Finite Group Databasebeta

← #28402342

Subgroups — #28402342

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#20299406C212yesmaximal, complemented
2#20299402C212yesmaximal, complemented
3#20298988C211yesmaximal, characteristic, complemented
4#20298866C212yesmaximal, complemented
5#20298864C212yesmaximal, complemented
6#20298775C211yesmaximal, characteristic, complemented
7#20298767C212yesmaximal, complemented
8#20298717C212yesmaximal, complemented
9#454×C113C211yesmaximal, characteristic, complemented
10#16813668C2212yescomplemented
11#16813667C2211yescharacteristic
12#16813653—22no—
13#16813653C2212yescomplemented
14#16813650C2212yes—
15#16813649—22no—
16#16813647—22no—
17#16813647C2212yescomplemented
18#16813646—22no—
19#16813643—22no—
20#16813642—22no—
21#16813642C2212yes—
22C2×#14268634—22no—
23C2×#14268634C2212yescomplemented
24C2×#14268632C2212yescomplemented
25D8×D113—22no—
26#16813630C2212yescomplemented
27#16813629C2212yescomplemented
28#16813619C2211yescharacteristic, complemented
29#16813557C2212yescomplemented
30#16813550C2212yescomplemented
31#16813539C2211yescharacteristic, complemented
32#16813538C2212yescomplemented
33#16813533C2211yescharacteristic, complemented
342−1+4×C113C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C113C2212yescomplemented
36#124×C113C2211yescharacteristic, complemented
37#104×C113C2212yescomplemented
38#101×C113C2211yescharacteristic, complemented
39#14268648—22no—
40#14268647—22no—
41#14268647C2312yes—
42#14268646—22no—
43C2×D452—22no—
44C2×D452—41no—
45C2×D452C2312yes—
46C2×C22⋊D113—41no—
47D4×D113—22no—
48D4×D113—22no—
49D4×D113—42no—
50#14268637—22no—
51#14268637D412yescomplemented
52#14268636—22no—
53#14268635D412yescomplemented
54#14268634—22no—
55#14268634—22no—
56#14268634—22no—
57#14268634—22no—
58#14268634D412yescomplemented
59D904—22no—
60#14268632—22no—
61#14268632—22no—
62#14268632—22no—
63#14268632D412yescomplemented
64#14268631—22no—
65#14268631C2312yes—
66#14268630—22no—
67C2×#11744751—22no—
68C2×#11744751C2312yes—
69C8×D113—22no—
70#41×C113—22no—
71#41×C113—41no—
72#41×C113C2312yes—
73C2×Q8×C113—22no—
74C2×Q8×C113C2311yescharacteristic
75C2×D4×C113C2311yescharacteristic
76SD16×C113—22no—
77D8×C113—22no—
78M16×C113C2312yes—
79C22⋊C4×C113—41no—
80C42×C113C2311yescharacteristic
81C23×C113—21no—
82C22×D113—22no—
83C22×D113—41no—
84C22×D113—42no—
85Q8×C113—22no—
86Q8×C113—41no—
87Q8×C113C2×D412yes—
88D4×C113—22no—
89D4×C113—22no—
90D4×C113—41no—
91D4×C113C2×D412yes—
92C2×C4×C113—22no—
93C2×C4×C113—21no—
94C2×C4×C113—21no—
95C2×C4×C113—42no—
96C2×C4×C113C2411yescharacteristic
97C22⋊D113—22no—
98C22⋊D113—42no—
99C22⋊D113—42no—
100C2×Dic113—41no—
101D452—22no—
102D452—22no—
103D452—42no—
104D452—82no—
105D452C2×D412yes—
106C4×D113—22no—
107C4×D113—42no—
108Dic226C2×D412yes—
109C8×C113—22no—
110#11744751—22no—
111#11744751C2×D412yes—
112#11744751C2×D412yes—
113C22×C113—22no—
114C22×C113—41no—
115C22×C113—81no—
116C22×C113H(5,𝔽2)11yescharacteristic
117C2×D113—22no—
118C2×D113—42no—
119C2×D113—82no—
120C2×D113—161no—
121C2×D113—162no—
122C4×C113—22no—
123C4×C113—41no—
124C4×C113—41no—
125C4×C113C22×D412yes—
126Dic113—22no—
127C2×C113—21no—
128C2×C113—42no—
129C2×C113—81no—
130C2×C113D4211yescharacteristic
131D113—42no—
132D113—82no—
133#3288—1131nomaximal, Sylow
134C113#328811yescharacteristic, complemented, Sylow
135#577—1132no—
136#576—1132no—
137#496—1131no—
138#472—1132no—
139#471—1132no—
140#455—1131no—
141#454D11311yescharacteristic, complemented
142#453—1132no—
143#444—1132no—
1442−1+4C2×D11311yescharacteristic, complemented
145H(5,𝔽2)—1132no—
146H(5,𝔽2)—1131no—
147#137—1132no—
148#137—2262no—
149AGL(1,ℤ/8)—1132no—
150AGL(1,ℤ/8)—1132no—
151AGL(1,ℤ/8)—2262no—
152AGL(1,ℤ/8)—2262no—
153AGL(1,ℤ/8)—2262no—
154AGL(1,ℤ/8)C2×D11312yescomplemented
155#135—1132no—
156#135—2262no—
157#135—2262no—
158C2×SD16—1132no—
159C2×SD16—2262no—
160C2×D8—1132no—
161C2×D8—2262no—
162#131—1132no—
163#131—1132no—
164#127—1131no—
165#124C2×D11311yescharacteristic, complemented
166#108—1132no—
167#104—1132no—
168#104C2×D11312yescomplemented
169#101C2×D11311yescharacteristic, complemented
170#100—1131no—
171#100—1132no—
172#97—1131no—
173#41—22no—
174#41—41no—
175#41—1132no—
176#41—2262no—
177#41—2262no—
178#41—2262no—
179#41C22×D11312yes—
180C2×Q8—22no—
181C2×Q8C22×D11311yescharacteristic
182C2×D4—1132no—
183C2×D4—4521no—
184C2×D4—4522no—
185C2×D4—4521no—
186C2×D4—2262no—
187C2×D4—2262no—
188C2×D4—2262no—
189C2×D4C22×D11311yescharacteristic
190Q16—1132no—
191Q16—2262no—
192SD16—22no—
193SD16—1132no—
194SD16—1132no—
195SD16—2262no—
196SD16—2262no—
197SD16—2262no—
198SD16—2262no—
199SD16—2262no—
200D8—22no—
201D8—1132no—
202D8—2262no—
203D8—2262no—
204D8—2262no—
205D8—2262no—
206M16—1132no—
207M16—2262no—
208M16—2262no—
209M16C22×D11312yes—
210C2×C8—1132no—
211C2×C8—2262no—
212C2×C8—2262no—
213C22⋊C4—41no—
214C42C22×D11311yescharacteristic
215C23—21no—
216C23—4522no—
217C23—4521no—
218C23—2262no—
219Q8—22no—
220Q8—41no—
221Q8—1132no—
222Q8D4×D11312yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—1132no—
227D4—9042no—
228D4—4522no—
229D4—4522no—
230D4—4522no—
231D4—2262no—
232D4—2262no—
233D4—2262no—
234D4D4×D11312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—4522no—
240C2×C4—4521no—
241C2×C4—2262no—
242C2×C4C23×D11311yescharacteristic
243C8—22no—
244C8—1132no—
245C8—1132no—
246C8—2262no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—18082no—
251C22—18081no—
252C22—9042no—
253C22—4522no—
254C22—2262no—
255C22#1681366711yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—2262no—
260C4C2×D4×D11312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—9042no—
265C2—4522no—
266C2#2029923011yescharacteristic
subgroupG/H
centerC2#20299230
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#454×C113C2
socleC2×C113D42
hypercenter#454D113
hypocenterC113#3288
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #20299406C2semidirect
2. #20299402C2semidirect
3. #20298988C2semidirect
4. #20298866C2semidirect
5. #20298864C2semidirect
6. #20298775C2semidirect
7. #20298767C2semidirect
8. #20298717C2semidirect
9. #454×C113C2semidirect
10. #16813668C22semidirect
13. #16813653C22semidirect
17. #16813647C22semidirect
23. C2×#14268634C22semidirect
24. C2×#14268632C22semidirect
26. #16813630C22semidirect
27. #16813629C22semidirect
28. #16813619C22semidirect
29. #16813557C22semidirect
30. #16813550C22semidirect
31. #16813539C22semidirect
32. #16813538C22semidirect
33. #16813533C22semidirect
34. 2−1+4×C113C22semidirect
35. AGL(1,ℤ/8)×C113C22semidirect
36. #124×C113C22semidirect
37. #104×C113C22semidirect
38. #101×C113C22semidirect
51. #14268637D4semidirect
53. #14268635D4semidirect
58. #14268634D4semidirect
63. #14268632D4semidirect
134. C113#3288semidirect
141. #454D113semidirect
144. 2−1+4C2×D113semidirect
154. AGL(1,ℤ/8)C2×D113semidirect
165. #124C2×D113semidirect
168. #104C2×D113semidirect
169. #101C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202994062
#202994022
#202989882
#202988662
#202988642
#202987752
#202987672
#202987172
#454×C1132
#3288113