Finite Group Databasebeta

← #27182928

Subgroups — #27182928

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#20265981C212yesmaximal, complemented
2#20265980C212yesmaximal, complemented
3#20265553C211yesmaximal, characteristic, complemented
4#20265446C212yesmaximal, complemented
5#20265445C212yesmaximal, complemented
6#20265346C212yesmaximal, complemented
7#20265344C211yesmaximal, characteristic, complemented
8#20265291C212yesmaximal, complemented
9#453×C101C211yesmaximal, characteristic, complemented
10#15692301C2212yescomplemented
11#15692300C2211yescharacteristic, complemented
12#15692283C2212yescomplemented
13#15692282—22no—
14#15692281—22no—
15#15692281C2212yescomplemented
16#15692280—22no—
17#15692280C2212yescomplemented
18#15692279—22no—
19#15692276—22no—
20#15692275—22no—
21#15692275C2212yes—
22C2×#13108059—22no—
23C2×#13108059C2212yescomplemented
24C2×#13108059C2212yescomplemented
25D8×D101—22no—
26#15692263C2212yescomplemented
27#15692262C2212yescomplemented
28#15692252C2211yescharacteristic, complemented
29#15692190C2212yescomplemented
30#15692183C2212yescomplemented
31#15692172C2211yescharacteristic, complemented
32#15692171C2212yescomplemented
33C4×#11717579C2211yescharacteristic, complemented
34H(5,𝔽2)×C101C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C101C2212yescomplemented
36#127×C101C2211yescharacteristic, complemented
37#104×C101C2212yescomplemented
38#100×C101C2211yescharacteristic, complemented
39#13108075—22no—
40#13108074—22no—
41#13108074C2312yes—
42#13108073—22no—
43C2×D404—22no—
44C2×D404—41no—
45C2×D404C2312yes—
46C2×C22⋊D101—41no—
47D4×D101—22no—
48D4×D101—22no—
49D4×D101—42no—
50#13108064D412yescomplemented
51#13108063—22no—
52#13108062—22no—
53#13108062D412yescomplemented
54#13108061—22no—
55#13108061D412yescomplemented
56D808—22no—
57#13108059—22no—
58#13108059—22no—
59#13108059—22no—
60#13108059—22no—
61#13108059—22no—
62#13108059—22no—
63#13108059D412yescomplemented
64#13108058—22no—
65#13108058C2312yes—
66#13108057—22no—
67C2×#11717579—22no—
68C2×#11717579C2312yes—
69C8×D101—22no—
70#41×C101—22no—
71#41×C101C2312yes—
72C2×D4×C101—22no—
73C2×D4×C101—41no—
74C2×D4×C101—41no—
75C2×D4×C101C2311yescharacteristic
76C2×D4×C101C2311yescharacteristic
77SD16×C101—22no—
78D8×C101—22no—
79M16×C101C2312yes—
80C42×C101C2311yescharacteristic
81C23×C101—21no—
82C23×C101—21no—
83C23×C101—41no—
84C22×D101—22no—
85C22×D101—41no—
86C22×D101—42no—
87Q8×C101C2×D412yes—
88D4×C101—22no—
89D4×C101—22no—
90D4×C101—22no—
91D4×C101—42no—
92D4×C101—82no—
93D4×C101C2×D412yes—
94C2×C4×C101—21no—
95C2×C4×C101—22no—
96C2×C4×C101—41no—
97C2×C4×C101C2411yescharacteristic
98C22⋊D101—22no—
99C22⋊D101—42no—
100C22⋊D101—42no—
101C2×Dic101—41no—
102D404—22no—
103D404—22no—
104D404—42no—
105D404—82no—
106D404C2×D412yes—
107C4×D101—22no—
108C4×D101—42no—
109Dic202C2×D412yes—
110C8×C101—22no—
111#11717579—22no—
112#11717579C2×D412yes—
113#11717579C2×D412yes—
114C22×C101—22no—
115C22×C101—41no—
116C22×C101—41no—
117C22×C101—81no—
118C22×C101—81no—
119C22×C101—161no—
120C22×C101H(5,𝔽2)11yescharacteristic
121C2×D101—22no—
122C2×D101—42no—
123C2×D101—82no—
124C2×D101—161no—
125C2×D101—162no—
126C4×C101—22no—
127C4×C101—22no—
128C4×C101C22×D412yes—
129Dic101—22no—
130C2×C101—21no—
131C2×C101—42no—
132C2×C101—81no—
133C2×C101—81no—
134C2×C101D4211yescharacteristic
135D101—42no—
136D101—82no—
137#3292—1011nomaximal, Sylow
138C101#329211yescharacteristic, complemented, Sylow
139#576—1012no—
140#576—1012no—
141#493—1011no—
142#472—1012no—
143#472—1012no—
144#453—1011no—
145#453—1012no—
146#453D10111yescharacteristic, complemented
147#443—1012no—
148H(5,𝔽2)—1012no—
149H(5,𝔽2)—1011no—
150H(5,𝔽2)C2×D10111yescharacteristic, complemented
151AGL(1,ℤ/8)—1012no—
152AGL(1,ℤ/8)—1012no—
153AGL(1,ℤ/8)—1012no—
154AGL(1,ℤ/8)—2022no—
155AGL(1,ℤ/8)—2022no—
156AGL(1,ℤ/8)—2022no—
157AGL(1,ℤ/8)—2022no—
158AGL(1,ℤ/8)C2×D10112yescomplemented
159#135—1012no—
160#135—2022no—
161#135—2022no—
162C2×D8—1012no—
163C2×D8—1012no—
164C2×D8—2022no—
165C2×D8—2022no—
166#131—1012no—
167#131—1012no—
168#127—1011no—
169#127C2×D10111yescharacteristic, complemented
170#108—1012no—
171#104—1012no—
172#104C2×D10112yescomplemented
173#100—1011no—
174#100—1012no—
175#100C2×D10111yescharacteristic, complemented
176C4×C8—1011no—
177#41—22no—
178#41—1012no—
179#41—2022no—
180#41—2022no—
181#41—2022no—
182#41C22×D10112yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—1012no—
187C2×D4—4041no—
188C2×D4—4042no—
189C2×D4—4041no—
190C2×D4—2022no—
191C2×D4—2022no—
192C2×D4—2022no—
193C2×D4C22×D10111yescharacteristic
194C2×D4C22×D10111yescharacteristic
195Q16—1012no—
196SD16—22no—
197SD16—1012no—
198SD16—1012no—
199SD16—2022no—
200SD16—2022no—
201SD16—2022no—
202D8—22no—
203D8—1012no—
204D8—2022no—
205D8—2022no—
206D8—2022no—
207D8—2022no—
208D8—2022no—
209D8—2022no—
210D8—2022no—
211M16—1012no—
212M16—2022no—
213M16—2022no—
214M16C22×D10112yes—
215C2×C8—1012no—
216C2×C8—2022no—
217C2×C8—2022no—
218C42C22×D10111yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—4042no—
223C23—4041no—
224C23—2022no—
225Q8—1012no—
226Q8D4×D10112yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—8082no—
233D4—1012no—
234D4—4042no—
235D4—4042no—
236D4—4042no—
237D4—2022no—
238D4—2022no—
239D4—2022no—
240D4D4×D10112yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—4042no—
245C2×C4—4041no—
246C2×C4—2022no—
247C2×C4C23×D10111yescharacteristic
248C8—22no—
249C8—1012no—
250C8—1012no—
251C8—2022no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—8082no—
259C22—16162no—
260C22—16161no—
261C22—4042no—
262C22—2022no—
263C22#1569230011yescharacteristic
264C4—22no—
265C4—22no—
266C4—2022no—
267C4C2×D4×D10112yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—8082no—
273C2—4042no—
274C2#2026580911yescharacteristic
subgroupG/H
centerC2#20265809
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#453×C101C2
socleC2×C101D42
hypercenter#453D101
hypocenterC101#3292
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20265981C2semidirect
2. #20265980C2semidirect
3. #20265553C2semidirect
4. #20265446C2semidirect
5. #20265445C2semidirect
6. #20265346C2semidirect
7. #20265344C2semidirect
8. #20265291C2semidirect
9. #453×C101C2semidirect
10. #15692301C22semidirect
11. #15692300C22semidirect
12. #15692283C22semidirect
15. #15692281C22semidirect
17. #15692280C22semidirect
23. C2×#13108059C22semidirect
24. C2×#13108059C22semidirect
26. #15692263C22semidirect
27. #15692262C22semidirect
28. #15692252C22semidirect
29. #15692190C22semidirect
30. #15692183C22semidirect
31. #15692172C22semidirect
32. #15692171C22semidirect
33. C4×#11717579C22semidirect
34. H(5,𝔽2)×C101C22semidirect
35. AGL(1,ℤ/8)×C101C22semidirect
36. #127×C101C22semidirect
37. #104×C101C22semidirect
38. #100×C101C22semidirect
50. #13108064D4semidirect
53. #13108062D4semidirect
55. #13108061D4semidirect
63. #13108059D4semidirect
138. C101#3292semidirect
146. #453D101semidirect
150. H(5,𝔽2)C2×D101semidirect
158. AGL(1,ℤ/8)C2×D101semidirect
169. #127C2×D101semidirect
172. #104C2×D101semidirect
175. #100C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659812
#202659802
#202655532
#202654462
#202654452
#202653462
#202653442
#202652912
#453×C1012
#3292101