Finite Group Databasebeta

← #27281804

Subgroups — #27281804

Order 13568.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283800C211yesmaximal, characteristic, complemented
2#20279714C211yesmaximal, characteristic
3#20273428C212yesmaximal
4#20272570C212yesmaximal, complemented
5C53×#1467C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7C2×#13133206C412yes—
8#16810754—24no—
9#16810680—21no—
10#16810680C2212yes—
11#16810676—21no—
12#16810540—21no—
13#16810540—21no—
14#16810525—21no—
15#16810525—21no—
16#16810524C412yes—
17C4×C4⋊Dic53C2211yescharacteristic
18#16810366—24no—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#432C2212yes—
21C53×#324—21no—
22C53×#324—21no—
23C22×D212—21no—
24C22×D212—21no—
25C22×D212C2×C412yes—
26#13133214—22no—
27#13133214—22no—
28#13133206—44no—
29C2×C4⋊Dic53—21no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×C4×Dic53C2×C412yes—
35C4⋊#88322—22no—
36C4⋊#88322—22no—
37C4⋊#88322D414yes—
38C22⋊#88322—22no—
39#13133128—22no—
40#13133128—22no—
41C4×#88322—22no—
42#13133124—21no—
43C22×Q8×C53C2×C411yescharacteristic
44C4×Q8×C53—22no—
45C4×Q8×C53—22no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52C4⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54#97×C53—22no—
55C23×D53—22no—
56C2×Q8×C53—22no—
57C2×Q8×C53—22no—
58C2×Q8×C53—41no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×Dic53—22no—
67C2×D212—24no—
68C2×D212—42no—
69C2×D212—42no—
70C2×D212—42no—
71C2×D212—42no—
72C2×D212—41no—
73C2×D212—42no—
74C2×D212—41no—
75C2×D212—82no—
76C2×C8×C53—42no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53C2×D412yes—
85C42×C53#4112yes—
86#11719921—44no—
87#11719921—44no—
88#11719921—82no—
89#11719921—82no—
90C4⋊Dic53—42no—
91C4×Dic53—42no—
92C2×#88322—24no—
93C23×C53C2×C22⋊C411yescharacteristic
94C22×D53—42no—
95C22×D53—42no—
96C22×D53—42no—
97C22×D53—42no—
98C22×D53—42no—
99C22×D53—42no—
100C22×D53—42no—
101Q8×C53—44no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#10012yes—
116C2×C4×C53C4×D412yes—
117C2×C4×C53#12011yescharacteristic
118C2×C4×C53#12112yes—
119C2×C4×C53#12311yescharacteristic
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123D212—84no—
124D212—82no—
125D212—84no—
126D212—82no—
127C8×C53—82no—
128#88322—44no—
129#9458—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53#38611yescharacteristic
133C22×C53C2×#10011yescharacteristic
134C22×C53#41311yescharacteristic
135C2×D53—82no—
136C2×D53—82no—
137C2×D53—82no—
138C2×D53—82no—
139C2×D53—82no—
140C2×D53—82no—
141C2×D53—82no—
142C4×C53—21no—
143C4×C53—22no—
144C4×C53—22no—
145C4×C53—22no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53#45412yes—
150Dic53—82no—
151#2983—531no—
152#2548—531no—
153#1555—532no—
154#1467D5311yescharacteristic, complemented
155#1285—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53#188811yescharacteristic
159C2×C53#190011yescharacteristic
160C2×C53#191011yescharacteristic
161D53—162no—
162C2×#127—531no—
163C2×#124—532no—
164C2×C4×Q8C2×D5311yescharacteristic
165#446—1064no—
166#432—532no—
167#432—1061no—
168#432C2×D5312yes—
169#431—1061no—
170#394—1061no—
171#394—1061no—
172#390—532no—
173#390—1061no—
174#390—1061no—
175C4×C4⋊C4—531no—
176#331—1064no—
177#324—21no—
178#324—21no—
179C53#945811yescharacteristic, complemented, Sylow
180C22×Q8C4×D5311yescharacteristic
181C22×D4—532no—
182C22×D4—1061no—
183C22×D4—1061no—
184#127—1062no—
185#127—1062no—
186#124—2124no—
187C4×Q8—22no—
188C4×Q8—22no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4C4×D5311yescharacteristic
191C2×C4⋊C4C22⋊D5311yescharacteristic
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C42—532no—
198C2×C42D21211yescharacteristic
199C2×C42D21211yescharacteristic
200C2×C42C22×D5311yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—534no—
203C4⋊C8—1062no—
204C4⋊C8—1062no—
205C22⋊C8—22no—
206C22⋊C8—1062no—
207#97—22no—
208#97—1062no—
209#97—1062no—
210C4×C8—1062no—
211#95—1061no—
212C24—1062no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×D4—1064no—
217C2×D4—4242no—
218C2×D4—2122no—
219C2×D4—2122no—
220C2×D4—2122no—
221C2×D4—2122no—
222C2×D4—2121no—
223C2×D4—2122no—
224C2×D4—2121no—
225C22×C4—1062no—
226C22×C4#1171992111yescharacteristic
227C22×C4#1171992111yescharacteristic
228C22×C4#1171992111yescharacteristic
229C22×C4#1171992111yescharacteristic
230C22×C4C2×C4×D5311yescharacteristic
231C22×C4C2×D21211yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233C2×C8—42no—
234C2×C8—1064no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—2122no—
239C22⋊C4—4242no—
240C22⋊C4—4242no—
241C22⋊C4—2124no—
242C22⋊C4—2124no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—2122no—
248C42D4×D5312yes—
249C42#1171994712yes—
250C23—2122no—
251C23—2122no—
252C23—2122no—
253C23—2122no—
254C23—2122no—
255C23—2122no—
256C23—2122no—
257C23C2×#1171992111yescharacteristic
258Q8—44no—
259D4—4244no—
260D4—4244no—
261D4—4242no—
262D4—4242no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4#1313313312yes—
280C2×C4#1313317712yes—
281C2×C4#1313318711yescharacteristic
282C2×C4#1313318912yes—
283C2×C4#1313320111yescharacteristic
284C8—82no—
285C8—2124no—
286C22—21no—
287C22—21no—
288C22—4242no—
289C22—4242no—
290C22—4242no—
291C22—4242no—
292C22—4242no—
293C22—4242no—
294C22—4242no—
295C22#1681050011yescharacteristic
296C22C2×#1313313311yescharacteristic
297C22#1681060511yescharacteristic
298C4—22no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—4242no—
306C4#1681079812yes—
307C2—21no—
308C2—21no—
309C2—8482no—
310C2#2027511611yescharacteristic
311C2#2027517611yescharacteristic
312C2#2027524011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1467C2
socleC22×C53#386
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1467D53
hypocenterC53#9458
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
1. #20283800C2semidirect
4. #20272570C2semidirect
5. C53×#1467C2semidirect
154. #1467D53semidirect
179. C53#9458semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202797142
#202734282
#202725702
C53×#14672
#945853