Finite Group Databasebeta

← #27281646

Subgroups — #27281646

Order 13568.

620 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285165C211yesmaximal, characteristic
2#20273429C211yesmaximal, characteristic, complemented
3#20273261C212yesmaximal, complemented
4#20272579C212yesmaximal, complemented
5C53×#2380C211yesmaximal, characteristic, complemented
6C2×#13133243—22no—
7#16811173C2211yescharacteristic
8#16811010—22no—
9#16810999C412yescomplemented
10#16810995C412yescomplemented
11#16810680C2212yes—
12C4×#11719917C2211yescharacteristic
13#16810367—22no—
14#16810356—24no—
15C2×#127×C53C2211yescharacteristic
16C53×#386—22no—
17C53×#386—24no—
18C53×#383C2212yes—
19C2×#11719945—21no—
20#13133243—44no—
21#13133243—44no—
22C2×#11719923—24no—
23C2×#11719922—24no—
24C2×#11719917—21no—
25#13133217C2×C411yescharacteristic, complemented
26#13133214C2×C411yescharacteristic, complemented
27#13133205—21no—
28#13133191—22no—
29C4×D212—22no—
30C4⋊#88322—22no—
31C22⋊#88322—22no—
32#13133128—21no—
33#13133128—22no—
34C4×#88322C2×C412yescomplemented
35C22×D4×C53—22no—
36C22×D4×C53D412yes—
37C23×C4×C53—22no—
38#127×C53—22no—
39#127×C53C2×C412yes—
40#127×C53D412yes—
41C2×C22⋊C4×C53—24no—
42C2×C22⋊C4×C53—22no—
43C2×C42×C53C2311yescharacteristic
44#95×C53—22no—
45#95×C53—22no—
46#95×C53D414yes—
47C24×C53—22no—
48C2×D4×C53—24no—
49C2×D4×C53—24no—
50C2×D4×C53—22no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—82no—
54C2×D4×C53—82no—
55C2×D4×C53C22⋊C414yes—
56C22×C4×C53—24no—
57C22×C4×C53—84no—
58C22×C4×C53—82no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D412yes—
61C2×C22⋊D53—41no—
62#11719945—42no—
63#11719945—42no—
64C2×D212—21no—
65C2×D212—22no—
66C2×C4×D53—41no—
67C2×Dic106—21no—
68C22⋊C4×C53—84no—
69C22⋊C4×C53—82no—
70C42×C53C22×C411yescharacteristic
71C42×C53C2×D411yescharacteristic
72C42×C53#4112yes—
73#11719923—48no—
74#11719922—48no—
75#11719921—42no—
76C4⋊Dic53—22no—
77#11719917—24no—
78C2×#88322—22no—
79C2×#88322—42no—
80C23×C53—24no—
81C23×C53—44no—
82C23×C53—44no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53#12011yescharacteristic
86C22×D53—41no—
87D4×C53—44no—
88D4×C53—82no—
89D4×C53—84no—
90D4×C53—84no—
91C2×C4×C53—22no—
92C2×C4×C53—42no—
93C2×C4×C53—48no—
94C2×C4×C53—44no—
95C2×C4×C53—88no—
96C2×C4×C53C2×C22⋊C411yescharacteristic
97C2×C4×C53C4×D412yes—
98C2×C4×C53#12112yes—
99C2×C4×C53#12311yescharacteristic
100C22⋊D53—82no—
101C2×Dic53—41no—
102D212—42no—
103D212—82no—
104C4×D53—81no—
105C4×D53—81no—
106Dic106—42no—
107#88322—24no—
108#88322—82no—
109#9428—531nomaximal, Sylow
110C22×C53—22no—
111C22×C53—44no—
112C22×C53—48no—
113C22×C53—82no—
114C22×C53—84no—
115C22×C53—82no—
116C22×C53#38611yescharacteristic
117C22×C53#45312yes—
118C2×D53—81no—
119C2×D53—82no—
120C4×C53—21no—
121C4×C53—21no—
122C4×C53—22no—
123C4×C53—22no—
124C4×C53—84no—
125Dic53—81no—
126#3151—531no—
127#2380D5311yescharacteristic, complemented
128#1555—531no—
129#1514—532no—
130#1288—532no—
131C2×C53—22no—
132C2×C53—84no—
133C2×C53#188711yescharacteristic
134C2×C53#190412yes—
135D53—161no—
136C2×AGL(1,ℤ/8)—1062no—
137#532—531no—
138C2×#127C2×D5311yescharacteristic
139#496—1062no—
140#493—532no—
141#492—532no—
142#432—532no—
143C4×M16—531no—
144#386—22no—
145#386—24no—
146#383C2×D5312yes—
147#331—1062no—
148#327—1064no—
149C53#942811yescharacteristic, complemented, Sylow
150C2×#41—1061no—
151C22×D4—22no—
152C22×D4C22⋊D5312yes—
153C23×C4—22no—
154AGL(1,ℤ/8)—2124no—
155AGL(1,ℤ/8)—2124no—
156C2×SD16—1064no—
157C2×D8—1064no—
158C2×M16—1061no—
159C4⋊Q8—531no—
160#127—22no—
161#127—531no—
162#127C4×D5312yes—
163#127D21212yes—
164#124—1061no—
165#121—1062no—
166C4×D4—1062no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C42C22×D5311yescharacteristic
170C4⋊C8—1062no—
171C22⋊C8—1062no—
172#97—1061no—
173#97—1062no—
174C4×C8—532no—
175#95—22no—
176#95—22no—
177#95C22⋊D5314yes—
178C24—22no—
179#41—2122no—
180#41—2122no—
181C2×Q8—1061no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—82no—
188C2×D4—82no—
189C2×D4—1061no—
190C2×D4—1062no—
191C2×D4—2121no—
192C2×D4#1171992114yes—
193C22×C4—24no—
194C22×C4—84no—
195C22×C4—82no—
196C22×C4—2121no—
197C22×C4C2×C22⋊D5311yescharacteristic
198C22×C4C2×C22⋊D5312yes—
199SD16—2128no—
200D8—2128no—
201M16—1064no—
202C2×C8—1062no—
203C2×C8—2122no—
204C4⋊C4—1062no—
205C22⋊C4—84no—
206C22⋊C4—82no—
207C22⋊C4—2122no—
208C42C2×C4×D5311yescharacteristic
209C42C2×D21211yescharacteristic
210C42#1171994512yes—
211C23—24no—
212C23—44no—
213C23—44no—
214C23—42no—
215C23—42no—
216C23—2121no—
217C23#1313318611yescharacteristic
218Q8—2122no—
219D4—44no—
220D4—82no—
221D4—84no—
222D4—84no—
223D4—4242no—
224D4—4242no—
225D4—2122no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—88no—
231C2×C4—4241no—
232C2×C4—4241no—
233C2×C4—2121no—
234C2×C4C2×#1171992111yescharacteristic
235C2×C4C4×C22⋊D5312yes—
236C2×C4#1313319112yes—
237C2×C4#1313320011yescharacteristic
238C8—1064no—
239C8—4242no—
240C22—22no—
241C22—48no—
242C22—44no—
243C22—82no—
244C22—82no—
245C22—84no—
246C22—4242no—
247C22—4241no—
248C22#1681050211yescharacteristic
249C22#1681079112yes—
250C4—22no—
251C4—22no—
252C4—21no—
253C4—21no—
254C4—84no—
255C4—4241no—
256C2—22no—
257C2—84no—
258C2—8481no—
259C2#2027511311yescharacteristic
260C2#2027520212yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2380C2
socleC22×C53#386
prime radical#20285165C2
cosocleC2×C42×C53C23
hypercenter#2380D53
hypocenterC53#9428
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2380

Complements

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

NH ≅ G/Ntype
2. #20273429C2semidirect
3. #20273261C2semidirect
4. #20272579C2semidirect
5. C53×#2380C2semidirect
9. #16810999C4semidirect
10. #16810995C4semidirect
25. #13133217C2×C4semidirect
26. #13133214C2×C4semidirect
34. C4×#88322C2×C4semidirect
127. #2380D53semidirect
149. C53#9428semidirect

Maximal subgroups

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

subgroupindex
#202851652
#202734292
#202732612
#202725792
C53×#23802
#942853