Finite Group Databasebeta

← #27153531

Subgroups — #27153531

Order 12416.

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

normal subgroups only

This group has 460 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#20261546C212yesmaximal, complemented
2#20261545C212yesmaximal, complemented
3#20261107C211yesmaximal, characteristic, complemented
4#20260997C212yesmaximal, complemented
5#20260996C212yesmaximal, complemented
6#20260919C212yesmaximal, complemented
7#20260902C211yesmaximal, characteristic, complemented
8#20260849C212yesmaximal, complemented
9#453×C97C211yesmaximal, characteristic, complemented
10#15690857C2212yes—
11#15690855C2211yescharacteristic, complemented
12#15690843C2212yescomplemented
13#15690840—22no—
14#15690840C2212yescomplemented
15#15690839—22no—
16#15690836—22no—
17#15690836C2212yes—
18#15690834—22no—
19#15690830—22no—
20#15690830C2212yes—
21#15690829—22no—
22C2×#13095499—22no—
23C2×#13095499C2212yescomplemented
24C2×#13095499C2212yescomplemented
25SD16×D97—22no—
26#15690818C2212yescomplemented
27#15690817C2212yescomplemented
28#15690810C2211yescharacteristic, complemented
29#15690745C2212yescomplemented
30#15690738C2212yescomplemented
31#15690728C2212yescomplemented
32#15690727C2211yescharacteristic, complemented
33C4×#11715895C2211yescharacteristic, complemented
34H(5,𝔽2)×C97C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C97C2212yes—
36#127×C97C2211yescharacteristic
37#104×C97C2212yescomplemented
38#100×C97C2211yescharacteristic, complemented
39#13095511—22no—
40#13095511C2312yes—
41#13095510—22no—
42#13095510—22no—
43#13095510—42no—
44C2×Dic194—22no—
45C2×Dic194C2312yes—
46Q8×D97—22no—
47C2×C22⋊D97—41no—
48D4×D97—22no—
49Dic388—22no—
50#13095501—22no—
51#13095501D412yescomplemented
52#13095500—22no—
53#13095499—22no—
54#13095499—22no—
55#13095499—22no—
56#13095499—22no—
57#13095499—22no—
58#13095499—22no—
59#13095499D412yescomplemented
60#13095498D412yescomplemented
61#13095496—22no—
62#13095496D412yescomplemented
63#13095495—22no—
64#13095495C2312yes—
65#13095494—22no—
66C2×#11715895—22no—
67C2×#11715895C2312yes—
68C8×D97—22no—
69C4⋊Dic97—41no—
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—41no—
85Q8×C97C2×D412yes—
86D4×C97—22no—
87D4×C97—22no—
88D4×C97—22no—
89D4×C97—42no—
90D4×C97—82no—
91D4×C97C2×D412yes—
92C2×C4×C97—21no—
93C2×C4×C97—22no—
94C2×C4×C97—41no—
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—22no—
97C22⋊D97—42no—
98C22⋊D97—42no—
99C2×Dic97—22no—
100C2×Dic97—41no—
101C2×Dic97—42no—
102D388C2×D412yes—
103C4×D97—22no—
104C4×D97—42no—
105Dic194—22no—
106Dic194—22no—
107Dic194—42no—
108Dic194C2×D412yes—
109C8×C97—22no—
110#11715895—22no—
111#11715895C2×D412yes—
112#11715895C2×D412yes—
113C22×C97—22no—
114C22×C97—41no—
115C22×C97—41no—
116C22×C97—81no—
117C22×C97—81no—
118C22×C97—161no—
119C22×C97H(5,𝔽2)11yescharacteristic
120C2×D97—22no—
121C2×D97—161no—
122C4×C97—22no—
123C4×C97—22no—
124C4×C97C22×D412yes—
125Dic97—22no—
126Dic97—42no—
127C2×C97—21no—
128C2×C97—42no—
129C2×C97—81no—
130C2×C97—81no—
131C2×C97D4211yescharacteristic
132D97—42no—
133#3291—971nomaximal, Sylow
134C97#329111yescharacteristic, complemented, Sylow
135#577—972no—
136#577—972no—
137#492—971no—
138#471—972no—
139#471—972no—
140#456—972no—
141#453—971no—
142#453D9711yescharacteristic, complemented
143#443—972no—
1442−1+4—972no—
145H(5,𝔽2)—971no—
146H(5,𝔽2)C2×D9711yescharacteristic, complemented
147#137—972no—
148#137—972no—
149#137—1942no—
150#137—1942no—
151AGL(1,ℤ/8)—972no—
152AGL(1,ℤ/8)—1942no—
153AGL(1,ℤ/8)—1942no—
154AGL(1,ℤ/8)C2×D9712yes—
155#135—972no—
156#135—1942no—
157#135—1942no—
158C2×SD16—972no—
159C2×SD16—972no—
160C2×SD16—1942no—
161C2×SD16—1942no—
162#131—972no—
163#131—972no—
164C4⋊Q8—971no—
165#127C2×D9711yescharacteristic
166#108—972no—
167#104—972no—
168#104C2×D9712yescomplemented
169#101—972no—
170#100—971no—
171#100C2×D9711yescharacteristic, complemented
172C4×C8—971no—
173#41—22no—
174#41—972no—
175#41—3882no—
176#41—1942no—
177#41—1942no—
178#41—1942no—
179#41C22×D9712yes—
180C2×Q8—972no—
181C2×Q8—1942no—
182C2×Q8—1942no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3881no—
187C2×D4—1942no—
188C2×D4C22×D9711yescharacteristic
189C2×D4C22×D9711yescharacteristic
190Q16—972no—
191Q16—1942no—
192Q16—1942no—
193SD16—22no—
194SD16—972no—
195SD16—972no—
196SD16—1942no—
197SD16—1942no—
198SD16—1942no—
199SD16—1942no—
200SD16—1942no—
201SD16—1942no—
202SD16—1942no—
203D8—22no—
204D8—972no—
205D8—1942no—
206M16—972no—
207M16—1942no—
208M16—1942no—
209M16C22×D9712yes—
210C2×C8—972no—
211C2×C8—1942no—
212C2×C8—1942no—
213C4⋊C4—3881no—
214C42C22×D9711yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—3881no—
219Q8—972no—
220Q8—3882no—
221Q8—1942no—
222Q8—1942no—
223Q8D4×D9712yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—972no—
230D4—3882no—
231D4—3882no—
232D4—1942no—
233D4D4×D9712yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—3882no—
238C2×C4—3882no—
239C2×C4—3881no—
240C2×C4—1942no—
241C2×C4—1942no—
242C2×C4C23×D9711yescharacteristic
243C8—22no—
244C8—972no—
245C8—972no—
246C8—1942no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—15521no—
254C22—1942no—
255C22#1569085511yescharacteristic
256C4—22no—
257C4—22no—
258C4—3882no—
259C4—1942no—
260C4C2×D4×D9712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—3882no—
266C2#2026136711yescharacteristic
subgroupG/H
centerC2#20261367
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#453×C97C2
socleC2×C97D42
hypercenter#453D97
hypocenterC97#3291
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. #20261546C2semidirect
2. #20261545C2semidirect
3. #20261107C2semidirect
4. #20260997C2semidirect
5. #20260996C2semidirect
6. #20260919C2semidirect
7. #20260902C2semidirect
8. #20260849C2semidirect
9. #453×C97C2semidirect
11. #15690855C22semidirect
12. #15690843C22semidirect
14. #15690840C22semidirect
23. C2×#13095499C22semidirect
24. C2×#13095499C22semidirect
26. #15690818C22semidirect
27. #15690817C22semidirect
28. #15690810C22semidirect
29. #15690745C22semidirect
30. #15690738C22semidirect
31. #15690728C22semidirect
32. #15690727C22semidirect
33. C4×#11715895C22semidirect
34. H(5,𝔽2)×C97C22semidirect
37. #104×C97C22semidirect
38. #100×C97C22semidirect
51. #13095501D4semidirect
59. #13095499D4semidirect
60. #13095498D4semidirect
62. #13095496D4semidirect
134. C97#3291semidirect
142. #453D97semidirect
146. H(5,𝔽2)C2×D97semidirect
168. #104C2×D97semidirect
171. #100C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615462
#202615452
#202611072
#202609972
#202609962
#202609192
#202609022
#202608492
#453×C972
#329197