Finite Group Databasebeta

← #28401576

Subgroups — #28401576

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×#16813611C211yesmaximal, characteristic
3#20298991C212yesmaximal, complemented
4#20298959C212yesmaximal, complemented
5#20298823C212yesmaximal, complemented
6#20298822C212yesmaximal, complemented
7#20298754C211yesmaximal, characteristic, complemented
8#20298751C211yesmaximal, characteristic
9#20298748C211yesmaximal, characteristic
10#20298652C211yesmaximal, characteristic, complemented
11#451×C113C211yesmaximal, characteristic, complemented
12C22×D452C2211yescharacteristic
13C2×#14268637C2212yescomplemented
14C2×#14268636C2212yes—
15C2×#14268634C2212yes—
16#16813621C2211yescharacteristic, complemented
17#16813620C2211yescharacteristic, complemented
18#16813612—21no—
19#16813611—21no—
20#16813611—21no—
21#16813611C2212yes—
22#16813611C2212yes—
23#16813601C2211yescharacteristic
24#16813599—21no—
25Q8×Dic113C2211yescharacteristic, complemented
26#16813587—21no—
27#16813587C2211yescharacteristic, complemented
28#16813578—21no—
29C2×#14268624—21no—
30C2×#14268624—21no—
31C2×C4×Dic113C2211yescharacteristic
32#16813551C2212yescomplemented
33#16813546C2212yescomplemented
34#16813544C2212yes—
35#16813542C2212yes—
36C22⋊#11744751C2211yescharacteristic
37#16813532C2212yescomplemented
38C22×Q8×C113C2211yescharacteristic
39C2×Q16×C113C2212yescomplemented
40C22⋊Q8×C113C2211yescharacteristic
41#103×C113C2212yescomplemented
42C22⋊C8×C113C2211yescharacteristic
43C23×D113—21no—
44C2×Dic226—21no—
45C2×Dic226C2311yescharacteristic
46C2×D452—21no—
47C2×D452—42no—
48C2×D452C2312yes—
49C22×Dic113D411yescharacteristic
50C22×Dic113D411yescharacteristic
51#14268637—42no—
52#14268636—42no—
53#14268634—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—
63#14268624—22no—
64#14268624—22no—
65#14268624—22no—
66#14268624—22no—
67C4×Dic113—21no—
68C4×Dic113—21no—
69C4×Dic113D411yescharacteristic
70C4×Dic113D411yescharacteristic
71C4×Dic113C2311yescharacteristic
72C4×Dic113C2311yescharacteristic
73C2×Q8×C113—21no—
74C2×Q8×C113—42no—
75C2×Q8×C113C2311yescharacteristic
76C2×Q8×C113C2312yes—
77C22×C4×C113—21no—
78C22×C4×C113C2311yescharacteristic
79Q16×C113—42no—
80C2×C8×C113C2312yes—
81C4⋊C4×C113—21no—
82C4⋊C4×C113C2311yescharacteristic
83C22⋊C4×C113—21no—
84C23×C113C2×D411yescharacteristic
85C22×D113—22no—
86C22×D113—42no—
87C22×D113—42no—
88C22×D113—42no—
89Q8×C113—22no—
90Q8×C113—22no—
91Q8×C113—22no—
92Q8×C113—42no—
93C2×C4×C113—22no—
94C2×C4×C113—21no—
95C2×C4×C113—21no—
96C2×C4×C113—22no—
97C2×C4×C113—42no—
98C2×C4×C113C2×D411yescharacteristic
99C2×C4×C113C2411yescharacteristic
100C2×Dic113—22no—
101C2×Dic113—21no—
102C2×Dic113—21no—
103C2×Dic113—41no—
104C2×Dic113C2×D411yescharacteristic
105C2×Dic113C2×D411yescharacteristic
106C2×Dic113C2×D411yescharacteristic
107C2×Dic113C2×D411yescharacteristic
108D452—22no—
109D452—22no—
110D452—42no—
111Dic226—22no—
112Dic226—42no—
113C8×C113—22no—
114#11744751—22no—
115C22×C113—21no—
116C22×C113—21no—
117C22×C113#13712yes—
118C22×C113C22×D411yescharacteristic
119C2×D113—42no—
120C2×D113—42no—
121C2×D113—42no—
122C2×D113—84no—
123C4×C113—21no—
124C4×C113—42no—
125C4×C113—41no—
126C4×C113—41no—
127C4×C113H(5,𝔽2)12yes—
128Dic113—21no—
129Dic113—21no—
130Dic113—21no—
131Dic113—21no—
132Dic113—41no—
133Dic113—41no—
134C2×C113—22no—
135C2×C113#53511yescharacteristic
136C2×C113C2×#13711yescharacteristic
137C2×C113#57511yescharacteristic
138D113—82no—
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#451D11311yescharacteristic, complemented
149#450—1131no—
150#450—1131no—
151#432—1131no—
152C22×Q8C2×D11311yescharacteristic
153C22×D4—1131no—
154C2×Q16—1132no—
155C2×Q16C2×D11312yescomplemented
156C2×SD16—1132no—
157C2×SD16—1132no—
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#103C2×D11312yescomplemented
178#102—1132no—
179#102—1132no—
180C22⋊C8—1131no—
181C22⋊C8C2×D11311yescharacteristic
182#97—1132no—
183C24—2261no—
184C2×Q8—21no—
185C2×Q8—42no—
186C2×Q8—1131no—
187C2×Q8—2261no—
188C2×Q8C22×D11312yes—
189C2×Q8C22×D11311yescharacteristic
190C2×D4—1132no—
191C2×D4—4522no—
192C2×D4—2261no—
193C22×C4—21no—
194C22×C4—1131no—
195C22×C4—1131no—
196C22×C4C22×D11311yescharacteristic
197Q16—42no—
198Q16—4522no—
199SD16—4522no—
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—4522no—
225C23—4522no—
226C23—4522no—
227C23—2262no—
228C23D4×D11311yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—22no—
232Q8—42no—
233Q8—4522no—
234Q8—2262no—
235D4—4522no—
236D4—2262no—
237D4—2262no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—1131no—
244C2×C4—1131no—
245C2×C4—1131no—
246C2×C4—1131no—
247C2×C4—4521no—
248C2×C4—2261no—
249C2×C4—2261no—
250C2×C4—2262no—
251C2×C4D4×D11311yescharacteristic
252C2×C4C23×D11311yescharacteristic
253C8—22no—
254C8—2262no—
255C22—21no—
256C22—21no—
257C22—9044no—
258C22—4522no—
259C22—4522no—
260C22—4522no—
261C22#1681365412yes—
262C22C2×D4×D11311yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—4521no—
268C4—4521no—
269C4—2261no—
270C4—2261no—
271C4—2261no—
272C4—2261no—
273C4#1681366712yes—
274C2—22no—
275C2—9042no—
276C2#2029916311yescharacteristic
277C2C2×#1681365411yescharacteristic
278C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#451×C113C2
socleC22×C113C22×D4
prime radicalC22×D452C22
hypercenter#451D113
hypocenterC113#3215
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #20299150C2semidirect
3. #20298991C2semidirect
4. #20298959C2semidirect
5. #20298823C2semidirect
6. #20298822C2semidirect
7. #20298754C2semidirect
10. #20298652C2semidirect
11. #451×C113C2semidirect
13. C2×#14268637C22semidirect
16. #16813621C22semidirect
17. #16813620C22semidirect
25. Q8×Dic113C22semidirect
27. #16813587C22semidirect
32. #16813551C22semidirect
33. #16813546C22semidirect
37. #16813532C22semidirect
39. C2×Q16×C113C22semidirect
41. #103×C113C22semidirect
140. C113#3215semidirect
148. #451D113semidirect
155. C2×Q16C2×D113semidirect
177. #103C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202991502
C2×#168136112
#202989912
#202989592
#202988232
#202988222
#202987542
#202987512
#202987482
#202986522
#451×C1132
#3215113