Finite Group Databasebeta

← #28401574

Subgroups — #28401574

Order 14464.

400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16813609C211yesmaximal, characteristic
3#20298989C212yesmaximal, complemented
4#20298956C212yesmaximal, complemented
5#20298819C212yesmaximal, complemented
6#20298818C212yesmaximal, complemented
7#20298756C211yesmaximal, characteristic, complemented
8#20298753C211yesmaximal, characteristic, complemented
9#20298749C211yesmaximal, characteristic
10#20298652C211yesmaximal, characteristic, complemented
11#450×C113C211yesmaximal, characteristic
12C22×Dic226C2211yescharacteristic
13C2×Dic452C2212yescomplemented
14C2×#14268637C2212yescomplemented
15C2×#14268635C2212yes—
16#16813621C2211yescharacteristic, complemented
17#16813620C2211yescharacteristic, complemented
18#16813612—21no—
19#16813609—21no—
20#16813609—21no—
21#16813609C2212yes—
22#16813609C2212yes—
23#16813601C2211yescharacteristic
24#16813599—21no—
25Q8×Dic113C2211yescharacteristic, complemented
26#16813587—21no—
27#16813587C2211yescharacteristic, complemented
28#16813578—21no—
29C2×C22⋊Dic113—21no—
30C2×C22⋊Dic113—21no—
31C2×C4×Dic113C2211yescharacteristic
32#16813551C2212yescomplemented
33#16813547C2212yescomplemented
34#16813545C2212yescomplemented
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×Dic226—21no—
45C2×Dic226—42no—
46C2×Dic226C2311yescharacteristic
47C2×Dic226C2312yes—
48C22×Dic113—21no—
49C22×Dic113D411yescharacteristic
50C22×Dic113D411yescharacteristic
51Dic452—42no—
52#14268637—42no—
53#14268635—42no—
54C2×#11744751C2312yes—
55C4⋊Dic113—21no—
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—
65C22⋊Dic113—22no—
66C22⋊Dic113—22no—
67C4×Dic113—21no—
68C4×Dic113—21no—
69C4×Dic113D411yescharacteristic
70C4×Dic113D411yescharacteristic
71C4×Dic113C2311yescharacteristic
72C4×Dic113C2311yescharacteristic
73C24×C113—21no—
74C2×Q8×C113C2311yescharacteristic
75C2×D4×C113—21no—
76C2×D4×C113—42no—
77C2×D4×C113C2312yes—
78C22×C4×C113C2311yescharacteristic
79SD16×C113—42no—
80C2×C8×C113C2312yes—
81C4⋊C4×C113—21no—
82C4⋊C4×C113C2311yescharacteristic
83C22⋊C4×C113—21no—
84C23×C113—22no—
85C23×C113—42no—
86C23×C113—42no—
87C23×C113—42no—
88C23×C113C2×D411yescharacteristic
89Q8×C113—22no—
90D4×C113—22no—
91D4×C113—22no—
92D4×C113—42no—
93C2×C4×C113—22no—
94C2×C4×C113—21no—
95C2×C4×C113—21no—
96C2×C4×C113C2×D411yescharacteristic
97C2×C4×C113C2411yescharacteristic
98C2×Dic113—22no—
99C2×Dic113—21no—
100C2×Dic113—21no—
101C2×Dic113—22no—
102C2×Dic113—41no—
103C2×Dic113—42no—
104C2×Dic113C2×D411yescharacteristic
105C2×Dic113C2×D411yescharacteristic
106C2×Dic113C2×D411yescharacteristic
107C2×Dic113C2×D411yescharacteristic
108Dic226—22no—
109Dic226—22no—
110Dic226—22no—
111Dic226—42no—
112Dic226—42no—
113C8×C113—22no—
114#11744751—22no—
115C22×C113—21no—
116C22×C113—21no—
117C22×C113—42no—
118C22×C113—42no—
119C22×C113—42no—
120C22×C113—84no—
121C22×C113#13712yes—
122C22×C113C22×D411yescharacteristic
123C4×C113—21no—
124C4×C113—41no—
125C4×C113—41no—
126C4×C113H(5,𝔽2)12yes—
127Dic113—21no—
128Dic113—21no—
129Dic113—21no—
130Dic113—21no—
131Dic113—42no—
132Dic113—41no—
133Dic113—41no—
134C2×C113—22no—
135C2×C113—82no—
136C2×C113#53511yescharacteristic
137C2×C113C2×#13711yescharacteristic
138C2×C113#57511yescharacteristic
139#3215—1131nomaximal, Sylow
140C113#321511yescharacteristic, complemented, Sylow
141#533—1131no—
142C2×#124—1131no—
143#497—1132no—
144#489—1132no—
145#464—1132no—
146#464—1132no—
147#451—1131no—
148#451—1131no—
149#450—1131no—
150#450D11311yescharacteristic
151#432—1131no—
152C22×Q8—1131no—
153C22×D4C2×D11311yescharacteristic
154C2×Q16—1132no—
155C2×Q16—1132no—
156C2×SD16—1132no—
157C2×SD16C2×D11312yes—
158C4⋊Q8—1131no—
159C4⋊Q8—1131no—
160#125—2261no—
161#124—1132no—
162#124—1132no—
163#124—2261no—
164#124—2261no—
165C22⋊Q8—1131no—
166C22⋊Q8—2261no—
167C22⋊Q8C2×D11311yescharacteristic
168C4×Q8—1131no—
169C4×Q8—1131no—
170C4×Q8—2261no—
171#117—2261no—
172C2×C22⋊C4—2261no—
173C2×C22⋊C4—2261no—
174C2×C42—1131no—
175C4⋊C8—1132no—
176#103—1132no—
177#103—1132no—
178#102—1132no—
179#102C2×D11312yes—
180C22⋊C8—1131no—
181C22⋊C8C2×D11311yescharacteristic
182#97—1132no—
183C24—21no—
184C2×Q8—1132no—
185C2×Q8—1131no—
186C2×Q8—4522no—
187C2×Q8—2261no—
188C2×Q8—2261no—
189C2×Q8C22×D11311yescharacteristic
190C2×D4—21no—
191C2×D4—42no—
192C2×D4C22×D11312yes—
193C22×C4—1131no—
194C22×C4—1131no—
195C22×C4—2261no—
196C22×C4C22×D11311yescharacteristic
197Q16—4522no—
198Q16—4522no—
199SD16—42no—
200SD16—4522no—
201C2×C8—1132no—
202C2×C8C22×D11312yes—
203C4⋊C4—21no—
204C4⋊C4—1131no—
205C4⋊C4—2261no—
206C4⋊C4—2261no—
207C4⋊C4—2261no—
208C4⋊C4—2261no—
209C4⋊C4—2261no—
210C4⋊C4—2261no—
211C4⋊C4C22×D11311yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—2262no—
214C22⋊C4—2262no—
215C22⋊C4—2262no—
216C22⋊C4—2262no—
217C22⋊C4—2261no—
218C42—1131no—
219C42—1131no—
220C42—1131no—
221C42—1131no—
222C42—2261no—
223C42—2261no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23D4×D11311yescharacteristic
229Q8—22no—
230Q8—4522no—
231Q8—4522no—
232Q8—2262no—
233Q8—2262no—
234Q8—2262no—
235D4—22no—
236D4—22no—
237D4—42no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—1131no—
242C2×C4—1131no—
243C2×C4—1131no—
244C2×C4—1131no—
245C2×C4—4522no—
246C2×C4—4521no—
247C2×C4—2261no—
248C2×C4—2261no—
249C2×C4—2262no—
250C2×C4—2262no—
251C2×C4D4×D11311yescharacteristic
252C2×C4C23×D11311yescharacteristic
253C8—22no—
254C8—2262no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—84no—
261C22#1681365112yes—
262C22C2×D4×D11311yescharacteristic
263C4—21no—
264C4—41no—
265C4—41no—
266C4—4521no—
267C4—4522no—
268C4—4521no—
269C4—2261no—
270C4—2261no—
271C4—2261no—
272C4—2261no—
273C4#1681366712yes—
274C2—22no—
275C2—82no—
276C2#2029916311yescharacteristic
277C2C2×#1681365111yescharacteristic
278C2#2029939411yescharacteristic
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#3215
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. #20298989C2semidirect
4. #20298956C2semidirect
5. #20298819C2semidirect
6. #20298818C2semidirect
7. #20298756C2semidirect
8. #20298753C2semidirect
10. #20298652C2semidirect
13. C2×Dic452C22semidirect
14. C2×#14268637C22semidirect
16. #16813621C22semidirect
17. #16813620C22semidirect
25. Q8×Dic113C22semidirect
27. #16813587C22semidirect
32. #16813551C22semidirect
33. #16813547C22semidirect
34. #16813545C22semidirect
37. #16813532C22semidirect
140. C113#3215semidirect

Maximal subgroups

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

subgroupindex
#202991502
C2×#168136092
#202989892
#202989562
#202988192
#202988182
#202987562
#202987532
#202987492
#202986522
#450×C1132
#3215113