Finite Group Databasebeta

← #28402340

Subgroups — #28402340

Order 14464.

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

normal subgroups only

This group has 464 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#20299404C212yesmaximal, complemented
2#20299399C212yesmaximal, complemented
3#20298865C212yesmaximal, complemented
4#20298858C212yesmaximal, complemented
5#20298775C211yesmaximal, characteristic, complemented
6#20298770C211yesmaximal, characteristic, complemented
7#20298765C212yesmaximal, complemented
8#20298721C212yesmaximal, complemented
9#496×C113C211yesmaximal, characteristic, complemented
10#16813670C2211yescharacteristic, complemented
11#16813667C2211yescharacteristic
12#16813667C2212yescomplemented
13#16813653C2212yescomplemented
14#16813651—22no—
15#16813649—22no—
16#16813648C2212yes—
17#16813648C2212yescomplemented
18#16813648C2212yescomplemented
19#16813646—22no—
20#16813646—22no—
21#16813644C2212yes—
22#16813643—22no—
23#16813641—22no—
24SD16×D113—22no—
25D8×D113—22no—
26#16813628C2212yescomplemented
27#16813628C2212yescomplemented
28#16813558C2212yescomplemented
29#16813550C2212yescomplemented
30#16813550C2212yescomplemented
31#16813541C2211yescharacteristic, complemented
32#16813539C2212yescomplemented
33#16813539C2211yescharacteristic, complemented
34C2×SD16×C113C2212yescomplemented
35C2×D8×C113C2212yescomplemented
36#127×C113C2211yescharacteristic, complemented
37#124×C113C2211yescharacteristic, complemented
38#97×C113C2211yescharacteristic, complemented
39#14268648—22no—
40#14268647—41no—
41#14268647C2312yes—
42#14268647C2312yes—
43#14268646—22no—
44#14268646—22no—
45#14268646—22no—
46Q8×D113—22no—
47C2×C22⋊D113—41no—
48C2×C22⋊D113—42no—
49D4×D113—22no—
50D4×D113—22no—
51D4×D113—22no—
52Dic452D412yescomplemented
53#14268637—22no—
54#14268636D412yescomplemented
55#14268636D412yescomplemented
56#14268635—22no—
57#14268635—22no—
58#14268635—22no—
59#14268634—22no—
60D904D412yescomplemented
61#14268632—22no—
62#14268632—22no—
63#14268632—22no—
64#14268631C2312yes—
65#14268631C2312yes—
66#14268630—22no—
67#14268630—22no—
68C8×D113—22no—
69C8×D113—22no—
70C2×Q8×C113C2311yescharacteristic
71C2×D4×C113—41no—
72C2×D4×C113C2312yes—
73C2×D4×C113C2311yescharacteristic
74SD16×C113—22no—
75SD16×C113—22no—
76D8×C113—22no—
77D8×C113—22no—
78C2×C8×C113C2312yes—
79C22⋊C4×C113—41no—
80C42×C113C2311yescharacteristic
81C23×C113—21no—
82C23×C113—22no—
83C22×D113—42no—
84C22×D113—41no—
85Q8×C113—22no—
86D4×C113—22no—
87D4×C113—22no—
88D4×C113—22no—
89D4×C113—82no—
90C2×C4×C113—21no—
91C2×C4×C113—21no—
92C2×C4×C113C2411yescharacteristic
93C22⋊D113—22no—
94C22⋊D113—22no—
95C22⋊D113—42no—
96C22⋊D113—42no—
97C22⋊D113—42no—
98C2×Dic113—42no—
99C2×Dic113—41no—
100D452—41no—
101D452C2×D412yes—
102D452C2×D412yes—
103C4×D113—22no—
104C4×D113—22no—
105C4×D113—42no—
106Dic226—41no—
107Dic226C2×D412yes—
108Dic226C2×D412yes—
109C8×C113C2×D412yes—
110C8×C113C2×D412yes—
111#11744751—22no—
112#11744751—22no—
113C22×C113—41no—
114C22×C113—42no—
115C22×C113—82no—
116C22×C113—81no—
117C22×C113H(5,𝔽2)11yescharacteristic
118C2×D113—22no—
119C2×D113—22no—
120C2×D113—162no—
121C2×D113—161no—
122C4×C113—41no—
123C4×C113—41no—
124C4×C113C22×D412yes—
125Dic113—22no—
126Dic113—22no—
127C2×C113—21no—
128C2×C113—82no—
129C2×C113—81no—
130C2×C113D4211yescharacteristic
131D113—42no—
132D113—42no—
133#3288—1131nomaximal, Sylow
134C113#328811yescharacteristic, complemented, Sylow
135#577—1132no—
136#576—1132no—
137#496D11311yescharacteristic, complemented
138#472—1132no—
139#471—1132no—
140#455—1131no—
141#454—1131no—
142#453—1132no—
143#444—1132no—
1442−1+4—1131no—
145H(5,𝔽2)—1131no—
146H(5,𝔽2)—1132no—
147#137—1132no—
148#137—2262no—
149AGL(1,ℤ/8)—1132no—
150AGL(1,ℤ/8)—1132no—
151AGL(1,ℤ/8)—1132no—
152AGL(1,ℤ/8)—2262no—
153AGL(1,ℤ/8)—2262no—
154AGL(1,ℤ/8)—2262no—
155#135—1132no—
156#135—2262no—
157#135—2262no—
158C2×SD16—2262no—
159C2×SD16C2×D11312yescomplemented
160C2×D8—2262no—
161C2×D8C2×D11312yescomplemented
162#131—1132no—
163#131—1132no—
164#127C2×D11311yescharacteristic, complemented
165#124C2×D11311yescharacteristic, complemented
166#108—1132no—
167#104—1132no—
168#104—1132no—
169#101—1131no—
170#100—1132no—
171#100—1131no—
172#97C2×D11311yescharacteristic, complemented
173#41—1132no—
174#41—1132no—
175#41—4521no—
176#41—2262no—
177#41—2262no—
178#41—2262no—
179#41—2262no—
180C2×Q8—2262no—
181C2×Q8C22×D11311yescharacteristic
182C2×D4—41no—
183C2×D4—4521no—
184C2×D4—4522no—
185C2×D4—2262no—
186C2×D4—2262no—
187C2×D4—2262no—
188C2×D4C22×D11311yescharacteristic
189C2×D4C22×D11312yes—
190Q16—1132no—
191Q16—2262no—
192SD16—22no—
193SD16—22no—
194SD16—1132no—
195SD16—1132no—
196SD16—2262no—
197SD16—2262no—
198SD16—2262no—
199SD16—2262no—
200D8—22no—
201D8—22no—
202D8—1132no—
203D8—2262no—
204D8—2262no—
205D8—2262no—
206M16—1132no—
207M16—1132no—
208M16—2262no—
209M16—2262no—
210C2×C8—2262no—
211C2×C8—2262no—
212C2×C8C22×D11312yes—
213C22⋊C4—41no—
214C42C22×D11311yescharacteristic
215C23—21no—
216C23—22no—
217C23—4522no—
218C23—4521no—
219Q8—22no—
220Q8—1132no—
221Q8—1132no—
222Q8—4521no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—1132no—
228D4—1132no—
229D4—4521no—
230D4—4522no—
231D4—4522no—
232D4—4522no—
233D4—2262no—
234D4—2262no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—4522no—
238C2×C4—4521no—
239C2×C4—4522no—
240C2×C4—2262no—
241C2×C4—2262no—
242C2×C4C23×D11311yescharacteristic
243C8—2262no—
244C8—2262no—
245C8D4×D11312yes—
246C8D4×D11312yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—18081no—
252C22—18082no—
253C22—2262no—
254C22—2262no—
255C22#1681366711yescharacteristic
256C4—41no—
257C4—41no—
258C4—2262no—
259C4—2262no—
260C4C2×D4×D11312yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—4522no—
265C2—4522no—
266C2#2029923311yescharacteristic
subgroupG/H
centerC2#20299233
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#496×C113C2
socleC2×C113D42
hypercenter#496D113
hypocenterC113#3288
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20299404C2semidirect
2. #20299399C2semidirect
3. #20298865C2semidirect
4. #20298858C2semidirect
5. #20298775C2semidirect
6. #20298770C2semidirect
7. #20298765C2semidirect
8. #20298721C2semidirect
9. #496×C113C2semidirect
10. #16813670C22semidirect
12. #16813667C22semidirect
13. #16813653C22semidirect
17. #16813648C22semidirect
18. #16813648C22semidirect
26. #16813628C22semidirect
27. #16813628C22semidirect
28. #16813558C22semidirect
29. #16813550C22semidirect
30. #16813550C22semidirect
31. #16813541C22semidirect
32. #16813539C22semidirect
33. #16813539C22semidirect
34. C2×SD16×C113C22semidirect
35. C2×D8×C113C22semidirect
36. #127×C113C22semidirect
37. #124×C113C22semidirect
38. #97×C113C22semidirect
52. Dic452D4semidirect
54. #14268636D4semidirect
55. #14268636D4semidirect
60. D904D4semidirect
134. C113#3288semidirect
137. #496D113semidirect
159. C2×SD16C2×D113semidirect
161. C2×D8C2×D113semidirect
164. #127C2×D113semidirect
165. #124C2×D113semidirect
172. #97C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202994042
#202993992
#202988652
#202988582
#202987752
#202987702
#202987652
#202987212
#496×C1132
#3288113