Finite Group Databasebeta

← #28320904

Subgroups — #28320904

Order 13568.

1786 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 328 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
1C24×#88324C211yesmaximal, characteristic, complemented
2C2×#16811650C213yesmaximal, complemented
3#20290690C213yesmaximal, complemented
4#20290659C213yesmaximal, complemented
5#20290537C211yesmaximal, characteristic, complemented
6C22⋊C4×#88324C213yesmaximal, complemented
7#20277214C211yesmaximal, characteristic
8C22×D53⋊C8C2211yescharacteristic, complemented
9C23×#88324—212no—
10C23×#88324C2213yes—
11C23×#88324C2213yes—
12#16811650—212no—
13C2×#13133284C2213yescomplemented
14C2×#13133274C2213yescomplemented
15C2×C22⋊#88324C2213yes—
16C2×C22⋊#88324C2213yes—
17C2×C4×#88324C2213yescomplemented
18#16811489C2216yescomplemented
19C24×Dic53C2211yescharacteristic
20C2×#13133186C411yescharacteristic
21C2×#13133177C2213yes—
22#16810500C413yes—
23#16810495C413yes—
24C22⋊C4×Dic53C2213yes—
25C53×#379C411yescharacteristic
26C2×D53⋊C8—26no—
27C22×#88324—232no—
28C22×#88324—26no—
29C22×#88324—212no—
30C22×#88324—212no—
31C22×#88324D4112yescomplemented
32C22×#88324C2313yes—
33C22×#88324C2311yescharacteristic
34C22×#88324C2313yes—
35C22×#88324C2311yescharacteristic
36#13133284—26no—
37#13133274—26no—
38C22⋊#88324—26no—
39C22⋊#88324—26no—
40C4×#88324—26no—
41C22×C22⋊D53C2×C413yes—
42C23×Dic53—26no—
43C23×Dic53—26no—
44C23×Dic53C2×C413yes—
45C23×Dic53C2313yes—
46C22×C4×D53C2311yescharacteristic
47#13133186C818yescomplemented
48#13133177—26no—
49#13133177—26no—
50C2×#11719919C2×C413yes—
51C2×C22⋊Dic53C2×C413yes—
52C2×#11719921C2×C413yes—
53C2×C4×Dic53C2313yes—
54#13133124C2×C416yes—
55C25×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53C2×C413yes—
57C2×C22⋊C4×C53C2×C413yes—
58C24×C53—26no—
59C24×C53—26no—
60C24×C53C2×C814yescomplemented
61C24×C53C22×C413yes—
62C23×D53C22×C411yescharacteristic
63C22×C4×C53C22×C413yes—
64C22×C4×C53C22×C411yescharacteristic
65C2×C22⋊D53—26no—
66C2×C22⋊D53—26no—
67C2×C22⋊D53C2×C8112yescomplemented
68C22×Dic53—224no—
69C22×Dic53—28no—
70C22×Dic53—26no—
71C22×Dic53—212no—
72C22×Dic53—26no—
73C22×Dic53—26no—
74C22×Dic53C22×C413yes—
75C22×Dic53C22×C413yes—
76C22×Dic53C2×D416yes—
77C22×Dic53#4116yes—
78C22×Dic53C2411yescharacteristic
79C2×C4×D53—23no—
80C2×C4×D53—23no—
81C2×#88324—224no—
82C2×#88324—26no—
83C2×#88324—224no—
84C2×#88324—23no—
85C2×#88324—23no—
86C2×#88324—26no—
87C2×#88324C2×D416yes—
88C2×#88324C2×D416yes—
89D53⋊C8—44no—
90C22⋊C4×C53—26no—
91C22⋊C4×C53—26no—
92C22⋊Dic53C2×C8112yescomplemented
93#11719921—26no—
94#11719919—26no—
95C4×Dic53—23no—
96C4×Dic53—23no—
97C23×C53—28no—
98C23×C53—224no—
99C23×C53—26no—
100C23×C53—26no—
101C23×C53—224no—
102C23×C53C4×D416yes—
103C23×C53C22×C816yes—
104C23×C53#13116yes—
105C23×C53C23×C411yescharacteristic
106C22×D53—23no—
107C22×D53—23no—
108C22×D53C22×C812yescomplemented
109C2×C4×C53—23no—
110C2×C4×C53—23no—
111C2×C4×C53—26no—
112C22⋊D53—224no—
113C2×Dic53—224no—
114C2×Dic53—26no—
115C2×Dic53—23no—
116C2×Dic53—23no—
117C2×Dic53—26no—
118C2×Dic53—26no—
119C2×Dic53—212no—
120C2×Dic53C4×D416yes—
121C2×Dic53C22×C816yescomplemented
122C2×Dic53C22×D413yes—
123C2×Dic53C2×#4113yes—
124C4×D53—41no—
125C4×D53—43no—
126#88324—26no—
127#88324—41no—
128#88324—43no—
129#88324#12014yescomplemented
130C8×#120—531nomaximal, Sylow
131C22×C53—232no—
132C22×C53—212no—
133C22×C53—26no—
134C22×C53—212no—
135C22×C53—26no—
136C22×C53C8×D4112yescomplemented
137C22×C53C2×C4×D413yes—
138C22×C53C23×C811yescharacteristic
139C22×C53C2×#13113yes—
140C2×D53—26no—
141C2×D53—43no—
142C2×D53—41no—
143C4×C53—43no—
144C4×C53—41no—
145Dic53—26no—
146Dic53—26no—
147Dic53C2×#12011yescharacteristic
148Dic53#52513yes—
149C24×C8—531no—
150C2×C8×D4—533no—
151C4×#120—531no—
152#1869—533no—
153#1850—533no—
154#1779—531no—
155C8×C22⋊C4—533no—
156C2×C53—212no—
157C2×C53C4×#12011yescharacteristic
158C2×C53C2×C8×D413yes—
159C2×C53#292913yes—
160D53—42no—
161C24×C4—531no—
162C23×C8—533no—
163C23×C8—533no—
164C23×C8—531no—
165C23×C8—10612no—
166C2×#120—531no—
167C2×C4×D4—533no—
168C8×D4—10612no—
169C2×C4⋊C8—533no—
170C2×C22⋊C8—533no—
171C2×C22⋊C8—533no—
172C2×C22⋊C8—533no—
173C2×C4×C8—533no—
174#386—533no—
175#385—533no—
176#379#615711yescharacteristic
177C4×C22⋊C4—533no—
178#336—536no—
179C53C8×#12011yescharacteristic, complemented, Sylow
180C25C2×#615711yescharacteristic
181C22×D4—533no—
182C23×C4—533no—
183C23×C4—533no—
184C23×C4—531no—
185C23×C4—1066no—
186C23×C4—1066no—
187C22×C8—531no—
188C22×C8—533no—
189C22×C8—5312no—
190C22×C8—531no—
191C22×C8—533no—
192C22×C8—10612no—
193C22×C8—10612no—
194C22×C8—1066no—
195C22×C8—1066no—
196C22×C8—10632no—
197#120—538no—
198C4×D4—1066no—
199C4×D4—1066no—
200C2×C4⋊C4—533no—
201C2×C22⋊C4—533no—
202C2×C22⋊C4—533no—
203C2×C22⋊C4C2×#615713yes—
204C2×C22⋊C4C2×#615713yes—
205C2×C42—533no—
206C4⋊C8—1066no—
207C22⋊C8—1066no—
208C22⋊C8—1066no—
209C22⋊C8—1066no—
210C4×C8—1066no—
211#95—536no—
212C24—26no—
213C24—26no—
214C24—531no—
215C24D53⋊C814yescomplemented
216C24C22×#615713yes—
217C2×D4—5312no—
218C2×D4—1066no—
219C2×D4—1066no—
220C22×C4—531no—
221C22×C4—536no—
222C22×C4—536no—
223C22×C4—533no—
224C22×C4—533no—
225C22×C4—1063no—
226C22×C4—1063no—
227C22×C4—1066no—
228C22×C4—1066no—
229C22×C4—10612no—
230C22×C4—1068no—
231C22×C4—10624no—
232C22×C4—1066no—
233C22×C4C22×#615713yes—
234C22×C4C22×#615711yescharacteristic
235C2×C8—536no—
236C2×C8—536no—
237C2×C8—10624no—
238C2×C8—1066no—
239C2×C8—10624no—
240C2×C8—1063no—
241C2×C8—1063no—
242C2×C8—1066no—
243C2×C8—2124no—
244C4⋊C4—1066no—
245C22⋊C4—26no—
246C22⋊C4—26no—
247C22⋊C4—5312no—
248C22⋊C4—1066no—
249C42—1063no—
250C42—1063no—
251C23—28no—
252C23—224no—
253C23—224no—
254C23—26no—
255C23—26no—
256C23—532no—
257C23—1063no—
258C23—1063no—
259C23D4×#615716yes—
260C23C2×D53⋊C816yes—
261C23#1313330416yes—
262C23C23×#615711yescharacteristic
263D4—10624no—
264C2×C4—23no—
265C2×C4—23no—
266C2×C4—26no—
267C2×C4—533no—
268C2×C4—533no—
269C2×C4—536no—
270C2×C4—536no—
271C2×C4—1066no—
272C2×C4—1066no—
273C2×C4—10612no—
274C2×C4—10624no—
275C2×C4—1063no—
276C2×C4—1066no—
277C2×C4—1063no—
278C2×C4—2121no—
279C2×C4—2123no—
280C8—534no—
281C8—1066no—
282C8—2123no—
283C8—2121no—
284C22—232no—
285C22—212no—
286C22—26no—
287C22—212no—
288C22—26no—
289C22—1066no—
290C22—2123no—
291C22—2121no—
292C22#16811650112yescomplemented
293C22C2×D4×#615713yes—
294C22C22×D53⋊C811yescharacteristic
295C22C2×#1313330413yes—
296C4—43no—
297C4—41no—
298C4—531no—
299C4—533no—
300C4—1066no—
301C4—1066no—
302C2—212no—
303C2—2122no—
304C2#120×#615711yescharacteristic
305C2C2×#1681165013yes—
306C2#2029143013yes—
subgroupG/H
center, FrattiniC23C23×#6157
commutatorC22×C53C23×C8
FittingC53×#379C4
socleC23×C53C23×C4
prime radicalC2×#13133186C4
cosocleC22×Dic53C24
hypercenter#379#6157
hypocenterC53C8×#120
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C24×#88324C2semidirect
2. C2×#16811650C2semidirect
3. #20290690C2semidirect
4. #20290659C2semidirect
5. #20290537C2semidirect
6. C22⋊C4×#88324C2semidirect
8. C22×D53⋊C8C22semidirect
13. C2×#13133284C22semidirect
14. C2×#13133274C22semidirect
17. C2×C4×#88324C22semidirect
18. #16811489C22semidirect
31. C22×#88324D4semidirect
47. #13133186C8semidirect
60. C24×C53C2×C8semidirect
67. C2×C22⋊D53C2×C8semidirect
92. C22⋊Dic53C2×C8semidirect
108. C22×D53C22×C8semidirect
121. C2×Dic53C22×C8semidirect
129. #88324#120semidirect
136. C22×C53C8×D4semidirect
179. C53C8×#120semidirect
215. C24D53⋊C8semidirect
292. C22#16811650semidirect

Maximal subgroups

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

subgroupindex
C24×#883242
C2×#168116502
#202906902
#202906592
#202905372
C22⋊C4×#883242
#202772142
C8×#12053