Finite Group Databasebeta

← #28400520

Subgroups — #28400520

Order 14464.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16813598C211yesmaximal, characteristic
2#117×D113C211yesmaximal, characteristic
3#20298938C212yesmaximal, complemented
4#20298937C212yesmaximal, complemented
5#20298932C212yesmaximal, complemented
6#20298927C212yesmaximal, complemented
7#20298641C211yesmaximal, characteristic, complemented
8#20298640C211yesmaximal, characteristic, complemented
9#20298607C211yesmaximal, characteristic, complemented
10#20298561C211yesmaximal, characteristic
11#419×C113C211yesmaximal, characteristic, complemented
12C22×C4×D113C2211yescharacteristic
13C2×#14268631C2211yescharacteristic
14#16813598—21no—
15#16813598—21no—
16#16813598C2212yes—
17#16813598C2212yes—
18#16813581C2211yescharacteristic
19#16813580C2211yescharacteristic
20#16813580C2211yescharacteristic
21#16813579—21no—
22#16813578—21no—
23C2×C4⋊Dic113C2211yescharacteristic
24C2×#14268626—21no—
25C4⋊C4×D113C2211yescharacteristic
26C4⋊C4×D113C2211yescharacteristic
27C2×#14268624—21no—
28C22⋊C4×D113—21no—
29C42×D113—21no—
30#16813556C2212yescomplemented
31#16813555C2212yescomplemented
32#16813554C2212yescomplemented
33#16813553C2212yescomplemented
34#16813546C2212yescomplemented
35#16813546C2212yescomplemented
36#16813534C2212yes—
37#16813534C2212yes—
38C22×Q8×C113C2211yescharacteristic
39C2×M16×C113C2211yescharacteristic
40#117×C113C2211yescharacteristic
41#103×C113C2212yescomplemented
42#103×C113C2212yescomplemented
43C23×D113D411yescharacteristic
44C22×Dic113—21no—
45C22×Dic113D411yescharacteristic
46C2×C4×D113—22no—
47C2×C4×D113—22no—
48C2×C4×D113—21no—
49C2×C4×D113—21no—
50C2×C4×D113D411yescharacteristic
51C2×C4×D113D411yescharacteristic
52C2×C4×D113C2311yescharacteristic
53C2×C4×D113C2311yescharacteristic
54#14268631—22no—
55C2×#11744751C2312yes—
56C4⋊Dic113—21no—
57C4⋊Dic113C2311yescharacteristic
58C4⋊Dic113C2311yescharacteristic
59C4⋊Dic113C2312yes—
60#14268626—21no—
61#14268626—21no—
62#14268626—22no—
63#14268626—22no—
64C22⋊Dic113—21no—
65#14268624—21no—
66#14268624—21no—
67#14268624—22no—
68#14268624—22no—
69C4×Dic113—21no—
70C4×Dic113—21no—
71C4×Dic113—21no—
72C2×Q8×C113—21no—
73C2×Q8×C113—22no—
74C2×Q8×C113—22no—
75C2×Q8×C113C2312yes—
76C22×C4×C113—21no—
77C22×C4×C113C2311yescharacteristic
78M16×C113—22no—
79C2×C8×C113C2312yes—
80C4⋊C4×C113C2311yescharacteristic
81C4⋊C4×C113C2311yescharacteristic
82C22⋊C4×C113—21no—
83C42×C113—21no—
84C23×C113C2×D411yescharacteristic
85C22×D113—22no—
86C22×D113—22no—
87C22×D113—41no—
88C22×D113C2×D411yescharacteristic
89C22×D113C2×D411yescharacteristic
90Q8×C113—22no—
91Q8×C113—22no—
92Q8×C113—42no—
93C2×C4×C113—21no—
94C2×C4×C113—21no—
95C2×C4×C113—22no—
96C2×C4×C113—42no—
97C2×C4×C113C2×D411yescharacteristic
98C2×C4×C113#4112yes—
99C2×C4×C113#4112yes—
100C2×C4×C113C2411yescharacteristic
101C2×Dic113—22no—
102C2×Dic113—21no—
103C2×Dic113—21no—
104C2×Dic113—22no—
105C2×Dic113—42no—
106C2×Dic113C2×D411yescharacteristic
107C2×Dic113C2×D411yescharacteristic
108C4×D113—22no—
109C4×D113—22no—
110C4×D113—21no—
111C4×D113—21no—
112C4×D113—21no—
113C4×D113—21no—
114C4×D113—42no—
115C4×D113—42no—
116C8×C113—22no—
117#11744751—22no—
118C22×C113—21no—
119C22×C113—21no—
120C22×C113#12312yes—
121C22×C113C22×D411yescharacteristic
122C2×D113—21no—
123C2×D113—21no—
124C2×D113—21no—
125C2×D113—21no—
126C2×D113—21no—
127C2×D113—44no—
128C2×D113#13712yes—
129C4×C113—42no—
130C4×C113—41no—
131C4×C113—41no—
132C4×C113#12312yes—
133C4×C113C2×#4111yescharacteristic
134C4×C113C2×#4111yescharacteristic
135Dic113—21no—
136Dic113—21no—
137Dic113—41no—
138Dic113—41no—
139Dic113—42no—
140C2×C113—22no—
141C2×C113C2×#12311yescharacteristic
142C2×C113C2×#13711yescharacteristic
143C2×C113#57511yescharacteristic
144D113—22no—
145D113—41no—
146#3095—1131nomaximal, Sylow
147C113#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—1131no—
149C2×#117—1131no—
150#485—1132no—
151#485—1132no—
152#484—1132no—
153#483—1132no—
154#428—1131no—
155#428—1131no—
156#419—1131no—
157#419D11311yescharacteristic, complemented
158#407—1131no—
159C22×Q8C2×D11311yescharacteristic
160C23×C4—1131no—
161C2×M16—1131no—
162C2×M16C2×D11311yescharacteristic
163C22⋊Q8—1132no—
164C22⋊Q8—1132no—
165C22⋊Q8—2261no—
166C22⋊Q8—2261no—
167#117—1131no—
168#117—1131no—
169#117—1131no—
170#117—2261no—
171#117—2261no—
172#117C2×D11311yescharacteristic
173C2×C4⋊C4—1131no—
174C2×C4⋊C4—1131no—
175C2×C4⋊C4—1131no—
176C2×C4⋊C4—2261no—
177C2×C22⋊C4—2261no—
178C2×C22⋊C4—2261no—
179C2×C42—2261no—
180#107—1132no—
181#107—1132no—
182#106—1132no—
183#106—1132no—
184#103—1132no—
185#103—1132no—
186#103C2×D11312yescomplemented
187#103C2×D11312yescomplemented
188C22⋊C8—1132no—
189C22⋊C8—1132no—
190C24—1131no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D11312yes—
195C22×C4—21no—
196C22×C4—1131no—
197C22×C4—1131no—
198C22×C4—1131no—
199C22×C4—1131no—
200C22×C4—1131no—
201C22×C4—2261no—
202C22×C4—2261no—
203C22×C4—2261no—
204C22×C4—2262no—
205C22×C4—2262no—
206C22×C4C22×D11311yescharacteristic
207M16—22no—
208M16—2262no—
209C2×C8—1132no—
210C2×C8C22×D11312yes—
211C4⋊C4—1131no—
212C4⋊C4—1131no—
213C4⋊C4—1132no—
214C4⋊C4—2261no—
215C4⋊C4—2261no—
216C4⋊C4—2262no—
217C4⋊C4—2262no—
218C4⋊C4—2261no—
219C4⋊C4C22×D11311yescharacteristic
220C4⋊C4C22×D11311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—2261no—
223C22⋊C4—2261no—
224C22⋊C4—2262no—
225C22⋊C4—2262no—
226C22⋊C4—2261no—
227C42—21no—
228C42—2261no—
229C42—2261no—
230C42—2261no—
231C23—1131no—
232C23—1131no—
233C23—4521no—
234C23—2262no—
235C23—2262no—
236C23D4×D11311yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1131no—
245C2×C4—1131no—
246C2×C4—4522no—
247C2×C4—4522no—
248C2×C4—4522no—
249C2×C4—2261no—
250C2×C4—2261no—
251C2×C4—2262no—
252C2×C4—2262no—
253C2×C4—2262no—
254C2×C4—2262no—
255C2×C4—2261no—
256C2×C4—2261no—
257C2×C4—2261no—
258C2×C4—2261no—
259C2×C4D4×D11311yescharacteristic
260C2×C4#1426864612yes—
261C2×C4#1426864712yes—
262C2×C4C23×D11311yescharacteristic
263C8—22no—
264C8—2262no—
265C22—21no—
266C22—21no—
267C22—1132no—
268C22—4524no—
269C22—2261no—
270C22—2261no—
271C22—2261no—
272C22—2261no—
273C22—2261no—
274C22#1681360412yes—
275C22C2×D4×D11311yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—4521no—
280C4—4521no—
281C4—4522no—
282C4—2261no—
283C4—2261no—
284C4#1681360412yes—
285C4C2×#1426864611yescharacteristic
286C4C2×#1426864711yescharacteristic
287C2—22no—
288C2—4521no—
289C2—2262no—
290C2C2×#1681360411yescharacteristic
291C2#137×D11311yescharacteristic
292C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#419×C113C2
socleC22×C113C22×D4
prime radicalC23×D113D4
hypercenter#419D113
hypocenterC113#3095
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #20298938C2semidirect
4. #20298937C2semidirect
5. #20298932C2semidirect
6. #20298927C2semidirect
7. #20298641C2semidirect
8. #20298640C2semidirect
9. #20298607C2semidirect
11. #419×C113C2semidirect
30. #16813556C22semidirect
31. #16813555C22semidirect
32. #16813554C22semidirect
33. #16813553C22semidirect
34. #16813546C22semidirect
35. #16813546C22semidirect
41. #103×C113C22semidirect
42. #103×C113C22semidirect
147. C113#3095semidirect
157. #419D113semidirect
186. #103C2×D113semidirect
187. #103C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168135982
#117×D1132
#202989382
#202989372
#202989322
#202989272
#202986412
#202986402
#202986072
#202985612
#419×C1132
#3095113