Finite Group Databasebeta

← #28313684

Subgroups — #28313684

Order 13568.

1530 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 272 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×#16811569C211yesmaximal, characteristic, complemented
2C2×#16811568C211yesmaximal, characteristic, complemented
3#20289900C216yesmaximal, complemented
4#20289898C216yesmaximal
5C2×#126×D53C211yesmaximal, characteristic
6C22×C4⋊#6157C2216yes—
7C22×C22⋊#6157C2214yes—
8#16811569—24no—
9#16811568—24no—
10C2×(C2×#6157)⋊C4C2216yes—
11C2×(C2×#6157)⋊C4C22112yes—
12C2×#13133212C413yescomplemented
13C2×#13133211C411yescharacteristic, complemented
14C2×#13133210C413yescomplemented
15#126×D53—24no—
16#126×D53—24no—
17C2×C4⋊C4×D53C2213yes—
18C2×C22⋊C4×D53C2213yes—
19C2×C42×D53C2211yescharacteristic
20C2×#126×C53C411yescharacteristic, complemented
21C23×#6157C2314yes—
22C23×#6157C2314yes—
23C2×C4⋊#6157—212no—
24C2×C4⋊#6157—412no—
25C2×C22⋊#6157—412no—
26(C2×#6157)⋊C4—212no—
27(C2×#6157)⋊C4—412no—
28C24×D53C2311yescharacteristic
29C22×C4×D53C2313yes—
30C22×C4×D53C2313yes—
31#13133212—212no—
32#13133211—24no—
33#13133210—212no—
34C2×C4⋊Dic53C2×C413yes—
35C2×#11719919C2×C413yes—
36C2×#11719919C2×C413yes—
37C4⋊C4×D53—26no—
38C4⋊C4×D53—26no—
39C2×C22⋊Dic53C2×C413yes—
40C2×#11719921C2×C413yes—
41C2×#11719921C2×C413yes—
42C22⋊C4×D53—26no—
43C22⋊C4×D53—26no—
44C2×C4×Dic53C2×C413yes—
45C42×D53D414yes—
46C42×D53D414yes—
47#126×C53—24no—
48C2×C4⋊C4×C53C2×C413yes—
49C2×C22⋊C4×C53C2×C413yes—
50C2×C42×C53C2×C411yescharacteristic
51C24×C53C22×C411yescharacteristic
52C23×D53—21no—
53C23×D53—21no—
54C23×D53—26no—
55C23×D53—26no—
56C23×D53C22×C411yescharacteristic
57C23×D53C2411yescharacteristic
58C22×#6157—24no—
59C22×#6157—24no—
60C22×#6157—212no—
61C22×#6157—212no—
62C22×#6157—212no—
63C22×#6157—212no—
64C22×C4×C53C22×C413yes—
65C22×C4×C53C22×C413yes—
66C22×Dic53C22×C413yes—
67C22×Dic53C22×C413yes—
68C2×C4×D53—26no—
69C2×C4×D53—26no—
70C2×C4×D53—23no—
71C2×C4×D53—23no—
72C2×C4×D53—26no—
73C2×C4×D53—26no—
74C2×C4×D53C2×D416yes—
75C2×C4×D53C2×D416yes—
76C22⋊#6157—416no—
77C4⋊#6157—424no—
78C4⋊C4×C53—26no—
79C22⋊C4×C53—26no—
80C42×C53C22⋊C414yescomplemented
81C22⋊Dic53—26no—
82#11719921—26no—
83#11719921—26no—
84C4⋊Dic53—26no—
85#11719919—26no—
86#11719919—26no—
87C4×Dic53C22⋊C4112yescomplemented
88C23×C53—21no—
89C23×C53—26no—
90C23×C53C23×C411yescharacteristic
91C22×D53—21no—
92C22×D53—26no—
93C22×D53—43no—
94C22×D53—43no—
95C22×D53—46no—
96C22×D53—44no—
97C22×D53—412no—
98C22×D53C22×D411yescharacteristic
99C22×D53C22×D411yescharacteristic
100C22×D53H(5,𝔽2)16yes—
101C22×D532−1+416yes—
102C2×#6157—412no—
103C2×#6157—412no—
104C2×#6157—44no—
105C2×#6157—44no—
106C2×#6157—412no—
107C2×#6157—412no—
108C2×C4×C53—26no—
109C2×C4×C53—23no—
110C2×C4×C53—26no—
111C2×C4×C53C2×C22⋊C416yes—
112C2×Dic53—26no—
113C2×Dic53—23no—
114C2×Dic53—26no—
115C2×Dic53C2×C22⋊C416yes—
116C4×D53—26no—
117C4×D53—26no—
118C4×D53—212no—
119C4×D53—43no—
120C4×D53—43no—
121C4×D53—46no—
122#14665—531nomaximal, Sylow
123C22×C53—43no—
124C22×C53—44no—
125C22×C53C22×C22⋊C411yescharacteristic
126C22×C53#52016yes—
127C2×D53—43no—
128C2×D53—44no—
129C2×D53—81no—
130C2×D53—83no—
131C2×D53—84no—
132C2×D53#53513yes—
133C2×D53#53613yes—
134C2×D53#53716yes—
135C2×D53#56014yes—
136C2×D53#563112yes—
137#6157—84no—
138#6157—84no—
139C4×C53—26no—
140C4×C53—43no—
141Dic53—26no—
142Dic53—43no—
143C22×#126—531no—
144C2×#384—531no—
145C2×#383—531no—
146#1441—536no—
147#1439—536no—
148C2×C53—81no—
149C2×C53#231113yes—
150C2×C53#238314yes—
151D53—81no—
152D53#281811yescharacteristic
153D53#282113yes—
154D53#282314yes—
155C2×#126—533no—
156C2×#126—533no—
157C2×#126—531no—
158C2×#126—1064no—
159C2×#126—1064no—
160C2×#126#615711yescharacteristic, complemented
161C22×C4⋊C4—533no—
162C22×C4⋊C4—536no—
163C22×C22⋊C4—533no—
164C22×C22⋊C4—534no—
165C22×C42—531no—
166#384—1064no—
167#383—1064no—
168C2×#95—536no—
169C2×#95—5312no—
170C53#1466511yescharacteristic, complemented, Sylow
171C25—531no—
172C23×C4—533no—
173C23×C4—533no—
174C23×C4—534no—
175C23×C4—534no—
176#126—24no—
177#126—10612no—
178#126—10612no—
179#126—1064no—
180C2×C4⋊C4—533no—
181C2×C4⋊C4—533no—
182C2×C4⋊C4—533no—
183C2×C4⋊C4—1066no—
184C2×C4⋊C4—1066no—
185C2×C4⋊C4—10612no—
186C2×C4⋊C4—21212no—
187C2×C4⋊C4C2×#615713yes—
188C2×C22⋊C4—533no—
189C2×C22⋊C4—533no—
190C2×C22⋊C4—533no—
191C2×C22⋊C4—1066no—
192C2×C22⋊C4—1066no—
193C2×C22⋊C4—21212no—
194C2×C22⋊C4C2×#615713yes—
195C2×C42—533no—
196C2×C42—534no—
197C2×C42—534no—
198C2×C42C2×#615711yescharacteristic
199#95—10612no—
200#95—21212no—
201C24—531no—
202C24—531no—
203C24—1061no—
204C24—1061no—
205C24—1066no—
206C24—1066no—
207C24C22×#615711yescharacteristic
208C22×C4—533no—
209C22×C4—536no—
210C22×C4—536no—
211C22×C4—533no—
212C22×C4—1066no—
213C22×C4—1066no—
214C22×C4—1066no—
215C22×C4—1063no—
216C22×C4—1063no—
217C22×C4—1066no—
218C22×C4—1064no—
219C22×C4—10612no—
220C22×C4—10612no—
221C22×C4—10612no—
222C22×C4—1064no—
223C22×C4—10612no—
224C22×C4C22×#615713yes—
225C22×C4C22×#615713yes—
226C4⋊C4—26no—
227C4⋊C4—1066no—
228C4⋊C4—1066no—
229C4⋊C4—1066no—
230C4⋊C4—21224no—
231C22⋊C4—26no—
232C22⋊C4—1066no—
233C22⋊C4—1066no—
234C22⋊C4—1066no—
235C22⋊C4—21216no—
236C42—5312no—
237C42C22⋊#615714yescomplemented
238C23—21no—
239C23—26no—
240C23—531no—
241C23—531no—
242C23—536no—
243C23—536no—
244C23—1061no—
245C23—1066no—
246C23—2124no—
247C23—2123no—
248C23—2126no—
249C23—2123no—
250C23—21212no—
251C23C23×#615711yescharacteristic
252C2×C4—26no—
253C2×C4—23no—
254C2×C4—26no—
255C2×C4—536no—
256C2×C4—1066no—
257C2×C4—10612no—
258C2×C4—1066no—
259C2×C4—1066no—
260C2×C4—1063no—
261C2×C4—1066no—
262C2×C4—2123no—
263C2×C4—2126no—
264C2×C4—2123no—
265C2×C4—2124no—
266C2×C4—2124no—
267C2×C4—21212no—
268C2×C4—21212no—
269C2×C4—21212no—
270C2×C4—21212no—
271C2×C4C2×C22⋊#615716yes—
272C22—43no—
273C22—44no—
274C22—533no—
275C22—536no—
276C22—533no—
277C22—534no—
278C22—5312no—
279C22—4241no—
280C22—4244no—
281C22—4243no—
282C22—2123no—
283C22—2124no—
284C22C22×C22⋊#615711yescharacteristic
285C22#1681169216yes—
286C4—26no—
287C4—43no—
288C4—1066no—
289C4—4244no—
290C4—4244no—
291C4—2123no—
292C2—81no—
293C2—531no—
294C2—534no—
295C2—533no—
296C2—4241no—
297C2#2029119013yes—
298C2#2029126214yes—
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC2×#126×C53C4
prime radicalC24×D53C23
cosocleC23×D53C24
hypercenterC2×#126#6157
hypocenterC53#14665
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#16811569C2semidirect
2. C2×#16811568C2semidirect
3. #20289900C2semidirect
12. C2×#13133212C4semidirect
13. C2×#13133211C4semidirect
14. C2×#13133210C4semidirect
20. C2×#126×C53C4semidirect
80. C42×C53C22⋊C4semidirect
87. C4×Dic53C22⋊C4semidirect
160. C2×#126#6157semidirect
170. C53#14665semidirect
237. C42C22⋊#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115692
C2×#168115682
#202899002
#202898982
C2×#126×D532
#1466553