Finite Group Databasebeta

← #27393356

Subgroups — #27393356

Order 13568.

1538 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 315 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#20277313C212yesmaximal, complemented
2#20275552C218yesmaximal, complemented
3#20274630C214yesmaximal, complemented
4C2×C53×#534C211yesmaximal, characteristic
5C2×#13133132C22116yescomplemented
6#16810516—28no—
7#16810483C2214yescomplemented
8#16810479C2214yes—
9C4×C22⋊Dic53C2214yescomplemented
10C23×D4×C53C2212yes—
11C53×#534—28no—
12C2×#123×C53C414yescomplemented
13C2×#121×C53C2214yes—
14C2×#120×C53C414yescomplemented
15C22×C22⋊C4×C53C2211yescharacteristic
16C2×C4⋊Dic53—24no—
17C2×C22⋊Dic53—24no—
18C2×C22⋊Dic53—24no—
19C2×C22⋊Dic53C2318yes—
20C2×C4×Dic53—24no—
21#13133132—416no—
22#13133124—24no—
23#13133124—24no—
24C25×C53C2×C412yes—
25C25×C53C2311yescharacteristic
26C22×D4×C53—28no—
27C22×D4×C53—48no—
28C22×D4×C53C2×C414yes—
29C22×D4×C53C2×C414yes—
30C22×D4×C53C2314yes—
31C23×C4×C53C2×C412yes—
32C23×C4×C53C2312yes—
33#123×C53—216no—
34#121×C53—28no—
35#121×C53—28no—
36#120×C53—216no—
37C2×C4⋊C4×C53C2×C414yes—
38C2×C22⋊C4×C53C2×C418yes—
39C2×C22⋊C4×C53C2×C414yes—
40C2×C22⋊C4×C53D418yes—
41C24×C53—24no—
42C24×C53—24no—
43C24×C53—44no—
44C24×C53—82no—
45C24×C53—84no—
46C24×C53C22×C414yes—
47C24×C53C22×C412yes—
48C24×C53C2×D414yes—
49C24×C53C2411yescharacteristic
50C2×D4×C53—28no—
51C2×D4×C53—28no—
52C2×D4×C53—216no—
53C2×D4×C53—28no—
54C2×D4×C53—48no—
55C2×D4×C53—432no—
56C2×D4×C53—84no—
57C22×C4×C53—24no—
58C22×C4×C53—44no—
59C22×C4×C53—82no—
60C22×C4×C53C22×C414yes—
61C22×C4×C53C22×C414yes—
62C22×C4×C53C2×D414yes—
63C22×C4×C53C2×D414yes—
64C22×Dic53—28no—
65C4⋊C4×C53—28no—
66C22⋊C4×C53—216no—
67C22⋊C4×C53C22⋊C4116yes—
68C22⋊Dic53—216no—
69C22⋊Dic53—84no—
70C22⋊Dic53—84no—
71C4⋊Dic53—84no—
72C4×Dic53—84no—
73C23×C53—24no—
74C23×C53—28no—
75C23×C53—24no—
76C23×C53—28no—
77C23×C53—42no—
78C23×C53—44no—
79C23×C53—44no—
80C23×C53—44no—
81C23×C53—48no—
82C23×C53—48no—
83C23×C53—88no—
84C23×C53—88no—
85C23×C53—816no—
86C23×C53C2×C22⋊C414yes—
87C23×C53C23×C411yescharacteristic
88C23×C53C22×D412yes—
89C23×C53H(5,𝔽2)14yes—
90D4×C53—416no—
91D4×C53—416no—
92D4×C53—88no—
93C2×C4×C53—24no—
94C2×C4×C53—28no—
95C2×C4×C53—44no—
96C2×C4×C53—44no—
97C2×C4×C53—44no—
98C2×C4×C53—48no—
99C2×C4×C53C2×C22⋊C418yes—
100C2×Dic53—48no—
101C2×Dic53—48no—
102C2×Dic53—48no—
103#21795—531nomaximal, Sylow
104C22×C53—22no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—24no—
108C22×C53—28no—
109C22×C53—44no—
110C22×C53—48no—
111C22×C53—44no—
112C22×C53—48no—
113C22×C53—88no—
114C22×C53—88no—
115C22×C53—816no—
116C22×C53C22×C22⋊C411yescharacteristic
117C22×C53#51812yes—
118C22×C53#53514yes—
119C4×C53—28no—
120C4×C53—44no—
121Dic53—88no—
122C2×#534D5311yescharacteristic
123#2312—532no—
124#1964—538no—
125#1781—534no—
126C2×C53—24no—
127C2×C53—22no—
128C2×C53—44no—
129C2×C53—84no—
130C2×C53#230911yescharacteristic
131C2×C53#288312yes—
132C23×D4C2×D5312yes—
133#534—28no—
134C2×#123Dic5314yescomplemented
135C2×#121C2×D5314yes—
136C2×#120Dic5314yescomplemented
137C22×C22⋊C4C2×D5311yescharacteristic
138C2×#99—5316no—
139#388—1068no—
140#380—534no—
141#379—534no—
142C4×C22⋊C4—534no—
143C53#2179511yescharacteristic, complemented, Sylow
144C25C2×Dic5312yes—
145C25C22×D5311yescharacteristic
146C22×D4—28no—
147C22×D4—48no—
148C22×D4C2×Dic5314yes—
149C22×D4C2×Dic5314yes—
150C22×D4C22×D5314yes—
151C23×C4C2×Dic5312yes—
152C23×C4C22×D5312yes—
153#123—216no—
154#121—28no—
155#121—28no—
156#120—216no—
157C2×C4⋊C4—1064no—
158C2×C4⋊C4C2×Dic5314yes—
159C2×C22⋊C4—538no—
160C2×C22⋊C4—1064no—
161C2×C22⋊C4—1064no—
162C2×C22⋊C4C2×Dic5318yes—
163C2×C22⋊C4C2×Dic5314yes—
164C2×C22⋊C4C22⋊D5318yes—
165C2×C42—1064no—
166#99—21216no—
167#95—1064no—
168#95—1064no—
169C24—24no—
170C24—24no—
171C24—44no—
172C24—82no—
173C24—84no—
174C24C22×Dic5314yes—
175C24C22×Dic5312yes—
176C24C2×C22⋊D5314yes—
177C24C23×D5311yescharacteristic
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—216no—
181C2×D4—28no—
182C2×D4—48no—
183C2×D4—432no—
184C2×D4—84no—
185C22×C4—24no—
186C22×C4—44no—
187C22×C4—82no—
188C22×C4—1068no—
189C22×C4C22×Dic5314yes—
190C22×C4C22×Dic5314yes—
191C22×C4C2×C22⋊D5314yes—
192C22×C4C2×C22⋊D5314yes—
193C4⋊C4—28no—
194C4⋊C4—4244no—
195C22⋊C4—216no—
196C22⋊C4—10616no—
197C22⋊C4—4244no—
198C22⋊C4—4244no—
199C22⋊C4C22⋊Dic53116yes—
200C42—4244no—
201C23—24no—
202C23—28no—
203C23—24no—
204C23—28no—
205C23—42no—
206C23—44no—
207C23—44no—
208C23—44no—
209C23—48no—
210C23—48no—
211C23—88no—
212C23—88no—
213C23—816no—
214C23C2×C22⋊Dic5314yes—
215C23C23×Dic5311yescharacteristic
216C23C22×C22⋊D5312yes—
217C23#1313326214yes—
218D4—416no—
219D4—416no—
220D4—88no—
221C2×C4—24no—
222C2×C4—28no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—48no—
227C2×C4—2128no—
228C2×C4—2128no—
229C2×C4—2128no—
230C2×C4C2×C22⋊Dic5318yes—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—28no—
235C22—24no—
236C22—44no—
237C22—48no—
238C22—48no—
239C22—44no—
240C22—88no—
241C22—88no—
242C22—816no—
243C22C22×C22⋊Dic5311yescharacteristic
244C22#1681109812yes—
245C22#1681119014yes—
246C4—28no—
247C4—44no—
248C4—4248no—
249C2—24no—
250C2—22no—
251C2—44no—
252C2—84no—
253C2#2027729711yescharacteristic
254C2#2028317612yes—
subgroupG/H
centerC22C22×C22⋊Dic53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
Fitting, prime radicalC2×C53×#534C2
socleC22×C53C22×C22⋊C4
cosocleC24×C53C24
hypercenterC2×#534D53
hypocenterC53#21795
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #20277313C2semidirect
2. #20275552C2semidirect
3. #20274630C2semidirect
5. C2×#13133132C22semidirect
7. #16810483C22semidirect
9. C4×C22⋊Dic53C22semidirect
12. C2×#123×C53C4semidirect
14. C2×#120×C53C4semidirect
134. C2×#123Dic53semidirect
136. C2×#120Dic53semidirect
143. C53#21795semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202755522
#202746302
C2×C53×#5342
#2179553