Finite Group Databasebeta

← #27153532

Subgroups — #27153532

Order 12416.

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#20261539C212yesmaximal, complemented
2#20261538C212yesmaximal, complemented
3#20261111C211yesmaximal, characteristic, complemented
4#20261004C212yesmaximal, complemented
5#20261003C212yesmaximal, complemented
6#20260904C212yesmaximal, complemented
7#20260902C211yesmaximal, characteristic, complemented
8#20260849C212yesmaximal, complemented
9#453×C97C211yesmaximal, characteristic, complemented
10#15690856C2212yescomplemented
11#15690855C2211yescharacteristic, complemented
12#15690838C2212yescomplemented
13#15690837—22no—
14#15690836—22no—
15#15690836C2212yescomplemented
16#15690835—22no—
17#15690835C2212yescomplemented
18#15690834—22no—
19#15690831—22no—
20#15690830—22no—
21#15690830C2212yes—
22C2×#13095496—22no—
23C2×#13095496C2212yescomplemented
24C2×#13095496C2212yescomplemented
25D8×D97—22no—
26#15690818C2212yescomplemented
27#15690817C2212yescomplemented
28#15690807C2211yescharacteristic, complemented
29#15690745C2212yescomplemented
30#15690738C2212yescomplemented
31#15690727C2211yescharacteristic, complemented
32#15690726C2212yescomplemented
33C4×#11715895C2211yescharacteristic, complemented
34H(5,𝔽2)×C97C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C97C2212yescomplemented
36#127×C97C2211yescharacteristic, complemented
37#104×C97C2212yescomplemented
38#100×C97C2211yescharacteristic, complemented
39#13095512—22no—
40#13095511—22no—
41#13095511C2312yes—
42#13095510—22no—
43C2×D388—22no—
44C2×D388—41no—
45C2×D388C2312yes—
46C2×C22⋊D97—41no—
47D4×D97—22no—
48D4×D97—22no—
49D4×D97—42no—
50#13095501D412yescomplemented
51#13095500—22no—
52#13095499—22no—
53#13095499D412yescomplemented
54#13095498—22no—
55#13095498D412yescomplemented
56D776—22no—
57#13095496—22no—
58#13095496—22no—
59#13095496—22no—
60#13095496—22no—
61#13095496—22no—
62#13095496—22no—
63#13095496D412yescomplemented
64#13095495—22no—
65#13095495C2312yes—
66#13095494—22no—
67C2×#11715895—22no—
68C2×#11715895C2312yes—
69C8×D97—22no—
70#41×C97—22no—
71#41×C97C2312yes—
72C2×D4×C97—22no—
73C2×D4×C97—41no—
74C2×D4×C97—41no—
75C2×D4×C97C2311yescharacteristic
76C2×D4×C97C2311yescharacteristic
77SD16×C97—22no—
78D8×C97—22no—
79M16×C97C2312yes—
80C42×C97C2311yescharacteristic
81C23×C97—21no—
82C23×C97—21no—
83C23×C97—41no—
84C22×D97—22no—
85C22×D97—41no—
86C22×D97—42no—
87Q8×C97C2×D412yes—
88D4×C97—22no—
89D4×C97—22no—
90D4×C97—22no—
91D4×C97—42no—
92D4×C97—82no—
93D4×C97C2×D412yes—
94C2×C4×C97—21no—
95C2×C4×C97—22no—
96C2×C4×C97—41no—
97C2×C4×C97C2411yescharacteristic
98C22⋊D97—22no—
99C22⋊D97—42no—
100C22⋊D97—42no—
101C2×Dic97—41no—
102D388—22no—
103D388—22no—
104D388—42no—
105D388—82no—
106D388C2×D412yes—
107C4×D97—22no—
108C4×D97—42no—
109Dic194C2×D412yes—
110C8×C97—22no—
111#11715895—22no—
112#11715895C2×D412yes—
113#11715895C2×D412yes—
114C22×C97—22no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97—81no—
118C22×C97—81no—
119C22×C97—161no—
120C22×C97H(5,𝔽2)11yescharacteristic
121C2×D97—22no—
122C2×D97—42no—
123C2×D97—82no—
124C2×D97—161no—
125C2×D97—162no—
126C4×C97—22no—
127C4×C97—22no—
128C4×C97C22×D412yes—
129Dic97—22no—
130C2×C97—21no—
131C2×C97—42no—
132C2×C97—81no—
133C2×C97—81no—
134C2×C97D4211yescharacteristic
135D97—42no—
136D97—82no—
137#3292—971nomaximal, Sylow
138C97#329211yescharacteristic, complemented, Sylow
139#576—972no—
140#576—972no—
141#493—971no—
142#472—972no—
143#472—972no—
144#453—971no—
145#453—972no—
146#453D9711yescharacteristic, complemented
147#443—972no—
148H(5,𝔽2)—972no—
149H(5,𝔽2)—971no—
150H(5,𝔽2)C2×D9711yescharacteristic, complemented
151AGL(1,ℤ/8)—972no—
152AGL(1,ℤ/8)—972no—
153AGL(1,ℤ/8)—972no—
154AGL(1,ℤ/8)—1942no—
155AGL(1,ℤ/8)—1942no—
156AGL(1,ℤ/8)—1942no—
157AGL(1,ℤ/8)—1942no—
158AGL(1,ℤ/8)C2×D9712yescomplemented
159#135—972no—
160#135—1942no—
161#135—1942no—
162C2×D8—972no—
163C2×D8—972no—
164C2×D8—1942no—
165C2×D8—1942no—
166#131—972no—
167#131—972no—
168#127—971no—
169#127C2×D9711yescharacteristic, complemented
170#108—972no—
171#104—972no—
172#104C2×D9712yescomplemented
173#100—971no—
174#100—972no—
175#100C2×D9711yescharacteristic, complemented
176C4×C8—971no—
177#41—22no—
178#41—972no—
179#41—1942no—
180#41—1942no—
181#41—1942no—
182#41C22×D9712yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—972no—
187C2×D4—3881no—
188C2×D4—3882no—
189C2×D4—3881no—
190C2×D4—1942no—
191C2×D4—1942no—
192C2×D4—1942no—
193C2×D4C22×D9711yescharacteristic
194C2×D4C22×D9711yescharacteristic
195Q16—972no—
196SD16—22no—
197SD16—972no—
198SD16—972no—
199SD16—1942no—
200SD16—1942no—
201SD16—1942no—
202D8—22no—
203D8—972no—
204D8—1942no—
205D8—1942no—
206D8—1942no—
207D8—1942no—
208D8—1942no—
209D8—1942no—
210D8—1942no—
211M16—972no—
212M16—1942no—
213M16—1942no—
214M16C22×D9712yes—
215C2×C8—972no—
216C2×C8—1942no—
217C2×C8—1942no—
218C42C22×D9711yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—3882no—
223C23—3881no—
224C23—1942no—
225Q8—972no—
226Q8D4×D9712yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—7762no—
233D4—972no—
234D4—3882no—
235D4—3882no—
236D4—3882no—
237D4—1942no—
238D4—1942no—
239D4—1942no—
240D4D4×D9712yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—3882no—
245C2×C4—3881no—
246C2×C4—1942no—
247C2×C4C23×D9711yescharacteristic
248C8—22no—
249C8—972no—
250C8—972no—
251C8—1942no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—7762no—
258C22—161no—
259C22—15522no—
260C22—15521no—
261C22—3882no—
262C22—1942no—
263C22#1569085511yescharacteristic
264C4—22no—
265C4—22no—
266C4—1942no—
267C4C2×D4×D9712yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—7762no—
273C2—3882no—
274C2#2026136711yescharacteristic
subgroupG/H
centerC2#20261367
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#453×C97C2
socleC2×C97D42
hypercenter#453D97
hypocenterC97#3292
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20261539C2semidirect
2. #20261538C2semidirect
3. #20261111C2semidirect
4. #20261004C2semidirect
5. #20261003C2semidirect
6. #20260904C2semidirect
7. #20260902C2semidirect
8. #20260849C2semidirect
9. #453×C97C2semidirect
10. #15690856C22semidirect
11. #15690855C22semidirect
12. #15690838C22semidirect
15. #15690836C22semidirect
17. #15690835C22semidirect
23. C2×#13095496C22semidirect
24. C2×#13095496C22semidirect
26. #15690818C22semidirect
27. #15690817C22semidirect
28. #15690807C22semidirect
29. #15690745C22semidirect
30. #15690738C22semidirect
31. #15690727C22semidirect
32. #15690726C22semidirect
33. C4×#11715895C22semidirect
34. H(5,𝔽2)×C97C22semidirect
35. AGL(1,ℤ/8)×C97C22semidirect
36. #127×C97C22semidirect
37. #104×C97C22semidirect
38. #100×C97C22semidirect
50. #13095501D4semidirect
53. #13095499D4semidirect
55. #13095498D4semidirect
63. #13095496D4semidirect
138. C97#3292semidirect
146. #453D97semidirect
150. H(5,𝔽2)C2×D97semidirect
158. AGL(1,ℤ/8)C2×D97semidirect
169. #127C2×D97semidirect
172. #104C2×D97semidirect
175. #100C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615392
#202615382
#202611112
#202610042
#202610032
#202609042
#202609022
#202608492
#453×C972
#329297