Finite Group Databasebeta

← #27276594

Subgroups — #27276594

Order 13568.

818 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 131 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#20274798C211yesmaximal, characteristic, complemented
2#20274792C211yesmaximal, characteristic, complemented
3#20273244C214yesmaximal, complemented
4C2×C53×#540C211yesmaximal, characteristic
5#16810587C2212yes—
6#16810486C2212yes—
7#16810392—22no—
8#16810390—22no—
9#16810357—28no—
10C53×#540—28no—
11C2×#127×C53C411yescharacteristic, complemented
12C2×#125×C53C411yescharacteristic, complemented
13C2×#123×C53—22no—
14C2×#121×C53—22no—
15C2×#121×C53C2212yes—
16C2×C4×D4×C53C412yescomplemented
17C2×#117×C53C2211yescharacteristic
18C2×#11719917—22no—
19C2×C22⋊Dic53—22no—
20C22⋊#88322—24no—
21#13133124—24no—
22C22×D4×C53—22no—
23C22×D4×C53—22no—
24C22×D4×C53C2×C412yes—
25C23×C4×C53—22no—
26C23×C4×C53C2311yescharacteristic
27#127×C53—24no—
28#125×C53—24no—
29#123×C53—216no—
30#121×C53—216no—
31#121×C53—24no—
32#121×C53D418yes—
33C4×D4×C53—24no—
34C4×D4×C53—24no—
35#117×C53—24no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53C2×C412yes—
38C2×C4⋊C4×C53D412yes—
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53D412yes—
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C24×C53—22no—
45C24×C53C2×D411yescharacteristic
46C2×D4×C53—28no—
47C2×D4×C53—28no—
48C2×D4×C53—22no—
49C2×D4×C53—44no—
50C2×D4×C53—84no—
51C2×D4×C53—84no—
52C2×D4×C53C22⋊C414yes—
53C22×C4×C53—22no—
54C22×C4×C53—28no—
55C22×C4×C53—42no—
56C22×C4×C53—84no—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D414yes—
62C22×Dic53—42no—
63C4⋊C4×C53—28no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—24no—
66C4⋊C4×C53C22⋊C414yes—
67C22⋊C4×C53—28no—
68C22⋊C4×C53—24no—
69C22⋊C4×C53—28no—
70C42×C53—22no—
71C42×C53—22no—
72C22⋊Dic53—44no—
73#11719917—44no—
74C2×#88322—42no—
75C23×C53—22no—
76C23×C53—24no—
77C23×C53—42no—
78C23×C53—44no—
79C23×C53—44no—
80C23×C53—44no—
81C23×C53C2×C22⋊C411yescharacteristic
82C23×C53#12012yes—
83D4×C53—44no—
84D4×C53—44no—
85D4×C53—88no—
86D4×C53—88no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—24no—
91C2×C4×C53—22no—
92C2×C4×C53—44no—
93C2×C4×C53—44no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—88no—
97C2×C4×C53C2×C22⋊C412yes—
98C2×C4×C53#12012yes—
99C2×Dic53—84no—
100C2×Dic53—82no—
101#88322—82no—
102#8518—531nomaximal, Sylow
103C22×C53—22no—
104C22×C53—24no—
105C22×C53—22no—
106C22×C53—44no—
107C22×C53—44no—
108C22×C53—44no—
109C22×C53—82no—
110C22×C53—88no—
111C22×C53#37911yescharacteristic
112C22×C53#41012yes—
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—44no—
116C4×C53—42no—
117C4×C53—42no—
118Dic53—162no—
119C2×#540D5311yescharacteristic
120#1822—531no—
121#1820—531no—
122#1510—534no—
123C2×C53—22no—
124C2×C53—42no—
125C2×C53—84no—
126C2×C53#178211yescharacteristic
127C2×C53#179512yes—
128#540—28no—
129C2×#127Dic5311yescharacteristic, complemented
130C2×#125Dic5311yescharacteristic, complemented
131C2×#123—22no—
132C2×#121—22no—
133C2×#121C2×D5312yes—
134C2×C4×D4Dic5312yescomplemented
135C2×#117C2×D5311yescharacteristic
136#407—532no—
137#381—532no—
138#343—1062no—
139#342—1062no—
140#327—1068no—
141C53#851811yescharacteristic, complemented, Sylow
142C22×D4—22no—
143C22×D4—22no—
144C22×D4C2×Dic5312yes—
145C23×C4—22no—
146C23×C4C22×D5311yescharacteristic
147C2×M16—1062no—
148#127—24no—
149#125—24no—
150#123—216no—
151#121—216no—
152#121—24no—
153#121C22⋊D5318yes—
154C4×D4—24no—
155C4×D4—24no—
156#117—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×Dic5312yes—
159C2×C4⋊C4C22⋊D5312yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4C22⋊D5312yes—
164C2×C42C2×Dic5311yescharacteristic
165C2×C42C2×Dic5311yescharacteristic
166C22⋊C8—1064no—
167#95—1064no—
168C24—22no—
169C24C2×C22⋊D5311yescharacteristic
170C2×D4—28no—
171C2×D4—28no—
172C2×D4—22no—
173C2×D4—44no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4C22⋊Dic5314yes—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—42no—
180C22×C4—84no—
181C22×C4—2122no—
182C22×C4C22⋊Dic5312yes—
183C22×C4C22⋊Dic5312yes—
184C22×C4C22×Dic5311yescharacteristic
185C22×C4C2×C22⋊D5311yescharacteristic
186C22×C4C2×C22⋊D5314yes—
187M16—2124no—
188C2×C8—2122no—
189C4⋊C4—28no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4C22⋊Dic5314yes—
193C22⋊C4—28no—
194C22⋊C4—24no—
195C22⋊C4—28no—
196C22⋊C4—2124no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—24no—
201C23—42no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23C2×C22⋊Dic5311yescharacteristic
206C23#1313318612yes—
207D4—44no—
208D4—44no—
209D4—88no—
210D4—88no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—88no—
221C2×C4—4242no—
222C2×C4—4244no—
223C2×C4C2×C22⋊Dic5312yes—
224C2×C4#1313318612yes—
225C8—4242no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—44no—
230C22—44no—
231C22—44no—
232C22—82no—
233C22—88no—
234C22#1681047911yescharacteristic
235C22#1681059712yes—
236C4—22no—
237C4—22no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—8482no—
242C2—22no—
243C2—42no—
244C2—84no—
245C2#2027463311yescharacteristic
246C2#2027468512yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#540C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenterC2×#540D53
hypocenterC53#8518
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. #20274798C2semidirect
2. #20274792C2semidirect
3. #20273244C2semidirect
11. C2×#127×C53C4semidirect
12. C2×#125×C53C4semidirect
16. C2×C4×D4×C53C4semidirect
129. C2×#127Dic53semidirect
130. C2×#125Dic53semidirect
134. C2×C4×D4Dic53semidirect
141. C53#8518semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202747922
#202732442
C2×C53×#5402
#851853