaximal, Sylow
154C101#329211yescharacteristic, complemented, Sylow
155#576—1012no—
156#576—1011no—
157#576—1011no—
158#493—1011no—
159#472—1011no—
160#472—1012no—
161#472D10111yescharacteristic, complemented
162#453—1012no—
163#453—1012no—
164#443—1011no—
165#443—1011no—
166H(5,𝔽2)—1012no—
167H(5,𝔽2)—1012no—
168AGL(1,ℤ/8)—1012no—
169AGL(1,ℤ/8)—1012no—
170AGL(1,ℤ/8)—1012no—
171AGL(1,ℤ/8)—2022no—
172AGL(1,ℤ/8)—2022no—
173AGL(1,ℤ/8)—2022no—
174AGL(1,ℤ/8)—2022no—
175AGL(1,ℤ/8)C2×D10112yescomplemented
176#135—1011no—
177#135—1011no—
178#135—2021no—
179#135—2021no—
180#135—2022no—
181C2×D8—1012no—
182C2×D8—1011no—
183C2×D8—2021no—
184C2×D8—2022no—
185C2×D8—2021no—
186C2×D8C2×D10111yescharacteristic, complemented
187#131—1012no—
188#131—1011no—
189#131C2×D10111yescharacteristic, complemented
190#127—1012no—
191#108—1011no—
192#108C2×D10111yescharacteristic, complemented
193#104—1011no—
194#104—1012no—
195#104—1011no—
196#100—1012no—
197#100C2×D10112yescomplemented
198C4×C8—1011no—
199#41—1011no—
200#41—1012no—
201#41—2022no—
202#41—2022no—
203#41—2022no—
204#41—2022no—
205#41C22×D10111yescharacteristic
206C2×D4—1012no—
207C2×D4—4042no—
208C2×D4—4042no—
209C2×D4—4042no—
210C2×D4—2022no—
211C2×D4—2022no—
212C2×D4—2022no—
213C2×D4—2022no—
214C2×D4C22×D10112yes—
215Q16—1011no—
216Q16—1011no—
217SD16—22no—
218SD16—1011no—
219SD16—1011no—
220SD16—1012no—
221SD16—2022no—
222SD16—2022no—
223SD16—2022no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—1011no—
228D8—1011no—
229D8—2022no—
230D8—2022no—
231D8—2022no—
232D8—2022no—
233D8—2022no—
234D8—2021no—
235D8—2021no—
236M16—21no—
237M16—1011no—
238M16—2022no—
239M16—2021no—
240M16C22×D10112yes—
241M16C22×D10111yescharacteristic
242C2×C8—21no—
243C2×C8—1011no—
244C2×C8—2022no—
245C2×C8—2021no—
246C2×C8C22×D10111yescharacteristic
247C42—1011no—
248C23—22no—
249C23—4042no—
250C23—4042no—
251C23—2022no—
252Q8—1011no—
253Q8—1012no—
254Q8D4×D10111yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—8084no—
259D4—1011no—
260D4—1012no—
261D4—4042no—
262D4—4042no—
263D4—4042no—
264D4—4042no—
265D4—2022no—
266D4—2021no—
267D4—2022no—
268D4—2022no—
269D4D4×D10111yescharacteristic
270C2×C4—21no—
271C2×C4—4042no—
272C2×C4—4042no—
273C2×C4—2021no—
274C2×C4—2022no—
275C2×C4—2021no—
276C2×C4C23×D10111yescharacteristic
277C8—21no—
278C8—22no—
279C8—1011no—
280C8—1011no—
281C8—2021no—
282C8D4×D10111yescharacteristic
283C8D4×D10111yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—8082no—
288C22—16162no—
289C22—16162no—
290C22—4042no—
291C22—2022no—
292C22—2021no—
293C22#1569230111yescharacteristic
294C4—21no—
295C4—2022no—
296C4—2022no—
297C4—2021no—
298C4C2×D4×D10111yescharacteristic
299C4C2×D4×D10111yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—8082no—
304C2—4042no—
305C2—4041no—
306C2#2026581111yescharacteristic
subgroupG/H
centerC2#20265811
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#472×C101C2
socleC2×C101D42
hypercenter#472D101
hypocenterC101#3292
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #20265982C2semidirect
2. #20265981C2semidirect
3. #20265978C2semidirect
4. #20265552C2semidirect
5. #20265448C2semidirect
6. #20265446C2semidirect
7. #20265347C2semidirect
8. #20265343C2semidirect
9. #20265293C2semidirect
10. #20265292C2semidirect
11. #472×C101C2semidirect
12. #15692301C22semidirect
13. #15692300C22semidirect
15. #15692283C22semidirect
16. #15692283C22semidirect
18. #15692281C22semidirect
27. C2×D808C22semidirect
28. C2×#13108059C22semidirect
31. #15692263C22semidirect
32. #15692261C22semidirect
33. #15692251C22semidirect
34. #15692190C22semidirect
35. #15692182C22semidirect
36. #15692181C22semidirect
37. #15692181C22semidirect
38. #15692171C22semidirect
39. C8×Dic101C22semidirect
40. AGL(1,ℤ/8)×C101C22semidirect
41. C2×D8×C101C22semidirect
42. #131×C101C22semidirect
43. #108×C101C22semidirect
44. #100×C101C22semidirect
59. Dic404D4semidirect
60. #13108064D4semidirect
63. #13108063D4semidirect
65. #13108062D4semidirect
66. #13108061D4semidirect
71. D808D4semidirect
75. #13108059D4semidirect
154. C101#3292semidirect
161. #472D101semidirect
175. AGL(1,ℤ/8)C2×D101semidirect
186. C2×D8C2×D101semidirect
189. #131C2×D101semidirect
192. #108C2×D101semidirect
197. #100C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659822
#202659812
#202659782
#202655522
#202654482
#202654462
#202653472
#202653432
#202652932
#202652922
#472×C1012
#3292101