Finite Group Databasebeta

← #27496191

Subgroups — #27496191

Order 13568.

578 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 179 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#20283078C211yesmaximal, characteristic
2#20276283C214yesmaximal
3#20276278C212yesmaximal
4#20276273C212yesmaximal
5#20275679C212yesmaximal
6#20275679C212yesmaximal
7#20274461C211yesmaximal, characteristic
8C53×#2864C211yesmaximal, characteristic
9C2×#13133209C2212yes—
10C2×C4×#88322C2212yes—
11#16810521—22no—
12#16810521—22no—
13#16810493C2211yescharacteristic
14#16810493C2211yescharacteristic
15#16810489C2212yes—
16#16810385C2214yes—
17#16810385C2214yes—
18#16810385C2214yes—
19#16810385C2214yes—
20#16810378C2212yes—
21#16810378C2212yes—
22C53×#389C2212yes—
23C53×#389C2212yes—
24C53×#387—22no—
25C53×#387—22no—
26C53×#382C2211yescharacteristic
27C53×#382C2211yescharacteristic
28C43×C53C2211yescharacteristic
29C22×#88322C2314yes—
30#13133209—28no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53C2312yes—
34C2×C4⋊Dic53C2312yes—
35C2×#11719919—22no—
36C2×#11719919—22no—
37C4×#88322—24no—
38#13133124—24no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53C2312yes—
42C2×C4⋊C4×C53C2312yes—
43C2×C42×C53D412yes—
44C2×C42×C53D412yes—
45C2×C42×C53C2311yescharacteristic
46C2×C42×C53C2311yescharacteristic
47C2×C42×C53C2311yescharacteristic
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—22no—
53#95×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53C2×D412yes—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D412yes—
60C22×C4×C53C2411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C4⋊C4×C53—24no—
64C4⋊C4×C53—24no—
65C4⋊C4×C53—44no—
66C4⋊C4×C53—44no—
67C42×C53—24no—
68C42×C53—22no—
69C42×C53—22no—
70C42×C53—42no—
71C42×C53#4114yes—
72C4⋊Dic53—24no—
73C4⋊Dic53—24no—
74C4⋊Dic53—44no—
75C4⋊Dic53—44no—
76#11719919—44no—
77#11719919—44no—
78C2×#88322—24no—
79C2×#88322—24no—
80C2×#88322—24no—
81C23×C53C22×D411yescharacteristic
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53#12412yes—
96C2×C4×C53#12412yes—
97C2×C4×C53#13514yes—
98C2×C4×C53C2×#4112yes—
99C2×C4×C532−1+412yes—
100C2×C4×C532−1+412yes—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—44no—
104C2×Dic53—44no—
105#88322—44no—
106#30231—531nomaximal, Sylow
107C22×C53C2×#12411yescharacteristic
108C22×C53#53611yescharacteristic
109C22×C53#53611yescharacteristic
110C22×C53C2×#13512yes—
111C22×C53#57512yes—
112C4×C53—22no—
113C4×C53—22no—
114C4×C53—24no—
115C4×C53—42no—
116C4×C53—82no—
117C4×C53—82no—
118C4×C53#54114yes—
119Dic53—82no—
120Dic53—82no—
121#2864D5311yescharacteristic
122#2859—531no—
123#2101—534no—
124#2100—532no—
125#2099—532no—
126#1991—532no—
127#1991—532no—
128#1744—531no—
129C2×C53#264211yescharacteristic
130C2×C53#313111yescharacteristic
131C2×C53#313511yescharacteristic
132C2×C53#323212yes—
133C2×C53#323312yes—
134C2×#125—532no—
135C2×C4×C8—532no—
136#389—1062no—
137#389—1062no—
138#389C2×D5312yes—
139#389C2×D5312yes—
140#387—22no—
141#387—22no—
142#384—531no—
143#384—531no—
144#382—532no—
145#382C2×D5311yescharacteristic
146#382C2×D5311yescharacteristic
147C43C2×D5311yescharacteristic
148#340—534no—
149#340—534no—
150#340—534no—
151#340—534no—
152#336—532no—
153#336—532no—
154C53#3023111yescharacteristic, complemented, Sylow
155C22×C8—534no—
156#125—1068no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4—1062no—
165C2×C4⋊C4C22×D5312yes—
166C2×C4⋊C4C22×D5312yes—
167C2×C42C22⋊D5312yes—
168C2×C42C22⋊D5312yes—
169C2×C42C22×D5311yescharacteristic
170C2×C42C22×D5311yescharacteristic
171C2×C42C22×D5311yescharacteristic
172C4×C8—1064no—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—22no—
178#95—22no—
179#95—1064no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—1062no—
183C22×C4—1062no—
184C22×C4C2×C22⋊D5312yes—
185C22×C4C2×C22⋊D5311yescharacteristic
186C22×C4C2×C22⋊D5311yescharacteristic
187C22×C4C2×C22⋊D5312yes—
188C22×C4C23×D5311yescharacteristic
189C2×C8—1064no—
190C2×C8—1064no—
191C2×C8—1064no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—1064no—
197C4⋊C4—1064no—
198C4⋊C4—2124no—
199C4⋊C4—2124no—
200C4⋊C4—2124no—
201C4⋊C4—2124no—
202C42—24no—
203C42—22no—
204C42—22no—
205C42—42no—
206C42#1171994914yes—
207C23C22×C22⋊D5311yescharacteristic
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—2122no—
222C2×C4—2122no—
223C2×C4—2124no—
224C2×C4—2124no—
225C2×C4#1313320612yes—
226C2×C4#1313320612yes—
227C2×C4#1313323714yes—
228C2×C4C2×#1171994912yes—
229C2×C4#1313326412yes—
230C2×C4#1313326412yes—
231C8—2124no—
232C22C2×#1313320611yescharacteristic
233C22#1681119411yescharacteristic
234C22#1681119411yescharacteristic
235C22C2×#1313323712yes—
236C22#1681142112yes—
237C4—22no—
238C4—22no—
239C4—24no—
240C4—42no—
241C4—82no—
242C4—82no—
243C4—4242no—
244C4—4242no—
245C4#1681123614yes—
246C2#2028066611yescharacteristic
247C2#2028504011yescharacteristic
248C2#2028506311yescharacteristic
249C2#2028587912yes—
250C2#2028589012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2864C2
socle, prime radicalC23×C53C22×D4
hypercenter#2864D53
hypocenterC53#30231
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #2864

Complements

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

NH ≅ G/Ntype
154. C53#30231semidirect

Maximal subgroups

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

subgroupindex
#202830782
#202762832
#202762782
#202762732
#202756792
#202744612
C53×#28642
#3023153