Finite Group Databasebeta

← #28401039

Subgroups — #28401039

Order 14464.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#16813651C212yesmaximal, complemented
2C2×#16813646C212yesmaximal, complemented
3#20299146C211yesmaximal, characteristic, complemented
4#124×D113C211yesmaximal, characteristic, complemented
5#20298993C211yesmaximal, characteristic, complemented
6#20298992C211yesmaximal, characteristic, complemented
7#20298985C211yesmaximal, characteristic, complemented
8#20298981C212yesmaximal, complemented
9#20298979C212yesmaximal, complemented
10#20298550C211yesmaximal, characteristic, complemented
11#496×C113C211yesmaximal, characteristic, complemented
12C2×#14268646C2212yes—
13C2×Q8×D113C2211yescharacteristic
14C2×D4×D113C2211yescharacteristic
15#16813651—28no—
16#16813646—28no—
17C2×Dic452C2212yescomplemented
18C2×#14268637C2212yescomplemented
19C2×#14268636C2212yes—
20C2×#14268635C2212yes—
21C2×#14268635C2212yescomplemented
22C2×#14268632C2212yes—
23C2×#14268630C2212yes—
24#16813622C2211yescharacteristic, complemented
25#16813611C2211yescharacteristic
26#16813610C2211yescharacteristic
27#16813609C2211yescharacteristic
28#16813609C2212yescomplemented
29#16813608—22no—
30#16813593—22no—
31D4×Dic113—22no—
32C22⋊C4×D113—22no—
33C42×D113C2211yescharacteristic
34#16813533C2211yescharacteristic, complemented
35C8×Dic113C2212yescomplemented
36C2×SD16×C113C2212yescomplemented
37C2×D8×C113C2212yes—
38#127×C113C2211yescharacteristic, complemented
39#124×C113C2211yescharacteristic
40#97×C113C2211yescharacteristic, complemented
41C23×D113—21no—
42#14268646—24no—
43#14268646—24no—
44C2×Dic226—21no—
45C2×Dic226C2311yescharacteristic
46C2×Dic226C2312yes—
47Q8×D113—24no—
48C2×D452C2311yescharacteristic
49C2×C22⋊D113—22no—
50C2×C22⋊D113—21no—
51D4×D113—24no—
52C22×Dic113—22no—
53C2×C4×D113—21no—
54C2×C4×D113D412yes—
55C2×C4×D113C2311yescharacteristic
56Dic452—24no—
57#14268637—24no—
58#14268636—24no—
59#14268635—24no—
60#14268635—24no—
61#14268632—24no—
62#14268630D418yescomplemented
63C2×#11744751C2312yes—
64C4⋊Dic113—22no—
65C22⋊Dic113—22no—
66C22⋊Dic113—24no—
67#14268624—22no—
68#14268624—22no—
69C4×Dic113D412yes—
70C4×Dic113C2311yescharacteristic
71C2×Q8×C113C2311yescharacteristic
72C2×D4×C113—22no—
73C2×D4×C113C2311yescharacteristic
74C2×D4×C113C2312yes—
75SD16×C113—24no—
76D8×C113—24no—
77C2×C8×C113C2312yes—
78C22⋊C4×C113—22no—
79C42×C113C2311yescharacteristic
80C23×C113—21no—
81C23×C113—22no—
82C22×D113—21no—
83C22×D113—44no—
84C22×D113—42no—
85C22×D113C2×D411yescharacteristic
86Q8×C113—22no—
87D4×C113—22no—
88D4×C113—22no—
89D4×C113—22no—
90D4×C113—44no—
91C2×C4×C113—21no—
92C2×C4×C113C2×D412yes—
93C2×C4×C113C2411yescharacteristic
94C22⋊D113—44no—
95C22⋊D113—42no—
96C2×Dic113—21no—
97C2×Dic113—22no—
98C2×Dic113—44no—
99C2×Dic113C2×D412yes—
100C2×Dic113C2×D411yescharacteristic
101D452—22no—
102C4×D113—42no—
103C4×D113—42no—
104C4×D113C2×D414yes—
105Dic226—22no—
106Dic226—22no—
107Dic226—22no—
108Dic226—42no—
109C8×C113C2×D414yes—
110#11744751C2×D414yes—
111C22×C113—42no—
112C22×C113—41no—
113C22×C113—44no—
114C22×C113—42no—
115C22×C113C22×D411yescharacteristic
116C2×D113—22no—
117C2×D113—41no—
118C2×D113—42no—
119C2×D113—84no—
120C2×D113#12712yes—
121C4×C113—22no—
122C4×C113—41no—
123C4×C113C22×D412yes—
124Dic113—22no—
125Dic113—41no—
126Dic113—42no—
127Dic113#12712yes—
128C2×C113—81no—
129C2×C113—82no—
130C2×C113C2×#12711yescharacteristic
131C2×C113#57512yes—
132D113—22no—
133D113—81no—
134#3153—1131nomaximal, Sylow
135C113#315311yescharacteristic, complemented, Sylow
136C2×#137—1132no—
137C2×AGL(1,ℤ/8)—1132no—
138#532—1131no—
139C2×#124—1131no—
140#497—1131no—
141#497—1131no—
142#496—1131no—
143#496D11311yescharacteristic, complemented
144#495—1132no—
145#495—1132no—
146C4×M16—1131no—
147C2×#41—1132no—
148C22×Q8—1131no—
149C22×D4—1131no—
150#137—2268no—
151AGL(1,ℤ/8)—2268no—
152C2×Q16—1132no—
153C2×Q16—1132no—
154C2×SD16—1132no—
155C2×SD16—1132no—
156C2×SD16—1132no—
157C2×SD16C2×D11312yescomplemented
158C2×D8—1132no—
159C2×D8C2×D11312yes—
160C2×M16—1132no—
161C4⋊Q8—1131no—
162#127C2×D11311yescharacteristic, complemented
163#124—1131no—
164#124—1131no—
165#124—1132no—
166#124—1131no—
167#124—2262no—
168#124C2×D11311yescharacteristic
169#121—2262no—
170C4×D4—2262no—
171C2×C22⋊C4—2262no—
172C2×C42—1131no—
173#97—1131no—
174#97C2×D11311yescharacteristic, complemented
175C4×C8—1132no—
176C24—2261no—
177#41—2264no—
178#41—2264no—
179C2×Q8—1131no—
180C2×Q8—1132no—
181C2×Q8—2261no—
182C2×Q8—2264no—
183C2×Q8C22×D11311yescharacteristic
184C2×D4—22no—
185C2×D4—1131no—
186C2×D4—2264no—
187C2×D4—2261no—
188C2×D4—2262no—
189C2×D4C22×D11311yescharacteristic
190C2×D4C22×D11312yes—
191C22×C4—1131no—
192C22×C4—1132no—
193C22×C4—2261no—
194C22×C4—2262no—
195Q16—2264no—
196Q16—2264no—
197SD16—24no—
198SD16—2264no—
199SD16—2264no—
200SD16—2264no—
201D8—24no—
202D8—2264no—
203M16—1138no—
204C2×C8—1132no—
205C2×C8C22×D11312yes—
206C4⋊C4—2262no—
207C22⋊C4—22no—
208C22⋊C4—2262no—
209C22⋊C4—2262no—
210C22⋊C4—2264no—
211C22⋊C4—2262no—
212C42—1132no—
213C42—1131no—
214C42C22×D11311yescharacteristic
215C23—21no—
216C23—22no—
217C23—1131no—
218C23—4522no—
219C23—4524no—
220C23—2261no—
221Q8—22no—
222Q8—4522no—
223Q8—2262no—
224Q8—2262no—
225Q8—2262no—
226D4—22no—
227D4—22no—
228D4—22no—
229D4—44no—
230D4—4522no—
231D4—4524no—
232D4—2262no—
233C2×C4—21no—
234C2×C4—1131no—
235C2×C4—1134no—
236C2×C4—1132no—
237C2×C4—4522no—
238C2×C4—4522no—
239C2×C4—4524no—
240C2×C4—2261no—
241C2×C4—2262no—
242C2×C4D4×D11312yes—
243C2×C4C23×D11311yescharacteristic
244C8—1134no—
245C8D4×D11314yes—
246C22—42no—
247C22—41no—
248C22—44no—
249C22—42no—
250C22—1132no—
251C22—9044no—
252C22—4521no—
253C22—4522no—
254C22—2262no—
255C22C2×D4×D11311yescharacteristic
256C4—22no—
257C4—41no—
258C4—1132no—
259C4—4522no—
260C4—4521no—
261C4—2262no—
262C4C2×D4×D11312yes—
263C2—81no—
264C2—82no—
265C2—9041no—
266C2—2262no—
267C2#127×D11311yescharacteristic
268C2#2029939412yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#496×C113C2
socleC22×C113C22×D4
hypercenter#496D113
hypocenterC113#3153
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16813651C2semidirect
2. C2×#16813646C2semidirect
3. #20299146C2semidirect
4. #124×D113C2semidirect
5. #20298993C2semidirect
6. #20298992C2semidirect
7. #20298985C2semidirect
8. #20298981C2semidirect
9. #20298979C2semidirect
10. #20298550C2semidirect
11. #496×C113C2semidirect
17. C2×Dic452C22semidirect
18. C2×#14268637C22semidirect
21. C2×#14268635C22semidirect
24. #16813622C22semidirect
28. #16813609C22semidirect
34. #16813533C22semidirect
35. C8×Dic113C22semidirect
36. C2×SD16×C113C22semidirect
38. #127×C113C22semidirect
40. #97×C113C22semidirect
62. #14268630D4semidirect
135. C113#3153semidirect
143. #496D113semidirect
157. C2×SD16C2×D113semidirect
162. #127C2×D113semidirect
174. #97C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136512
C2×#168136462
#202991462
#124×D1132
#202989932
#202989922
#202989852
#202989812
#202989792
#202985502
#496×C1132
#3153113