Finite Group Databasebeta

← #28400509

Subgroups — #28400509

Order 14464.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D113C211yesmaximal, characteristic
2C2×#16813580C211yesmaximal, characteristic, complemented
3#20298925C212yesmaximal, complemented
4#20298921C212yesmaximal, complemented
5#20298917C211yesmaximal, characteristic, complemented
6#20298914C211yesmaximal, characteristic, complemented
7#20298911C211yesmaximal, characteristic, complemented
8#20298639C212yesmaximal, complemented
9#20298601C212yesmaximal, complemented
10#20298560C211yesmaximal, characteristic, complemented
11#480×C113C211yesmaximal, characteristic, complemented
12C2×D4×D113—21no—
13C2×D4×D113C2211yescharacteristic
14C22×C4×D113C2211yescharacteristic
15C2×#14268630C2212yes—
16#16813597C2211yescharacteristic
17#16813596C2211yescharacteristic
18#16813595—21no—
19#16813594—21no—
20#16813593C2211yescharacteristic
21#16813580—21no—
22#16813580—21no—
23#16813580C2212yes—
24#16813580C2212yescomplemented
25C2×C4⋊Dic113C2211yescharacteristic, complemented
26C4⋊C4×D113C2211yescharacteristic
27C2×#14268624—21no—
28C22⋊C4×D113—21no—
29C2×C4×Dic113—21no—
30#16813555C2212yescomplemented
31#16813554C2212yescomplemented
32#16813553C2212yescomplemented
33#16813544C2212yescomplemented
34#16813543C2212yescomplemented
35#16813542C2212yescomplemented
36C22⋊#11744751C2211yescharacteristic, complemented
37#16813534C2212yescomplemented
38#121×C113C2211yescharacteristic
39C2×C4⋊C4×C113C2211yescharacteristic, complemented
40#107×C113C2212yescomplemented
41#102×C113C2212yescomplemented
42C22⋊C8×C113C2211yescharacteristic, complemented
43C23×D113—21no—
44C23×D113D411yescharacteristic
45C2×D452—21no—
46C2×D452C2311yescharacteristic
47C2×C22⋊D113—21no—
48C2×C22⋊D113—21no—
49C2×C22⋊D113—21no—
50D4×D113—24no—
51D4×D113—44no—
52C22×Dic113—21no—
53C22×Dic113D411yescharacteristic
54C2×C4×D113—24no—
55C2×C4×D113—21no—
56C2×C4×D113D411yescharacteristic
57C2×C4×D113D411yescharacteristic
58C2×C4×D113C2311yescharacteristic
59C2×C4×D113C2311yescharacteristic
60#14268630—24no—
61C2×#11744751C2312yes—
62C4⋊Dic113—21no—
63C4⋊Dic113C2311yescharacteristic
64C4⋊Dic113C2312yes—
65#14268626—21no—
66C22⋊Dic113—21no—
67#14268624—22no—
68#14268624—22no—
69#14268624—21no—
70#14268624—21no—
71C4×Dic113—22no—
72C4×Dic113—22no—
73C2×D4×C113—21no—
74C2×D4×C113C2311yescharacteristic
75C22×C4×C113—21no—
76C22×C4×C113C2311yescharacteristic
77C2×C8×C113C2312yes—
78C4⋊C4×C113—21no—
79C4⋊C4×C113C2311yescharacteristic
80C4⋊C4×C113C2312yes—
81C22⋊C4×C113—21no—
82C23×C113—21no—
83C23×C113C2×D411yescharacteristic
84C22×D113—22no—
85C22×D113—22no—
86C22×D113—21no—
87C22×D113—41no—
88C22×D113—42no—
89C22×D113—42no—
90C22×D113—42no—
91C22×D113C2×D411yescharacteristic
92C22×D113C2×D411yescharacteristic
93D4×C113—22no—
94D4×C113—42no—
95C2×C4×C113—22no—
96C2×C4×C113—21no—
97C2×C4×C113—22no—
98C2×C4×C113—42no—
99C2×C4×C113C2×D411yescharacteristic
100C2×C4×C113C2411yescharacteristic
101C22⋊D113—42no—
102C22⋊D113—42no—
103C22⋊D113—42no—
104C2×Dic113—22no—
105C2×Dic113—22no—
106C2×Dic113—21no—
107C2×Dic113—42no—
108C2×Dic113C2×D411yescharacteristic
109C2×Dic113C2×D411yescharacteristic
110D452—22no—
111D452—42no—
112C4×D113—22no—
113C4×D113—22no—
114C4×D113—22no—
115C4×D113—42no—
116C4×D113#4114yes—
117C8×C113—22no—
118#11744751—22no—
119C22×C113—21no—
120C22×C113—21no—
121C22×C113—41no—
122C22×C113—41no—
123C22×C113—41no—
124C22×C113AGL(1,ℤ/8)12yes—
125C22×C113C22×D411yescharacteristic
126C2×D113—21no—
127C2×D113—21no—
128C2×D113—21no—
129C2×D113—21no—
130C2×D113—21no—
131C2×D113—44no—
132C2×D113—41no—
133C2×D113—41no—
134C2×D113—41no—
135C2×D113—84no—
136C2×D113#12312yes—
137C4×C113—21no—
138C4×C113—41no—
139C4×C113—42no—
140C4×C113C2×#4112yes—
141Dic113—21no—
142Dic113—41no—
143Dic113—42no—
144Dic113#12312yes—
145C2×C113—22no—
146C2×C113—81no—
147C2×C113C2×#12311yescharacteristic
148C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C113#57511yescharacteristic
150D113—22no—
151D113—41no—
152D113—81no—
153#3094—1131nomaximal, Sylow
154C113#309411yescharacteristic, complemented, Sylow
155C2×#121—1131no—
156C2×#117—1131no—
157#482—1132no—
158#482—1132no—
159#481—1131no—
160#481—1131no—
161#480—1131no—
162#480D11311yescharacteristic, complemented
163#428—1132no—
164#418—1132no—
165#407—1131no—
166C22×D4—1131no—
167C22×D4—2261no—
168C23×C4—1131no—
169C2×M16—1132no—
170#121—1131no—
171#121—1131no—
172#121—1131no—
173#121—2261no—
174#121—2261no—
175#121C2×D11311yescharacteristic
176#117—1132no—
177#117—1132no—
178#117—2261no—
179#117—2261no—
180C2×C4⋊C4—1131no—
181C2×C4⋊C4—1131no—
182C2×C4⋊C4C2×D11311yescharacteristic, complemented
183C2×C22⋊C4—2261no—
184C2×C22⋊C4—2261no—
185C2×C42—2261no—
186#107—1132no—
187#107C2×D11312yescomplemented
188#106—1132no—
189#106—1132no—
190#102—1132no—
191#102—1132no—
192#102—1132no—
193#102C2×D11312yescomplemented
194C22⋊C8—1131no—
195C22⋊C8—1132no—
196C22⋊C8C2×D11311yescharacteristic, complemented
197C24—1131no—
198C24—2261no—
199C2×D4—21no—
200C2×D4—1131no—
201C2×D4—4524no—
202C2×D4—2261no—
203C2×D4—2264no—
204C2×D4—2261no—
205C2×D4—2261no—
206C2×D4—2261no—
207C2×D4C22×D11311yescharacteristic
208C22×C4—21no—
209C22×C4—1131no—
210C22×C4—1131no—
211C22×C4—1131no—
212C22×C4—1131no—
213C22×C4—1131no—
214C22×C4—2264no—
215C22×C4—2261no—
216C22×C4—2261no—
217C22×C4C22×D11311yescharacteristic
218M16—2264no—
219C2×C8—1132no—
220C2×C8C22×D11312yes—
221C4⋊C4—21no—
222C4⋊C4—1132no—
223C4⋊C4—1131no—
224C4⋊C4—2261no—
225C4⋊C4—2261no—
226C4⋊C4C22×D11312yes—
227C4⋊C4C22×D11311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2261no—
230C22⋊C4—2262no—
231C22⋊C4—2262no—
232C22⋊C4—2261no—
233C22⋊C4—2261no—
234C42—2262no—
235C42—2262no—
236C23—21no—
237C23—1131no—
238C23—1131no—
239C23—4521no—
240C23—4522no—
241C23—4522no—
242C23—4522no—
243C23—2262no—
244C23—2261no—
245C23—2262no—
246C23D4×D11311yescharacteristic
247D4—22no—
248D4—42no—
249D4—4522no—
250D4—4522no—
251D4—4522no—
252D4—4522no—
253D4—2262no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—1131no—
259C2×C4—1134no—
260C2×C4—1131no—
261C2×C4—4522no—
262C2×C4—4522no—
263C2×C4—2262no—
264C2×C4—2262no—
265C2×C4—2262no—
266C2×C4—2262no—
267C2×C4—2261no—
268C2×C4—2262no—
269C2×C4D4×D11311yescharacteristic
270C2×C4C23×D11311yescharacteristic
271C8—22no—
272C8—2262no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—1132no—
279C22—9044no—
280C22—4524no—
281C22—4521no—
282C22—4521no—
283C22—4521no—
284C22—2261no—
285C22—2261no—
286C22—2261no—
287C22—2261no—
288C22—2261no—
289C22#1681364612yes—
290C22C2×D4×D11311yescharacteristic
291C4—21no—
292C4—42no—
293C4—41no—
294C4—1132no—
295C4—4521no—
296C4—4522no—
297C4—2261no—
298C4#41×D11312yes—
299C2—22no—
300C2—81no—
301C2—9041no—
302C2—4521no—
303C2—2262no—
304C2#123×D11311yescharacteristic
305C2C2×#1681364611yescharacteristic
306C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#480×C113C2
socleC22×C113C22×D4
prime radical#121×D113C2
hypercenter#480D113
hypocenterC113#3094
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#16813580C2semidirect
3. #20298925C2semidirect
4. #20298921C2semidirect
5. #20298917C2semidirect
6. #20298914C2semidirect
7. #20298911C2semidirect
8. #20298639C2semidirect
9. #20298601C2semidirect
10. #20298560C2semidirect
11. #480×C113C2semidirect
24. #16813580C22semidirect
25. C2×C4⋊Dic113C22semidirect
30. #16813555C22semidirect
31. #16813554C22semidirect
32. #16813553C22semidirect
33. #16813544C22semidirect
34. #16813543C22semidirect
35. #16813542C22semidirect
36. C22⋊#11744751C22semidirect
37. #16813534C22semidirect
39. C2×C4⋊C4×C113C22semidirect
40. #107×C113C22semidirect
41. #102×C113C22semidirect
42. C22⋊C8×C113C22semidirect
154. C113#3094semidirect
162. #480D113semidirect
182. C2×C4⋊C4C2×D113semidirect
187. #107C2×D113semidirect
193. #102C2×D113semidirect
196. C22⋊C8C2×D113semidirect

Maximal subgroups

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

subgroupindex
#121×D1132
C2×#168135802
#202989252
#202989212
#202989172
#202989142
#202989112
#202986392
#202986012
#202985602
#480×C1132
#3094113