Finite Group Databasebeta

← #28402196

Subgroups — #28402196

Order 14464.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16813650C212yesmaximal, complemented
2#20299269C211yesmaximal, characteristic, complemented
3D4×D452C211yesmaximal, characteristic, complemented
4#20298975C211yesmaximal, characteristic, complemented
5#20298851C212yesmaximal, complemented
6#20298799C211yesmaximal, characteristic, complemented
7#20298794C211yesmaximal, characteristic, complemented
8#20298745C212yesmaximal, complemented
9#20298734C212yesmaximal, complemented
10C4×#14268636C211yesmaximal, characteristic, complemented
11#436×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2212yescomplemented
13C22×D452C2212yes—
14C2×D4×D113—21no—
15#16813650—28no—
16C2×#14268636C2211yescharacteristic
17C2×#14268636C2212yescomplemented
18C2×D904C2212yes—
19C2×D904C2212yescomplemented
20C2×D904C2212yescomplemented
21#16813619C2211yescharacteristic
22#16813619C2211yescharacteristic, complemented
23#16813618—21no—
24#16813597—21no—
25#16813596C2212yes—
26#16813596C2212yescomplemented
27#16813595—22no—
28#16813592—22no—
29C4×Dic226C2211yescharacteristic, complemented
30C4×D452C2211yescharacteristic
31#16813584—21no—
32#16813554C2211yescharacteristic, complemented
33#16813547C2211yescharacteristic, complemented
34#16813544C2211yescharacteristic
35#16813544C2212yescomplemented
36#16813544C2212yescomplemented
37C2×M16×C113C2212yescomplemented
38C4×D4×C113C2211yescharacteristic
39C4⋊C8×C113C2211yescharacteristic, complemented
40C22⋊C8×C113C2212yescomplemented
41C4×C8×C113C2211yescharacteristic, complemented
42C23×D113—22no—
43#14268647—24no—
44#14268647—42no—
45C2×Dic226C2311yescharacteristic
46C2×D452—24no—
47C2×D452—21no—
48C2×D452—22no—
49C2×D452—21no—
50C2×D452—21no—
51C2×D452—21no—
52C2×D452—42no—
53C2×D452C2312yes—
54C2×D452C2312yes—
55C2×D452C2311yescharacteristic
56C2×C22⋊D113—22no—
57C2×C22⋊D113—22no—
58C2×C22⋊D113—22no—
59D4×D113—44no—
60C2×C4×D113—22no—
61C2×C4×D113—21no—
62#14268636—24no—
63#14268636D414yescomplemented
64D904—24no—
65D904—24no—
66D904—42no—
67C4⋊Dic113C2311yescharacteristic
68C4⋊Dic113C2311yescharacteristic
69#14268626—21no—
70#14268624—22no—
71#14268624—21no—
72#14268624—22no—
73C4×Dic113—21no—
74C2×D4×C113D411yescharacteristic
75C22×C4×C113C2312yes—
76M16×C113—24no—
77C2×C8×C113C2312yes—
78C2×C8×C113C2311yescharacteristic
79C2×C8×C113C2311yescharacteristic
80C4⋊C4×C113D411yescharacteristic
81C22⋊C4×C113D412yes—
82C42×C113C2311yescharacteristic
83C23×C113C2×D412yes—
84C22×D113—21no—
85C22×D113—22no—
86C22×D113—22no—
87C22×D113—44no—
88C22×D113—44no—
89C22×D113—82no—
90D4×C113—22no—
91C2×C4×C113—22no—
92C2×C4×C113—22no—
93C2×C4×C113C2×D411yescharacteristic
94C2×C4×C113C2×D412yes—
95C2×C4×C113C2×D411yescharacteristic
96C2×C4×C113C2411yescharacteristic
97C22⋊D113—44no—
98C22⋊D113—44no—
99C22⋊D113—82no—
100C2×Dic113—21no—
101C2×Dic113—21no—
102C2×Dic113—21no—
103D452—22no—
104D452—22no—
105D452—21no—
106D452—22no—
107D452—22no—
108D452—44no—
109D452—42no—
110D452—42no—
111D452—81no—
112D452—81no—
113D452C2×D412yes—
114C4×D113—44no—
115C4×D113—42no—
116Dic226—21no—
117Dic226C2×D412yescomplemented
118C8×C113—22no—
119C8×C113—21no—
120C8×C113C2×D412yescomplemented
121C22×C113—22no—
122C22×C113—22no—
123C22×C113—22no—
124C22×C113C22×D411yescharacteristic
125C2×D113—22no—
126C2×D113—41no—
127C2×D113—41no—
128C2×D113—42no—
129C2×D113—42no—
130C2×D113—42no—
131C2×D113—42no—
132C2×D113—42no—
133C2×D113—42no—
134C2×D113—88no—
135C4×C113—22no—
136C4×C113—21no—
137C4×C113AGL(1,ℤ/8)12yescomplemented
138C4×C113C22×D411yescharacteristic
139C4×C113H(5,𝔽2)11yescharacteristic
140Dic113—22no—
141Dic113—41no—
142Dic113—41no—
143C2×C113—42no—
144C2×C113D4211yescharacteristic
145C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C113#57611yescharacteristic
147D113—42no—
148D113—82no—
149D113—82no—
150#3274—1131nomaximal, Sylow
151C113#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1132no—
153#550—1131no—
154D42—1131no—
155#493—1131no—
156#469—1132no—
157#460—1131no—
158#459—1131no—
159#449—1132no—
160#447—1132no—
161C4×SD16—1131no—
162#436D11311yescharacteristic, complemented
163C2×#41—1132no—
164C22×D4—1132no—
165C22×D4—2261no—
166AGL(1,ℤ/8)—2268no—
167C2×SD16—1131no—
168C2×SD16—1132no—
169C2×D8—1132no—
170C2×D8—1132no—
171C2×D8—1132no—
172C2×M16C2×D11312yescomplemented
173#127—1131no—
174#127—1131no—
175#127—2261no—
176#121—1132no—
177#121—1132no—
178#121—2261no—
179#121—2262no—
180#120—2262no—
181C4×Q8—1131no—
182C4×D4—1131no—
183C4×D4—2261no—
184C4×D4C2×D11311yescharacteristic
185#106—1131no—
186C4⋊C8C2×D11311yescharacteristic, complemented
187#103—1131no—
188#102—1131no—
189#102—1132no—
190#102—1132no—
191C22⋊C8C2×D11312yescomplemented
192C4×C8C2×D11311yescharacteristic, complemented
193C24—2262no—
194#41—4522no—
195#41—2264no—
196C2×Q8—1131no—
197C2×D4—1131no—
198C2×D4—1132no—
199C2×D4—1132no—
200C2×D4—4522no—
201C2×D4—4524no—
202C2×D4—2264no—
203C2×D4—2262no—
204C2×D4—2262no—
205C2×D4—2262no—
206C2×D4—2261no—
207C2×D4—2261no—
208C2×D4—2261no—
209C2×D4—2262no—
210C2×D4—2261no—
211C2×D4D45211yescharacteristic
212C22×C4—2261no—
213C22×C4—2262no—
214C22×C4C22×D11312yes—
215SD16—1134no—
216SD16—2264no—
217D8—4522no—
218D8—2264no—
219D8—2264no—
220M16—24no—
221C2×C8C22×D11311yescharacteristic
222C2×C8C22×D11312yes—
223C2×C8C22×D11311yescharacteristic
224C4⋊C4—1131no—
225C4⋊C4—1131no—
226C4⋊C4—2261no—
227C4⋊C4D45211yescharacteristic
228C22⋊C4—2261no—
229C22⋊C4—2262no—
230C22⋊C4—2262no—
231C22⋊C4D45212yes—
232C42—2261no—
233C42C22×D11311yescharacteristic
234C23—9042no—
235C23—4524no—
236C23—4524no—
237C23—2262no—
238C23—2262no—
239C23—2261no—
240C23C2×D45212yes—
241Q8—1132no—
242Q8—2261no—
243D4—22no—
244D4—1132no—
245D4—9041no—
246D4—9041no—
247D4—9042no—
248D4—4524no—
249D4—4524no—
250D4—4524no—
251D4—4522no—
252D4—4522no—
253D4—2262no—
254D4—2262no—
255D4—2262no—
256D4—2262no—
257D4—2261no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—4524no—
261C2×C4—4522no—
262C2×C4—2261no—
263C2×C4—2261no—
264C2×C4—2261no—
265C2×C4C2×D45211yescharacteristic
266C2×C4C2×D45212yes—
267C2×C4C2×D45211yescharacteristic
268C2×C4C23×D11311yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D11312yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—9048no—
276C22—4522no—
277C22—4522no—
278C22—4522no—
279C22—4522no—
280C22—4522no—
281C22—4522no—
282C22—4521no—
283C22—4521no—
284C22—2262no—
285C22C22×D45211yescharacteristic
286C4—21no—
287C4—22no—
288C4—4521no—
289C4—4521no—
290C4—2262no—
291C4#1681365012yescomplemented
292C4C2×D4×D11311yescharacteristic
293C4#1681366811yescharacteristic
294C2—42no—
295C2—9042no—
296C2—9042no—
297C2—4522no—
298C2D4×D45211yescharacteristic
299C2C2×#1681365011yescharacteristic
300C2#2029940311yescharacteristic
subgroupG/H
centerC22C22×D452
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#436×C113C2
socleC22×C113C22×D4
hypercenter#436D113
hypocenterC113#3274
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#16813650C2semidirect
2. #20299269C2semidirect
3. D4×D452C2semidirect
4. #20298975C2semidirect
5. #20298851C2semidirect
6. #20298799C2semidirect
7. #20298794C2semidirect
8. #20298745C2semidirect
9. #20298734C2semidirect
10. C4×#14268636C2semidirect
11. #436×C113C2semidirect
12. C2×#14268647C22semidirect
17. C2×#14268636C22semidirect
19. C2×D904C22semidirect
20. C2×D904C22semidirect
22. #16813619C22semidirect
26. #16813596C22semidirect
29. C4×Dic226C22semidirect
32. #16813554C22semidirect
33. #16813547C22semidirect
35. #16813544C22semidirect
36. #16813544C22semidirect
37. C2×M16×C113C22semidirect
39. C4⋊C8×C113C22semidirect
40. C22⋊C8×C113C22semidirect
41. C4×C8×C113C22semidirect
63. #14268636D4semidirect
117. Dic226C2×D4semidirect
120. C8×C113C2×D4semidirect
137. C4×C113AGL(1,ℤ/8)semidirect
151. C113#3274semidirect
162. #436D113semidirect
172. C2×M16C2×D113semidirect
186. C4⋊C8C2×D113semidirect
191. C22⋊C8C2×D113semidirect
192. C4×C8C2×D113semidirect
271. C8D4×D113semidirect
291. C4#16813650semidirect

Maximal subgroups

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

subgroupindex
C2×#168136502
#202992692
D4×D4522
#202989752
#202988512
#202987992
#202987942
#202987452
#202987342
C4×#142686362
#436×C1132
#3274113