Finite Group Databasebeta

← #28401572

Subgroups — #28401572

Order 14464.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#20299153C211yesmaximal, characteristic, complemented
2#124×D113C211yesmaximal, characteristic
3#20298990C211yesmaximal, characteristic, complemented
4#20298958C211yesmaximal, characteristic, complemented
5#20298957C211yesmaximal, characteristic, complemented
6#20298822C212yesmaximal, complemented
7#20298818C212yesmaximal, complemented
8#20298752C212yesmaximal, complemented
9#20298747C212yesmaximal
10#20298653C211yesmaximal, characteristic, complemented
11#497×C113C211yesmaximal, characteristic, complemented
12C2×Q8×D113C2211yescharacteristic
13C2×D4×D113C2211yescharacteristic
14C2×#14268637C2212yescomplemented
15C2×#14268634C2212yes—
16#16813622C2211yescharacteristic, complemented
17#16813613—21no—
18#16813611C2211yescharacteristic
19#16813610C2211yescharacteristic
20#16813609C2211yescharacteristic
21#16813608—21no—
22#16813608—21no—
23#16813602—21no—
24#16813598C2212yes—
25Q8×Dic113C2212yescomplemented
26#16813587—21no—
27#16813580—21no—
28C22⋊C4×D113—21no—
29C22⋊C4×D113—21no—
30C42×D113C2211yescharacteristic
31#16813551C2212yescomplemented
32#16813547C2212yescomplemented
33#16813546C2212yescomplemented
34#16813544C2212yes—
35#16813543C2212yes—
36#16813534C2212yes—
37#16813533C2211yescharacteristic, complemented
38C2×Q16×C113C2212yescomplemented
39C2×SD16×C113C2212yes—
40C4⋊Q8×C113C2211yescharacteristic, complemented
41#124×C113C2211yescharacteristic
42#97×C113C2211yescharacteristic, complemented
43C23×D113—21no—
44C2×Dic226—21no—
45C2×Dic226—21no—
46C2×Dic226C2311yescharacteristic
47Q8×D113—42no—
48C2×D452C2311yescharacteristic
49C2×C22⋊D113—21no—
50D4×D113—42no—
51C2×C4×D113—21no—
52C2×C4×D113D411yescharacteristic
53C2×C4×D113D411yescharacteristic
54C2×C4×D113C2311yescharacteristic
55#14268637—42no—
56#14268634—42no—
57C2×#11744751C2312yes—
58C4⋊Dic113—21no—
59C4⋊Dic113—22no—
60C4⋊Dic113C2312yes—
61#14268626—22no—
62C22⋊Dic113—21no—
63C22⋊Dic113—21no—
64#14268624—22no—
65#14268624—21no—
66#14268624—21no—
67#14268624—21no—
68#14268624—21no—
69C4×Dic113—22no—
70C4×Dic113D411yescharacteristic
71C4×Dic113D411yescharacteristic
72C4×Dic113C2311yescharacteristic
73C2×Q8×C113C2312yes—
74C2×Q8×C113C2311yescharacteristic
75C2×D4×C113C2311yescharacteristic
76Q16×C113—42no—
77SD16×C113—42no—
78C2×C8×C113C2312yes—
79C4⋊C4×C113—21no—
80C4⋊C4×C113—21no—
81C22⋊C4×C113—21no—
82C22⋊C4×C113—21no—
83C42×C113C2311yescharacteristic
84C23×C113—21no—
85C22×D113—21no—
86C22×D113—42no—
87C22×D113—42no—
88C22×D113—42no—
89C22×D113C2×D411yescharacteristic
90Q8×C113—22no—
91Q8×C113—22no—
92Q8×C113—22no—
93D4×C113—22no—
94C2×C4×C113—21no—
95C2×C4×C113—22no—
96C2×C4×C113C2×D411yescharacteristic
97C2×C4×C113C2×D411yescharacteristic
98C2×C4×C113C2411yescharacteristic
99C22⋊D113—42no—
100C2×Dic113—22no—
101C2×Dic113—21no—
102C2×Dic113C2×D411yescharacteristic
103C2×Dic113C2×D411yescharacteristic
104C2×Dic113C2×D411yescharacteristic
105D452—22no—
106C4×D113—22no—
107C4×D113—22no—
108C4×D113—41no—
109C4×D113—42no—
110Dic226—22no—
111Dic226—42no—
112Dic226—42no—
113C8×C113—22no—
114#11744751—22no—
115C22×C113—41no—
116C22×C113—41no—
117C22×C113—41no—
118C22×C113C22×D411yescharacteristic
119C2×D113—21no—
120C2×D113—21no—
121C2×D113—41no—
122C2×D113—41no—
123C2×D113—41no—
124C2×D113—84no—
125C2×D113#13712yes—
126C4×C113—21no—
127C4×C113—21no—
128C4×C113—41no—
129C4×C113—42no—
130C4×C113H(5,𝔽2)12yes—
131Dic113—21no—
132Dic113—21no—
133Dic113—21no—
134Dic113—41no—
135Dic113—42no—
136C2×C113—81no—
137C2×C113#53511yescharacteristic
138C2×C113C2×#13711yescharacteristic
139C2×C113#57511yescharacteristic
140D113—22no—
141D113—81no—
142#3215—1131nomaximal, Sylow
143C113#321511yescharacteristic, complemented, Sylow
144#533—1131no—
145C2×#124—1131no—
146#497—1131no—
147#497D11311yescharacteristic, complemented
148#489—1131no—
149#489—1131no—
150#464—1132no—
151#464—1132no—
152#451—1132no—
153#450—1132no—
154#432—1131no—
155C22×Q8—1131no—
156C22×D4—1131no—
157C2×Q16—1132no—
158C2×Q16C2×D11312yescomplemented
159C2×SD16—1132no—
160C2×SD16C2×D11312yes—
161C4⋊Q8—1131no—
162C4⋊Q8C2×D11311yescharacteristic, complemented
163#125—2261no—
164#124—1131no—
165#124—1131no—
166#124—1131no—
167#124—2261no—
168#124—2261no—
169#124C2×D11311yescharacteristic
170C22⋊Q8—1132no—
171C22⋊Q8—2261no—
172C4×Q8—1132no—
173C4×Q8—2261no—
174#117—2261no—
175C2×C22⋊C4—2261no—
176C2×C22⋊C4—2261no—
177C2×C42—1131no—
178C4⋊C8—1132no—
179#103—1132no—
180#103—1132no—
181#102—1132no—
182#102—1132no—
183C22⋊C8—1132no—
184#97—1131no—
185#97C2×D11311yescharacteristic, complemented
186C24—2261no—
187C2×Q8—1131no—
188C2×Q8—4522no—
189C2×Q8—2261no—
190C2×Q8—2261no—
191C2×Q8C22×D11311yescharacteristic
192C2×Q8C22×D11312yes—
193C2×D4—1131no—
194C2×D4—4522no—
195C2×D4—2261no—
196C2×D4C22×D11311yescharacteristic
197C22×C4—1131no—
198C22×C4—1131no—
199C22×C4—1131no—
200C22×C4—2261no—
201Q16—42no—
202Q16—4522no—
203SD16—42no—
204SD16—4522no—
205C2×C8—1132no—
206C2×C8C22×D11312yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—1132no—
210C4⋊C4—2262no—
211C4⋊C4—2262no—
212C4⋊C4—2261no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—2261no—
216C22⋊C4—2261no—
217C22⋊C4—2262no—
218C22⋊C4—2261no—
219C22⋊C4—2261no—
220C22⋊C4—2261no—
221C22⋊C4—2261no—
222C42—1131no—
223C42—1131no—
224C42—1131no—
225C42—2262no—
226C42C22×D11311yescharacteristic
227C23—21no—
228C23—1131no—
229C23—4522no—
230C23—4522no—
231C23—4522no—
232C23—2261no—
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—4522no—
237Q8—4522no—
238Q8—2262no—
239D4—22no—
240D4—4522no—
241D4—2262no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—1131no—
245C2×C4—1131no—
246C2×C4—1131no—
247C2×C4—4521no—
248C2×C4—4522no—
249C2×C4—2262no—
250C2×C4—2262no—
251C2×C4—2261no—
252C2×C4—2262no—
253C2×C4D4×D11311yescharacteristic
254C2×C4D4×D11311yescharacteristic
255C2×C4C23×D11311yescharacteristic
256C8—22no—
257C8—2262no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—1132no—
262C22—9044no—
263C22—4521no—
264C22—4521no—
265C22—4521no—
266C22—2261no—
267C22—2261no—
268C22C2×D4×D11311yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—4522no—
274C4—4521no—
275C4—2261no—
276C4—2261no—
277C4—2261no—
278C4#1681366712yes—
279C2—81no—
280C2—9041no—
281C2—2262no—
282C2#2029916111yescharacteristic
283C2#137×D11311yescharacteristic
284C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#497×C113C2
socleC22×C113C22×D4
prime radicalC2×D4×D113C22
hypercenter#497D113
hypocenterC113#3215
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #20299153C2semidirect
3. #20298990C2semidirect
4. #20298958C2semidirect
5. #20298957C2semidirect
6. #20298822C2semidirect
7. #20298818C2semidirect
8. #20298752C2semidirect
10. #20298653C2semidirect
11. #497×C113C2semidirect
14. C2×#14268637C22semidirect
16. #16813622C22semidirect
25. Q8×Dic113C22semidirect
31. #16813551C22semidirect
32. #16813547C22semidirect
33. #16813546C22semidirect
37. #16813533C22semidirect
38. C2×Q16×C113C22semidirect
40. C4⋊Q8×C113C22semidirect
42. #97×C113C22semidirect
143. C113#3215semidirect
147. #497D113semidirect
158. C2×Q16C2×D113semidirect
162. C4⋊Q8C2×D113semidirect
185. #97C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202991532
#124×D1132
#202989902
#202989582
#202989572
#202988222
#202988182
#202987522
#202987472
#202986532
#497×C1132
#3215113