Finite Group Databasebeta

← #28402193

Subgroups — #28402193

Order 14464.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16813646C212yesmaximal, complemented
2#20299268C211yesmaximal, characteristic, complemented
3#20299233C211yesmaximal, characteristic, complemented
4#20298848C212yesmaximal, complemented
5#20298795C211yesmaximal, characteristic, complemented
6#20298790C211yesmaximal, characteristic, complemented
7#20298740C212yesmaximal, complemented
8#20298730C212yesmaximal, complemented
9C4×#14268636C211yesmaximal, characteristic, complemented
10#20298677C211yesmaximal, characteristic, complemented
11#493×C113C211yesmaximal, characteristic, complemented
12C2×#14268646C2212yescomplemented
13C2×D4×D113—21no—
14C2×D4×D113C2212yes—
15#16813646—28no—
16C2×#14268636C2211yescharacteristic
17C2×#14268635C2212yescomplemented
18C2×#14268632C2212yes—
19C2×#14268630C2212yescomplemented
20#16813618—21no—
21#16813594—22no—
22#16813593—21no—
23#16813593C2212yes—
24#16813593C2212yescomplemented
25#16813590—22no—
26C4×Dic226C2211yescharacteristic, complemented
27C4×D452C2211yescharacteristic
28C4×D452C2211yescharacteristic
29D4×Dic113—21no—
30#16813554C2211yescharacteristic, complemented
31C4⋊#11744751C2211yescharacteristic, complemented
32#16813547C2211yescharacteristic, complemented
33#16813544C2211yescharacteristic
34#16813543C2212yescomplemented
35#16813543C2212yescomplemented
36#16813534C2212yescomplemented
37C2×D8×C113C2212yescomplemented
38C2×D8×C113C2212yescomplemented
39#127×C113C2211yescharacteristic
40#127×C113C2211yescharacteristic, complemented
41C4×C8×C113C2211yescharacteristic, complemented
42C23×D113—22no—
43#14268646—24no—
44#14268646—42no—
45C2×Dic226C2311yescharacteristic
46C2×D452D411yescharacteristic
47C2×D452C2311yescharacteristic
48C2×C22⋊D113—22no—
49C2×C22⋊D113—22no—
50C2×C22⋊D113—22no—
51C2×C22⋊D113—22no—
52D4×D113—24no—
53D4×D113—42no—
54D4×D113—44no—
55C22×Dic113—22no—
56C2×C4×D113—21no—
57C2×C4×D113C2312yes—
58#14268636D414yescomplemented
59#14268635—24no—
60#14268632—24no—
61#14268630—24no—
62C2×#11744751C2312yes—
63C4⋊Dic113D411yescharacteristic
64C4⋊Dic113C2311yescharacteristic
65C4⋊Dic113C2311yescharacteristic
66#14268626—21no—
67C22⋊Dic113—22no—
68C22⋊Dic113—22no—
69#14268624—21no—
70#14268624D412yes—
71C4×Dic113—21no—
72C2×D4×C113—21no—
73C2×D4×C113—21no—
74C2×D4×C113—21no—
75C2×D4×C113—21no—
76C2×D4×C113C2312yes—
77C2×D4×C113C2312yes—
78D8×C113—24no—
79D8×C113—42no—
80C2×C8×C113C2311yescharacteristic
81C2×C8×C113C2311yescharacteristic
82C42×C113C2311yescharacteristic
83C23×C113—22no—
84C23×C113—22no—
85C22×D113—21no—
86C22×D113—44no—
87C22×D113—44no—
88C22×D113—82no—
89C22×D113C2×D412yes—
90D4×C113—22no—
91D4×C113—22no—
92D4×C113—22no—
93D4×C113—22no—
94D4×C113—42no—
95D4×C113—42no—
96D4×C113—81no—
97D4×C113—81no—
98C2×C4×C113C2×D411yescharacteristic
99C2×C4×C113C2×D411yescharacteristic
100C2×C4×C113C2411yescharacteristic
101C22⋊D113—44no—
102C22⋊D113—44no—
103C22⋊D113—44no—
104C22⋊D113—82no—
105C2×Dic113—21no—
106C2×Dic113—21no—
107C2×Dic113—21no—
108C2×Dic113—44no—
109C2×Dic113C2×D412yes—
110D452—22no—
111D452—21no—
112D452C2×D412yes—
113C4×D113—22no—
114C4×D113—22no—
115C4×D113—42no—
116Dic226—21no—
117Dic226C2×D412yescomplemented
118C8×C113—21no—
119C8×C113C2×D412yescomplemented
120#11744751—22no—
121C22×C113—42no—
122C22×C113—42no—
123C22×C113—42no—
124C22×C113—42no—
125C22×C113—42no—
126C22×C113—42no—
127C22×C113C22×D411yescharacteristic
128C2×D113—22no—
129C2×D113—22no—
130C2×D113—22no—
131C2×D113—22no—
132C2×D113—41no—
133C2×D113—41no—
134C2×D113—88no—
135C4×C113—21no—
136C4×C113AGL(1,ℤ/8)12yescomplemented
137C4×C113C22×D411yescharacteristic
138C4×C113H(5,𝔽2)11yescharacteristic
139Dic113—22no—
140Dic113—22no—
141Dic113—41no—
142Dic113—41no—
143C2×C113—82no—
144C2×C113—82no—
145C2×C113D4211yescharacteristic
146C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C113#57611yescharacteristic
148D113—42no—
149D113—42no—
150#3274—1131nomaximal, Sylow
151C113#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1132no—
153#550—1131no—
154D42—1131no—
155#493D11311yescharacteristic, complemented
156#469—1132no—
157#460—1131no—
158#459—1131no—
159#449—1132no—
160#447—1132no—
161C4×SD16—1131no—
162#436—1131no—
163C2×#41—1132no—
164C22×D4—1132no—
165C22×D4—2261no—
166AGL(1,ℤ/8)—2268no—
167C2×SD16—1132no—
168C2×SD16—1131no—
169C2×D8—1132no—
170C2×D8C2×D11312yescomplemented
171C2×D8C2×D11312yescomplemented
172C2×M16—1132no—
173#127—2261no—
174#127C2×D11311yescharacteristic
175#127C2×D11311yescharacteristic, complemented
176#121—1132no—
177#121—1132no—
178#121—2262no—
179#121—2261no—
180#120—2262no—
181C4×Q8—1131no—
182C4×D4—1131no—
183C4×D4—1131no—
184C4×D4—2261no—
185#106—1131no—
186C4⋊C8—1131no—
187#103—1131no—
188#102—1132no—
189#102—1132no—
190#102—1131no—
191C22⋊C8—1132no—
192C4×C8C2×D11311yescharacteristic, complemented
193C24—2262no—
194#41—4522no—
195#41—2264no—
196C2×Q8—1131no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1131no—
202C2×D4—1131no—
203C2×D4—4524no—
204C2×D4—4522no—
205C2×D4—2262no—
206C2×D4—2264no—
207C2×D4—2262no—
208C2×D4—2262no—
209C2×D4—2262no—
210C2×D4C22×D11312yes—
211C2×D4C22×D11312yes—
212C22×C4—1132no—
213C22×C4—2262no—
214C22×C4—2261no—
215SD16—1134no—
216SD16—2264no—
217D8—24no—
218D8—42no—
219D8—2264no—
220M16—2264no—
221C2×C8—1132no—
222C2×C8C22×D11311yescharacteristic
223C2×C8C22×D11311yescharacteristic
224C4⋊C4—1131no—
225C4⋊C4—1131no—
226C4⋊C4—1131no—
227C4⋊C4—2261no—
228C22⋊C4—1132no—
229C22⋊C4—2262no—
230C22⋊C4—2262no—
231C22⋊C4—2261no—
232C42—2261no—
233C42C22×D11311yescharacteristic
234C23—22no—
235C23—22no—
236C23—1132no—
237C23—9042no—
238C23—4524no—
239C23—4524no—
240C23—2261no—
241Q8—1132no—
242Q8—2261no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4—42no—
248D4—42no—
249D4—81no—
250D4—81no—
251D4—1132no—
252D4—9042no—
253D4—4524no—
254D4—4524no—
255D4—4524no—
256D4—2262no—
257D4—2261no—
258C2×C4—1132no—
259C2×C4—4522no—
260C2×C4—4524no—
261C2×C4—2261no—
262C2×C4—2261no—
263C2×C4—2261no—
264C2×C4—2262no—
265C2×C4—2262no—
266C2×C4D4×D11311yescharacteristic
267C2×C4D4×D11311yescharacteristic
268C2×C4C23×D11311yescharacteristic
269C8—21no—
270C8—2262no—
271C8D4×D11312yescomplemented
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—9048no—
279C22—4521no—
280C22—4521no—
281C22—2262no—
282C22—2262no—
283C22—2262no—
284C22—2262no—
285C22C2×D4×D11311yescharacteristic
286C4—21no—
287C4—4521no—
288C4—4521no—
289C4—2262no—
290C4—2262no—
291C4#1681364612yescomplemented
292C4C2×D4×D11311yescharacteristic
293C4#1681366711yescharacteristic
294C2—82no—
295C2—82no—
296C2—4522no—
297C2—4522no—
298C2#2029923311yescharacteristic
299C2C2×#1681364611yescharacteristic
300C2#2029939911yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#493×C113C2
socleC22×C113C22×D4
hypercenter#493D113
hypocenterC113#3274
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#16813646C2semidirect
2. #20299268C2semidirect
3. #20299233C2semidirect
4. #20298848C2semidirect
5. #20298795C2semidirect
6. #20298790C2semidirect
7. #20298740C2semidirect
8. #20298730C2semidirect
9. C4×#14268636C2semidirect
10. #20298677C2semidirect
11. #493×C113C2semidirect
12. C2×#14268646C22semidirect
17. C2×#14268635C22semidirect
19. C2×#14268630C22semidirect
24. #16813593C22semidirect
26. C4×Dic226C22semidirect
30. #16813554C22semidirect
31. C4⋊#11744751C22semidirect
32. #16813547C22semidirect
34. #16813543C22semidirect
35. #16813543C22semidirect
36. #16813534C22semidirect
37. C2×D8×C113C22semidirect
38. C2×D8×C113C22semidirect
40. #127×C113C22semidirect
41. C4×C8×C113C22semidirect
58. #14268636D4semidirect
117. Dic226C2×D4semidirect
119. C8×C113C2×D4semidirect
136. C4×C113AGL(1,ℤ/8)semidirect
151. C113#3274semidirect
155. #493D113semidirect
170. C2×D8C2×D113semidirect
171. C2×D8C2×D113semidirect
175. #127C2×D113semidirect
192. C4×C8C2×D113semidirect
271. C8D4×D113semidirect
291. C4#16813646semidirect

Maximal subgroups

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

subgroupindex
C2×#168136462
#202992682
#202992332
#202988482
#202987952
#202987902
#202987402
#202987302
C4×#142686362
#202986772
#493×C1132
#3274113