Finite Group Databasebeta

← #27276595

Subgroups — #27276595

Order 13568.

818 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#16811228C211yesmaximal, characteristic
2#20274792C211yesmaximal, characteristic, complemented
3#20273245C214yesmaximal, complemented
4C53×#1822C211yesmaximal, characteristic, complemented
5#16811228—28no—
6C2×#13133214C411yescharacteristic, complemented
7C2×#13133209C411yescharacteristic, complemented
8C2×#13133202—22no—
9C2×#13133191C2212yes—
10C2×#13133190—22no—
11C2×C4×D212C412yescomplemented
12#16810486C2212yes—
13#16810390—22no—
14#16810358—28no—
15C2×#117×C53C2211yescharacteristic
16C53×#407C2212yes—
17C53×#343—22no—
18C22×D212—22no—
19C22×D212C2×C412yes—
20C22×C22⋊D53—22no—
21C22×C4×D53—22no—
22#13133214—24no—
23#13133209—24no—
24#13133202—216no—
25#13133191—24no—
26#13133191D418yes—
27#13133190—216no—
28C4×D212—24no—
29C4×D212—24no—
30C2×C4⋊Dic53C2×C412yes—
31C2×#11719919—22no—
32C2×C22⋊Dic53—22no—
33C2×#11719921—22no—
34C2×#11719921—22no—
35#13133124—24no—
36C23×C4×C53C2311yescharacteristic
37C2×M16×C53—22no—
38#117×C53—24no—
39C2×C4⋊C4×C53D412yes—
40C2×C22⋊C4×C53D412yes—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C22⋊C8×C53—24no—
44C24×C53C2×D411yescharacteristic
45C23×D53—22no—
46C22×C4×C53—42no—
47C22×C4×C53C22⋊C412yes—
48C22×C4×C53C22⋊C412yes—
49C22×C4×C53C22×C411yescharacteristic
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53C2×D414yes—
52C2×C22⋊D53—28no—
53C2×C22⋊D53—84no—
54C22×Dic53—22no—
55C22×Dic53—42no—
56C2×D212—22no—
57C2×D212—28no—
58C2×D212—44no—
59C2×D212—84no—
60C2×D212C22⋊C414yes—
61C2×C4×D53—28no—
62C2×C4×D53—84no—
63M16×C53—44no—
64C2×C8×C53—42no—
65C4⋊C4×C53—24no—
66C22⋊C4×C53—24no—
67C42×C53—22no—
68C42×C53—22no—
69C22⋊Dic53—44no—
70#11719921—28no—
71#11719921—28no—
72C4⋊Dic53—22no—
73C4⋊Dic53C22⋊C414yes—
74#11719919—28no—
75C23×C53—22no—
76C23×C53—42no—
77C23×C53C2×C22⋊C411yescharacteristic
78C23×C53#12012yes—
79C22×D53—24no—
80C22×D53—44no—
81C22×D53—44no—
82C22×D53—44no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—44no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53C2×C22⋊C412yes—
91C2×C4×C53#12012yes—
92C22⋊D53—88no—
93C2×Dic53—24no—
94C2×Dic53—44no—
95C2×Dic53—84no—
96C2×Dic53—82no—
97D212—44no—
98D212—44no—
99D212—88no—
100C4×D53—88no—
101C8×C53—82no—
102#8518—531nomaximal, Sylow
103C22×C53—22no—
104C22×C53—24no—
105C22×C53—22no—
106C22×C53—82no—
107C22×C53#37911yescharacteristic
108C22×C53#41012yes—
109C2×D53—44no—
110C2×D53—44no—
111C2×D53—44no—
112C2×D53—88no—
113C4×C53—22no—
114C4×C53—22no—
115C4×C53—42no—
116C4×C53—42no—
117Dic53—44no—
118Dic53—162no—
119C2×#540—531no—
120#1822D5311yescharacteristic, complemented
121#1820—531no—
122#1510—534no—
123C2×C53—22no—
124C2×C53—42no—
125C2×C53#178211yescharacteristic
126C2×C53#179512yes—
127D53—84no—
128#540—1068no—
129C2×#127—531no—
130C2×#125—531no—
131C2×#123—1062no—
132C2×#121—532no—
133C2×#121—1062no—
134C2×C4×D4—532no—
135C2×#117C2×D5311yescharacteristic
136#407C2×D5312yes—
137#381—532no—
138#343—22no—
139#342—1062no—
140#327—1068no—
141C53#851811yescharacteristic, complemented, Sylow
142C22×D4—532no—
143C22×D4—1062no—
144C22×D4—1062no—
145C23×C4—1062no—
146C23×C4C22×D5311yescharacteristic
147C2×M16—22no—
148#127—1064no—
149#125—1064no—
150#123—10616no—
151#121—538no—
152#121—10616no—
153#121—1064no—
154C4×D4—1064no—
155C4×D4—1064no—
156#117—24no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—1062no—
159C2×C4⋊C4C22⋊D5312yes—
160C2×C22⋊C4—1062no—
161C2×C22⋊C4—1062no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4D21212yes—
164C2×C42C4×D5311yescharacteristic
165C2×C42C4×D5311yescharacteristic
166C22⋊C8—24no—
167#95—1064no—
168C24—1062no—
169C24C2×D21211yescharacteristic
170C2×D4—534no—
171C2×D4—1068no—
172C2×D4—1068no—
173C2×D4—1062no—
174C2×D4—4244no—
175C2×D4—4244no—
176C2×D4—2124no—
177C22×C4—42no—
178C22×C4—1068no—
179C22×C4—1062no—
180C22×C4—4244no—
181C22×C4—2122no—
182C22×C4#1171992112yes—
183C22×C4#1171992112yes—
184C22×C4C2×C4×D5311yescharacteristic
185C22×C4D4×D5314yes—
186C22×C4C2×C22⋊D5311yescharacteristic
187M16—44no—
188C2×C8—42no—
189C4⋊C4—24no—
190C4⋊C4—534no—
191C4⋊C4—1068no—
192C4⋊C4—1062no—
193C22⋊C4—24no—
194C22⋊C4—1068no—
195C22⋊C4—1068no—
196C22⋊C4—2124no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—42no—
201C23—1064no—
202C23—2124no—
203C23—2124no—
204C23—2124no—
205C23C2×#1171992111yescharacteristic
206C23#1313318712yes—
207D4—4248no—
208D4—4248no—
209D4—2124no—
210D4—2124no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—1064no—
219C2×C4—4248no—
220C2×C4—4242no—
221C2×C4—4244no—
222C2×C4—2124no—
223C2×C4C22⋊C4×D5312yes—
224C2×C4#1313318512yes—
225C8—82no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—82no—
230C22—4248no—
231C22—2124no—
232C22—2124no—
233C22—2124no—
234C22#1681047811yescharacteristic
235C22#1681059612yes—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—8482no—
241C4—2124no—
242C2—22no—
243C2—42no—
244C2—4244no—
245C2#2027463211yescharacteristic
246C2#2027468612yes—
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1822C2
socleC22×C53#379
prime radicalC2×#16811228C2
cosocleC23×C4×C53C23
hypercenter#1822D53
hypocenterC53#8518
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #20274792C2semidirect
3. #20273245C2semidirect
4. C53×#1822C2semidirect
6. C2×#13133214C4semidirect
7. C2×#13133209C4semidirect
11. C2×C4×D212C4semidirect
120. #1822D53semidirect
141. C53#8518semidirect

Maximal subgroups

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

subgroupindex
C2×#168112282
#202747922
#202732452
C53×#18222
#851853