Finite Group Databasebeta

← #28400668

Subgroups — #28400668

Order 14464.

444 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 123 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#20299150C211yesmaximal, characteristic, complemented
2C2×D4×Dic113C211yesmaximal, characteristic
3#20298934C211yesmaximal, characteristic, complemented
4#20298931C211yesmaximal, characteristic, complemented
5#20298876C212yesmaximal, complemented
6#20298801C212yesmaximal, complemented
7#20298749C211yesmaximal, characteristic
8#20298699C212yesmaximal, complemented
9#20298695C212yesmaximal, complemented
10#20298652C211yesmaximal, characteristic, complemented
11#450×C113C211yesmaximal, characteristic
12C23×Dic113—21no—
13C2×#14268635C2212yes—
14#16813621C2211yescharacteristic, complemented
15#16813620C2211yescharacteristic, complemented
16#16813612—21no—
17#16813601C2211yescharacteristic
18#16813599—21no—
19Q8×Dic113C2211yescharacteristic, complemented
20#16813587—21no—
21#16813587C2211yescharacteristic, complemented
22D4×Dic113—21no—
23D4×Dic113—21no—
24D4×Dic113C2212yes—
25D4×Dic113C2212yes—
26#16813578—21no—
27C2×C4⋊Dic113C2211yescharacteristic
28C2×C22⋊Dic113—21no—
29C2×C4×Dic113C2211yescharacteristic
30#16813556C2212yescomplemented
31#16813555C2212yescomplemented
32#16813551C2212yescomplemented
33#16813547C2212yescomplemented
34#16813546C2212yescomplemented
35#16813543C2212yes—
36C22⋊#11744751C2211yescharacteristic
37#16813532C2212yescomplemented
38C22×D4×C113C2211yescharacteristic
39C2×SD16×C113C2212yes—
40C22⋊Q8×C113C2211yescharacteristic
41#102×C113C2212yes—
42C22⋊C8×C113C2211yescharacteristic
43C2×Dic226—21no—
44C2×Dic226C2311yescharacteristic
45C22×Dic113—21no—
46C22×Dic113—22no—
47C22×Dic113—22no—
48C22×Dic113—44no—
49C22×Dic113D411yescharacteristic
50C22×Dic113D411yescharacteristic
51#14268635—24no—
52C2×#11744751C2312yes—
53C4⋊Dic113—21no—
54C4⋊Dic113—21no—
55C4⋊Dic113C2312yes—
56C4⋊Dic113C2311yescharacteristic
57#14268626—21no—
58#14268626—21no—
59#14268626—21no—
60#14268626—21no—
61#14268626—21no—
62C22⋊Dic113—21no—
63C22⋊Dic113—22no—
64C22⋊Dic113—22no—
65C4×Dic113—21no—
66C4×Dic113—21no—
67C4×Dic113D411yescharacteristic
68C4×Dic113D411yescharacteristic
69C4×Dic113C2311yescharacteristic
70C4×Dic113C2311yescharacteristic
71C24×C113—21no—
72C2×Q8×C113C2311yescharacteristic
73C2×D4×C113—24no—
74C2×D4×C113—21no—
75C2×D4×C113C2312yes—
76C22×C4×C113C2311yescharacteristic
77SD16×C113—24no—
78C2×C8×C113C2312yes—
79C4⋊C4×C113—21no—
80C4⋊C4×C113C2311yescharacteristic
81C22⋊C4×C113—21no—
82C23×C113—24no—
83C23×C113—22no—
84C23×C113—42no—
85C23×C113—42no—
86C23×C113C2×D411yescharacteristic
87Q8×C113—22no—
88D4×C113—24no—
89D4×C113—22no—
90D4×C113#4114yes—
91C2×C4×C113—22no—
92C2×C4×C113—21no—
93C2×C4×C113—21no—
94C2×C4×C113C2×D411yescharacteristic
95C2×C4×C113C2411yescharacteristic
96C2×Dic113—22no—
97C2×Dic113—22no—
98C2×Dic113—21no—
99C2×Dic113—21no—
100C2×Dic113—22no—
101C2×Dic113—44no—
102C2×Dic113—42no—
103C2×Dic113—44no—
104C2×Dic113C2×D411yescharacteristic
105C2×Dic113C2×D411yescharacteristic
106C2×Dic113C2×D411yescharacteristic
107C2×Dic113C2×D411yescharacteristic
108Dic226—22no—
109Dic226—42no—
110C8×C113—22no—
111#11744751—22no—
112C22×C113—21no—
113C22×C113—21no—
114C22×C113—24no—
115C22×C113—48no—
116C22×C113—42no—
117C22×C113—42no—
118C22×C113#13712yes—
119C22×C113C22×D411yescharacteristic
120C4×C113—21no—
121C4×C113—41no—
122C4×C113—41no—
123C4×C113C2×#4112yes—
124Dic113—21no—
125Dic113—21no—
126Dic113—21no—
127Dic113—41no—
128Dic113—41no—
129Dic113—42no—
130Dic113AGL(1,ℤ/8)12yes—
131C2×C113—22no—
132C2×C113—44no—
133C2×C113#52511yescharacteristic
134C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C113C2×#13711yescharacteristic
136#3110—1131nomaximal, Sylow
137C113#311011yescharacteristic, complemented, Sylow
138#533—1131no—
139C2×C4×D4—1131no—
140#484—1131no—
141#484—1131no—
142#474—1132no—
143#461—1132no—
144#450—1131no—
145#450D11311yescharacteristic
146#440—1132no—
147#440—1132no—
148#432—1131no—
149C22×D4C2×D11311yescharacteristic
150C23×C4—2261no—
151C2×SD16—1132no—
152C2×SD16C2×D11312yes—
153C4⋊Q8—1131no—
154C4⋊Q8—1131no—
155#125—2261no—
156C22⋊Q8—1131no—
157C22⋊Q8—2261no—
158C22⋊Q8C2×D11311yescharacteristic
159C4×Q8—1131no—
160C4×Q8—1131no—
161C4×Q8—2261no—
162C4×D4—1132no—
163C4×D4—1132no—
164C4×D4—2261no—
165C4×D4—2261no—
166#117—2261no—
167C2×C4⋊C4—1131no—
168C2×C22⋊C4—2261no—
169C2×C42—1131no—
170#107—1132no—
171#107—1132no—
172C4⋊C8—1132no—
173#103—1132no—
174#103—1132no—
175#102—1132no—
176#102C2×D11312yes—
177C22⋊C8—1131no—
178C22⋊C8C2×D11311yescharacteristic
179#97—1132no—
180C24—21no—
181C2×Q8—1131no—
182C2×Q8—2261no—
183C2×Q8C22×D11311yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D11312yes—
187C22×C4—1131no—
188C22×C4—1131no—
189C22×C4—4524no—
190C22×C4—2262no—
191C22×C4—2262no—
192C22×C4—2261no—
193C22×C4C22×D11311yescharacteristic
194SD16—24no—
195SD16—2264no—
196C2×C8—1132no—
197C2×C8C22×D11312yes—
198C4⋊C4—21no—
199C4⋊C4—1131no—
200C4⋊C4—1132no—
201C4⋊C4—2261no—
202C4⋊C4—2261no—
203C4⋊C4—2261no—
204C4⋊C4—2261no—
205C4⋊C4—2261no—
206C4⋊C4—2261no—
207C4⋊C4—2261no—
208C4⋊C4C22×D11311yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—2262no—
211C22⋊C4—2262no—
212C22⋊C4—2261no—
213C42—1131no—
214C42—1131no—
215C42—1131no—
216C42—1131no—
217C42—2261no—
218C42—2261no—
219C23—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D11311yescharacteristic
224Q8—22no—
225Q8—4522no—
226Q8—2262no—
227D4—24no—
228D4—22no—
229D4#1426864614yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—1131no—
234C2×C4—1131no—
235C2×C4—1131no—
236C2×C4—1131no—
237C2×C4—4524no—
238C2×C4—4522no—
239C2×C4—4524no—
240C2×C4—2262no—
241C2×C4—2261no—
242C2×C4—2261no—
243C2×C4—2262no—
244C2×C4—2262no—
245C2×C4D4×D11311yescharacteristic
246C2×C4C23×D11311yescharacteristic
247C8—22no—
248C8—2262no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22#1681365112yes—
256C22C2×D4×D11311yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—1132no—
261C4—4521no—
262C4—4522no—
263C4—4521no—
264C4—2261no—
265C4—2261no—
266C4—2261no—
267C4C2×#1426864612yes—
268C2—22no—
269C2—44no—
270C2#2029910411yescharacteristic
271C2AGL(1,ℤ/8)×D11311yescharacteristic
272C2C2×#1681365111yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#450×C113C2
socleC22×C113C22×D4
prime radicalC22×D4×C113C22
hypercenter#450D113
hypocenterC113#3110
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20299150C2semidirect
3. #20298934C2semidirect
4. #20298931C2semidirect
5. #20298876C2semidirect
6. #20298801C2semidirect
8. #20298699C2semidirect
9. #20298695C2semidirect
10. #20298652C2semidirect
14. #16813621C22semidirect
15. #16813620C22semidirect
19. Q8×Dic113C22semidirect
21. #16813587C22semidirect
30. #16813556C22semidirect
31. #16813555C22semidirect
32. #16813551C22semidirect
33. #16813547C22semidirect
34. #16813546C22semidirect
37. #16813532C22semidirect
137. C113#3110semidirect

Maximal subgroups

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

subgroupindex
#202991502
C2×D4×Dic1132
#202989342
#202989312
#202988762
#202988012
#202987492
#202986992
#202986952
#202986522
#450×C1132
#3110113