Finite Group Databasebeta

← #27496185

Subgroups — #27496185

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#20283102C211yesmaximal, characteristic
2#20283078C211yesmaximal, characteristic
3#20276284C214yesmaximal
4#20276279C212yesmaximal
5#20276274C212yesmaximal
6#20275680C212yesmaximal
7#20275680C212yesmaximal
8C53×#1744C211yesmaximal, characteristic
9C2×#13133209C2212yes—
10#16810521—22no—
11#16810521—22no—
12#16810521C2212yes—
13#16810521C2212yes—
14#16810507—22no—
15#16810507—22no—
16#16810493C2211yescharacteristic
17#16810493C2211yescharacteristic
18#16810489C2212yes—
19#16810489C2211yescharacteristic
20#16810489C2211yescharacteristic
21#16810386C2214yes—
22#16810386C2214yes—
23#16810386C2214yes—
24#16810386C2214yes—
25C2×C4×C8×C53C2212yes—
26C43×C53C2211yescharacteristic
27C53×#336C2212yes—
28C53×#336C2212yes—
29#13133209—28no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53C2312yes—
33C2×C4⋊Dic53C2312yes—
34C2×C4⋊Dic53C2312yes—
35C2×C4⋊Dic53C2312yes—
36C2×#11719919—22no—
37C2×#11719919—22no—
38C2×#11719919—22no—
39C2×#11719919—22no—
40#13133124—22no—
41#13133124—22no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—24no—
47C22×C8×C53C2314yes—
48C2×C42×C53D412yes—
49C2×C42×C53D412yes—
50C2×C42×C53C2311yescharacteristic
51C2×C42×C53C2311yescharacteristic
52C2×C42×C53C2311yescharacteristic
53C4×C8×C53—24no—
54C22×C4×C53C2×D412yes—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C22×C4×C53C2411yescharacteristic
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C2×C8×C53—24no—
64C2×C8×C53—24no—
65C2×C8×C53—24no—
66C42×C53—24no—
67C42×C53—22no—
68C42×C53—22no—
69C42×C53—42no—
70C42×C53#4114yes—
71C4⋊Dic53—24no—
72C4⋊Dic53—24no—
73C4⋊Dic53—24no—
74C4⋊Dic53—24no—
75C4⋊Dic53—44no—
76C4⋊Dic53—44no—
77#11719919—44no—
78#11719919—44no—
79#11719919—44no—
80#11719919—44no—
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#12412yes—
90C2×C4×C53#12412yes—
91C2×C4×C53#13514yes—
92C2×C4×C53C2×#4112yes—
93C2×C4×C532−1+412yes—
94C2×C4×C532−1+412yes—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—44no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—44no—
105C8×C53—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#54114yes—
117Dic53—82no—
118Dic53—82no—
119Dic53—82no—
120Dic53—82no—
121#2864—531no—
122#2859—531no—
123#2101—534no—
124#2100—532no—
125#2099—532no—
126#1991—532no—
127#1991—532no—
128#1744D5311yescharacteristic
129C2×C53#264211yescharacteristic
130C2×C53#313111yescharacteristic
131C2×C53#313511yescharacteristic
132C2×C53#323212yes—
133C2×C53#323312yes—
134C2×#125—532no—
135C2×C4×C8C2×D5312yes—
136#389—532no—
137#389—532no—
138#389—1062no—
139#389—1062no—
140#387—1062no—
141#387—1062no—
142#384—531no—
143#384—531no—
144#382—532no—
145#382—531no—
146#382—531no—
147C43C2×D5311yescharacteristic
148#340—534no—
149#340—534no—
150#340—534no—
151#340—534no—
152#336C2×D5312yes—
153#336C2×D5312yes—
154C53#3023111yescharacteristic, complemented, Sylow
155C22×C8C22×D5314yes—
156#125—1068no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—532no—
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⋊C4—1062no—
166C2×C4⋊C4—1062no—
167C2×C42D21212yes—
168C2×C42D21212yes—
169C2×C42C22×D5311yescharacteristic
170C2×C42C22×D5311yescharacteristic
171C2×C42C22×D5311yescharacteristic
172C4×C8—24no—
173#95—1061no—
174#95—1061no—
175#95—1061no—
176#95—1061no—
177#95—1064no—
178#95—1062no—
179#95—1062no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4—1062no—
183C22×C4—1062no—
184C22×C4C2×D21212yes—
185C22×C4C2×D21211yescharacteristic
186C22×C4C2×D21211yescharacteristic
187C22×C4C2×D21212yes—
188C22×C4C23×D5311yescharacteristic
189C2×C8—24no—
190C2×C8—24no—
191C2×C8—24no—
192C4⋊C4—1064no—
193C4⋊C4—1064no—
194C4⋊C4—1064no—
195C4⋊C4—1064no—
196C4⋊C4—2124no—
197C4⋊C4—2124no—
198C4⋊C4—2124no—
199C4⋊C4—2124no—
200C4⋊C4—2124no—
201C4⋊C4—2124no—
202C42—24no—
203C42—22no—
204C42—22no—
205C42—42no—
206C42#1171994514yes—
207C23C22×D21211yescharacteristic
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—2122no—
216C2×C4—2122no—
217C2×C4—2124no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4—2124no—
221C2×C4—2122no—
222C2×C4—2122no—
223C2×C4—2122no—
224C2×C4—2122no—
225C2×C4#1313320512yes—
226C2×C4#1313320512yes—
227C2×C4#1313323914yes—
228C2×C4C2×#1171994512yes—
229C2×C4#1313326412yes—
230C2×C4#1313326412yes—
231C8—44no—
232C22C2×#1313320511yescharacteristic
233C22#1681119911yescharacteristic
234C22#1681119911yescharacteristic
235C22C2×#1313323912yes—
236C22#1681142312yes—
237C4—22no—
238C4—22no—
239C4—24no—
240C4—42no—
241C4—4242no—
242C4—4242no—
243C4—4242no—
244C4—4242no—
245C4#1681123514yes—
246C2#2028066311yescharacteristic
247C2#2028503811yescharacteristic
248C2#2028506111yescharacteristic
249C2#2028588512yes—
250C2#2028589612yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1744C2
socle, prime radicalC23×C53C22×D4
hypercenter#1744D53
hypocenterC53#30231
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #1744

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
#202831022
#202830782
#202762842
#202762792
#202762742
#202756802
C53×#17442
#3023153