Finite Group Databasebeta

← #28401342

Subgroups — #28401342

Order 14464.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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#20299111C211yesmaximal, characteristic, complemented
2C2×#16813593C211yesmaximal, characteristic, complemented
3#20298931C211yesmaximal, characteristic, complemented
4#20298921C212yesmaximal, complemented
5#20298917C211yesmaximal, characteristic, complemented
6#20298833C212yesmaximal, complemented
7#20298824C212yesmaximal, complemented
8#20298742C212yesmaximal, complemented
9#20298732C211yesmaximal, characteristic, complemented
10C22⋊C8×D113C211yesmaximal, characteristic, complemented
11#450×C113C211yesmaximal, characteristic, complemented
12C2×#14268648C2211yescharacteristic, complemented
13C22×C22⋊D113—21no—
14C22×C4×D113C2211yescharacteristic
15C2×#14268632C2212yescomplemented
16C2×C8×D113C2212yescomplemented
17#16813607—21no—
18#16813598C2211yescharacteristic, complemented
19#16813597C2211yescharacteristic, complemented
20#16813596C2211yescharacteristic
21#16813593—21no—
22#16813593—21no—
23#16813593C2212yes—
24#16813593C2212yes—
25#16813592—21no—
26#16813584—21no—
27#16813582—21no—
28#16813580C2211yescharacteristic, complemented
29C2×C4⋊Dic113C2211yescharacteristic
30C2×C22⋊Dic113—21no—
31#16813555C2212yescomplemented
32#16813554C2212yescomplemented
33#16813546C2212yescomplemented
34#16813544C2212yescomplemented
35#16813543C2212yescomplemented
36C22⋊#11744751C2211yescharacteristic
37#16813534C2212yescomplemented
38C22×D4×C113C2211yescharacteristic
39C2×SD16×C113C2212yescomplemented
40C22⋊Q8×C113C2211yescharacteristic
41#102×C113C2212yescomplemented
42C22⋊C8×C113C2211yescharacteristic
43C23×D113D411yescharacteristic
44#14268648—42no—
45C2×D452—21no—
46C2×D452—21no—
47C2×D452C2311yescharacteristic
48C2×C22⋊D113—21no—
49C2×C22⋊D113—22no—
50C2×C22⋊D113—22no—
51C2×C22⋊D113—44no—
52C22×Dic113—21no—
53C22×Dic113D411yescharacteristic
54C2×C4×D113—21no—
55C2×C4×D113—42no—
56C2×C4×D113D411yescharacteristic
57C2×C4×D113D411yescharacteristic
58C2×C4×D113C2311yescharacteristic
59C2×C4×D113C2311yescharacteristic
60#14268632—42no—
61C2×#11744751C2312yes—
62C8×D113—42no—
63C4⋊Dic113—21no—
64C4⋊Dic113C2312yes—
65C4⋊Dic113C2311yescharacteristic
66#14268626—21no—
67C22⋊Dic113—22no—
68C22⋊Dic113—22no—
69#14268624—21no—
70#14268624—21no—
71#14268624—21no—
72#14268624—21no—
73C4×Dic113—21no—
74C24×C113—21no—
75C2×Q8×C113C2311yescharacteristic
76C2×D4×C113—21no—
77C2×D4×C113—42no—
78C2×D4×C113C2312yes—
79C22×C4×C113C2311yescharacteristic
80SD16×C113—42no—
81C2×C8×C113C2312yes—
82C4⋊C4×C113—21no—
83C4⋊C4×C113C2311yescharacteristic
84C22⋊C4×C113—21no—
85C23×C113—22no—
86C23×C113—42no—
87C23×C113—42no—
88C23×C113—42no—
89C23×C113C2×D411yescharacteristic
90C22×D113—22no—
91C22×D113—22no—
92C22×D113—21no—
93C22×D113—41no—
94C22×D113C2×D411yescharacteristic
95C22×D113C2×D411yescharacteristic
96Q8×C113—22no—
97D4×C113—22no—
98D4×C113—22no—
99D4×C113—42no—
100C2×C4×C113—22no—
101C2×C4×C113—21no—
102C2×C4×C113—21no—
103C2×C4×C113C2×D411yescharacteristic
104C2×C4×C113C2411yescharacteristic
105C22⋊D113—44no—
106C22⋊D113—42no—
107C22⋊D113—44no—
108C2×Dic113—21no—
109C2×Dic113—22no—
110C2×Dic113—41no—
111C2×Dic113—42no—
112C2×Dic113C2×D411yescharacteristic
113C2×Dic113C2×D411yescharacteristic
114D452—22no—
115D452—42no—
116D452—42no—
117C4×D113—22no—
118C4×D113—22no—
119C4×D113—41no—
120C4×D113—41no—
121C4×D113—42no—
122C8×C113—22no—
123#11744751—22no—
124C22×C113—21no—
125C22×C113—21no—
126C22×C113—42no—
127C22×C113—42no—
128C22×C113—42no—
129C22×C113—84no—
130C22×C113#13512yes—
131C22×C113C22×D411yescharacteristic
132C2×D113—21no—
133C2×D113—21no—
134C2×D113—21no—
135C2×D113—21no—
136C2×D113—21no—
137C2×D113—44no—
138C2×D113—41no—
139C2×D113—41no—
140C2×D113—41no—
141C2×D113AGL(1,ℤ/8)12yes—
142C4×C113—21no—
143C4×C113—41no—
144C4×C113—41no—
145C4×C113H(5,𝔽2)12yes—
146Dic113—21no—
147Dic113—21no—
148Dic113—41no—
149Dic113—42no—
150C2×C113—22no—
151C2×C113—82no—
152C2×C113#53411yescharacteristic
153C2×C113C2×#13511yescharacteristic
154C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
155D113—22no—
156D113—41no—
157D113—81no—
158#3188—1131nomaximal, Sylow
159C113#318811yescharacteristic, complemented, Sylow
160#525—1131no—
161C2×#121—1131no—
162#484—1131no—
163#482—1132no—
164#481—1131no—
165#466—1132no—
166#465—1132no—
167#450D11311yescharacteristic, complemented
168#449—1132no—
169#447—1131no—
170C2×C22⋊C8—1131no—
171C2×#41—1131no—
172C22×D4—2261no—
173C22×D4C2×D11311yescharacteristic
174C23×C4—1131no—
175C2×SD16C2×D11312yescomplemented
176C2×D8—1132no—
177C22×C8—1132no—
178#123—2261no—
179C22⋊Q8—1131no—
180C22⋊Q8C2×D11311yescharacteristic
181#121—1132no—
182#121—1132no—
183#121—1131no—
184#121—1131no—
185#121—2261no—
186#121—2261no—
187#120—2261no—
188C4×D4—2261no—
189C4×D4—2261no—
190#117—1131no—
191C2×C4⋊C4—1131no—
192C2×C22⋊C4—2261no—
193#107—1132no—
194#106—1132no—
195#103—1132no—
196#102—1132no—
197#102—1132no—
198#102C2×D11312yescomplemented
199C22⋊C8—1131no—
200C22⋊C8—1132no—
201C22⋊C8C2×D11311yescharacteristic
202C24—21no—
203C24—1131no—
204#41—4522no—
205C2×Q8C22×D11311yescharacteristic
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—1131no—
209C2×D4—4524no—
210C2×D4—2261no—
211C2×D4—2261no—
212C2×D4—2262no—
213C2×D4—2262no—
214C2×D4—2261no—
215C2×D4C22×D11312yes—
216C22×C4—1131no—
217C22×C4—1131no—
218C22×C4—1131no—
219C22×C4—1131no—
220C22×C4—1131no—
221C22×C4—4522no—
222C22×C4—2261no—
223C22×C4—2261no—
224C22×C4C22×D11311yescharacteristic
225SD16—42no—
226D8—4522no—
227C2×C8—1132no—
228C2×C8—4522no—
229C2×C8C22×D11312yes—
230C4⋊C4—21no—
231C4⋊C4—1131no—
232C4⋊C4—1132no—
233C4⋊C4—2261no—
234C4⋊C4—2261no—
235C4⋊C4C22×D11311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—2261no—
238C22⋊C4—2261no—
239C22⋊C4—2261no—
240C22⋊C4—2262no—
241C22⋊C4—2262no—
242C22⋊C4—2261no—
243C42—2261no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—1131no—
249C23—1131no—
250C23—4521no—
251C23—2262no—
252C23—2261no—
253C23—2262no—
254C23D4×D11311yescharacteristic
255Q8—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—4522no—
260D4—4522no—
261D4—4524no—
262D4—4524no—
263D4—4522no—
264D4—2262no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—1131no—
269C2×C4—1131no—
270C2×C4—4521no—
271C2×C4—4521no—
272C2×C4—4522no—
273C2×C4—4522no—
274C2×C4—4521no—
275C2×C4—2261no—
276C2×C4—2262no—
277C2×C4—2262no—
278C2×C4—2262no—
279C2×C4D4×D11311yescharacteristic
280C2×C4C23×D11311yescharacteristic
281C8—22no—
282C8—2262no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—1132no—
290C22—4524no—
291C22—4521no—
292C22—4521no—
293C22—4521no—
294C22—2261no—
295C22—2261no—
296C22—2261no—
297C22—2261no—
298C22—2261no—
299C22#1681364312yes—
300C22C2×D4×D11311yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—4521no—
305C4—4522no—
306C4—2261no—
307C4—2261no—
308C4#1681366712yes—
309C2—22no—
310C2—82no—
311C2—9041no—
312C2—4521no—
313C2—2262no—
314C2#2029915811yescharacteristic
315C2C2×#1681364311yescharacteristic
316C2AGL(1,ℤ/8)×D11311yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#450×C113C2
socleC22×C113C22×D4
hypercenter#450D113
hypocenterC113#3188
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. #20299111C2semidirect
2. C2×#16813593C2semidirect
3. #20298931C2semidirect
4. #20298921C2semidirect
5. #20298917C2semidirect
6. #20298833C2semidirect
7. #20298824C2semidirect
8. #20298742C2semidirect
9. #20298732C2semidirect
10. C22⋊C8×D113C2semidirect
11. #450×C113C2semidirect
12. C2×#14268648C22semidirect
15. C2×#14268632C22semidirect
16. C2×C8×D113C22semidirect
18. #16813598C22semidirect
19. #16813597C22semidirect
28. #16813580C22semidirect
31. #16813555C22semidirect
32. #16813554C22semidirect
33. #16813546C22semidirect
34. #16813544C22semidirect
35. #16813543C22semidirect
37. #16813534C22semidirect
39. C2×SD16×C113C22semidirect
41. #102×C113C22semidirect
159. C113#3188semidirect
167. #450D113semidirect
175. C2×SD16C2×D113semidirect
198. #102C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202991112
C2×#168135932
#202989312
#202989212
#202989172
#202988332
#202988242
#202987422
#202987322
C22⋊C8×D1132
#450×C1132
#3188113