Finite Group Databasebeta

← #27151093

Subgroups — #27151093

Order 12416.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#20261533C212yesmaximal, complemented
2#20261302C211yesmaximal, characteristic, complemented
3#20260913C214yesmaximal, complemented
4#20260903C214yesmaximal, complemented
5#20260752C212yesmaximal, complemented
6C2×#15690727C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic, complemented
9#15690841—22no—
10#15690840C2214yescomplemented
11#15690836—22no—
12#15690835C2214yescomplemented
13#15690830—24no—
14C2×#13095495C2212yescomplemented
15#15690806C2212yescomplemented
16#15690797C2212yescomplemented
17#15690781—22no—
18#15690779—22no—
19#15690769C2211yescharacteristic, complemented
20#15690737C2218yescomplemented
21#15690727C2214yescomplemented
22H(5,𝔽2)×C97—24no—
23H(5,𝔽2)×C97—22no—
24H(5,𝔽2)×C97C2214yes—
25C2×#41×C97—22no—
26C2×#41×C97C2212yes—
27C22×D4×C97—22no—
28C22×D4×C97—22no—
29C22×D4×C97C2211yescharacteristic
30#13095511—22no—
31#13095511—22no—
32C2×Dic194C2311yescharacteristic, complemented
33C2×D388C2311yescharacteristic, complemented
34C2×C22⋊D97—21no—
35C2×C22⋊D97—24no—
36C2×C4×D97—21no—
37#13095501—24no—
38#13095499—24no—
39#13095498—24no—
40#13095496—24no—
41#13095495—22no—
42#13095495C2314yescomplemented
43C2×#11715895—22no—
44C4⋊Dic97—21no—
45C22⋊Dic97—21no—
46C22⋊Dic97—24no—
47C4×Dic97C2312yescomplemented
48C24×C97—22no—
49C24×C97D412yes—
50#41×C97—28no—
51#41×C97—28no—
52#41×C97—22no—
53#41×C97—22no—
54#41×C97D414yescomplemented
55#41×C97C2314yes—
56C2×Q8×C97D412yescomplemented
57C2×D4×C97—24no—
58C2×D4×C97—28no—
59C2×D4×C97—28no—
60C2×D4×C97—22no—
61C2×D4×C97—28no—
62C2×D4×C97—28no—
63C2×D4×C97—22no—
64C2×D4×C97—22no—
65C2×D4×C97D412yes—
66C2×D4×C97D412yescomplemented
67C2×D4×C97C2312yes—
68C22×C4×C97—22no—
69C22×C4×C97—22no—
70C22×C4×C97C2311yescharacteristic
71C23×C97—28no—
72C23×C97—22no—
73C23×C97—22no—
74C23×C97—22no—
75C23×C97—82no—
76C23×C97—84no—
77C23×C97C2×D414yes—
78C23×C97C2×D411yescharacteristic
79C22×D97—21no—
80Q8×C97—22no—
81Q8×C97C2×D414yes—
82D4×C97—216no—
83D4×C97—24no—
84D4×C97—24no—
85D4×C97—24no—
86D4×C97—22no—
87D4×C97—24no—
88D4×C97C2×D414yes—
89C2×C4×C97—28no—
90C2×C4×C97—22no—
91C2×C4×C97—24no—
92C2×C4×C97—24no—
93C2×C4×C97C2×D411yescharacteristic
94C2×C4×C97C2×D412yes—
95C2×C4×C97C2×D412yes—
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—41no—
98C22⋊D97—41no—
99C22⋊D97—84no—
100C2×Dic97—22no—
101C2×Dic97—21no—
102D388—22no—
103C4×D97—42no—
104Dic194—22no—
105#11715895—24no—
106C22×C97—24no—
107C22×C97—24no—
108C22×C97—41no—
109C22×C97—44no—
110C22×C97—44no—
111C22×C97—44no—
112C22×C97—82no—
113C22×C97—88no—
114C22×C97#12011yescharacteristic
115C22×C97#12011yescharacteristic
116C22×C97C22×D411yescharacteristic
117C2×D97—42no—
118C2×D97—41no—
119C4×C97—24no—
120C4×C97#12012yes—
121C4×C97C22×D412yes—
122Dic97—42no—
123Dic97—41no—
124C2×C97—21no—
125C2×C97—21no—
126C2×C97—21no—
127C2×C97—44no—
128C2×C97—44no—
129C2×C97C2×#12011yescharacteristic
130D97—81no—
131#3019—971nomaximal, Sylow
132C97#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)D9711yescharacteristic, complemented
134#575—972no—
135#535—971no—
136#455—974no—
137#453—974no—
138#421—972no—
139C2×#100—971no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D9714yes—
143C2×#41—22no—
144C2×#41—971no—
145C2×#41C2×D9712yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D9711yescharacteristic
149#137—974no—
150#137—1942no—
151AGL(1,ℤ/8)—974no—
152AGL(1,ℤ/8)—1942no—
153#135—1944no—
154C2×M16—972no—
155#127—972no—
156#124—972no—
157#121—1942no—
158#120—1942no—
159#117—971no—
160#104—978no—
161#100—974no—
162C24—22no—
163C24C22⋊D9712yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—1942no—
169#41—1942no—
170#41C22⋊D9714yescomplemented
171#41C22×D9714yes—
172C2×Q8—971no—
173C2×Q8C22⋊D9712yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—971no—
183C2×D4—1944no—
184C2×D4—1941no—
185C2×D4C22⋊D9712yes—
186C2×D4C22⋊D9712yescomplemented
187C2×D4C22×D9712yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—1941no—
191C22×C4C22×D9711yescharacteristic
192Q16—1944no—
193SD16—1944no—
194SD16—1944no—
195D8—1944no—
196M16—974no—
197M16—1942no—
198C2×C8—1942no—
199C4⋊C4—1941no—
200C22⋊C4—1941no—
201C22⋊C4—1944no—
202C42—972no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—1941no—
210C23C2×C22⋊D9714yes—
211C23C2×C22⋊D9711yescharacteristic
212Q8—22no—
213Q8—1942no—
214Q8C2×C22⋊D9714yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—7764no—
222D4—3881no—
223D4—3881no—
224D4—1942no—
225D4C2×C22⋊D9714yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—3882no—
231C2×C4—1942no—
232C2×C4—1941no—
233C2×C4C2×C22⋊D9711yescharacteristic
234C2×C4C2×C22⋊D9712yes—
235C2×C4C2×C22⋊D9712yes—
236C2×C4C23×D9711yescharacteristic
237C8—1944no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—3882no—
247C22—3881no—
248C22#1569077911yescharacteristic
249C22#1569077911yescharacteristic
250C22C22×C22⋊D9711yescharacteristic
251C4—24no—
252C4—3882no—
253C4—3881no—
254C4#1569077912yes—
255C4C22×C22⋊D9712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—7761no—
262C2C2×#1569077911yescharacteristic
subgroupG/H
centerC2C2×#15690779
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×H(5,𝔽2)×C97C2
socleC2×C97C2×#120
hypercenterC2×H(5,𝔽2)D97
hypocenterC97#3019
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20261533C2semidirect
2. #20261302C2semidirect
3. #20260913C2semidirect
4. #20260903C2semidirect
5. #20260752C2semidirect
6. C2×#15690727C2semidirect
7. C2×H(5,𝔽2)×C97C2semidirect
8. C2×#13095511C22semidirect
10. #15690840C22semidirect
12. #15690835C22semidirect
14. C2×#13095495C22semidirect
15. #15690806C22semidirect
16. #15690797C22semidirect
19. #15690769C22semidirect
20. #15690737C22semidirect
21. #15690727C22semidirect
32. C2×Dic194C23semidirect
33. C2×D388C23semidirect
42. #13095495C23semidirect
47. C4×Dic97C23semidirect
54. #41×C97D4semidirect
56. C2×Q8×C97D4semidirect
66. C2×D4×C97D4semidirect
132. C97#3019semidirect
133. C2×H(5,𝔽2)D97semidirect
170. #41C22⋊D97semidirect
173. C2×Q8C22⋊D97semidirect
186. C2×D4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202615332
#202613022
#202609132
#202609032
#202607522
C2×#156907272
C2×H(5,𝔽2)×C972
#301997