Finite Group Databasebeta

← #27469706

Subgroups — #27469706

Order 13568.

810 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 211 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#20283025C211yesmaximal, characteristic
2#20276272C214yesmaximal, complemented
3#20275683C212yesmaximal, complemented
4#20274850C212yesmaximal, complemented
5#20274805C214yesmaximal
6#20274477C211yesmaximal, characteristic
7C2×C53×#384C211yesmaximal, characteristic
8C2×#13133209C2211yescharacteristic
9C2×C22⋊#88322C2212yes—
10C2×#13133128C2212yes—
11#16810521C2212yes—
12#16810507—22no—
13#16810496—22no—
14#16810489C2211yescharacteristic
15#16810483C2212yes—
16#16810385C2218yes—
17#16810385C2218yes—
18#16810378C2212yes—
19C22×C4⋊C4×C53—22no—
20C22×C4⋊C4×C53C2212yes—
21C22×C42×C53C2211yescharacteristic
22C53×#384—22no—
23C53×#384C2212yes—
24C53×#384C2212yes—
25C22×#88322C2314yes—
26#13133209—24no—
27C2×C4⋊Dic53C2312yes—
28C2×C4⋊Dic53C2312yes—
29C2×#11719919—22no—
30C2×#11719919—22no—
31C2×C22⋊Dic53—22no—
32C22⋊#88322—24no—
33#13133128—24no—
34#13133124—22no—
35#13133124—22no—
36#13133124—22no—
37C23×C4×C53—22no—
38C23×C4×C53D412yes—
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—24no—
41C2×C4⋊C4×C53—24no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—28no—
44C2×C4⋊C4×C53—48no—
45C2×C4⋊C4×C53C2314yes—
46C2×C42×C53—24no—
47C2×C42×C53D412yes—
48C2×C42×C53C2311yescharacteristic
49C2×C42×C53C2311yescharacteristic
50C24×C53C2×D411yescharacteristic
51C22×C4×C53—24no—
52C22×C4×C53—24no—
53C22×C4×C53—24no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—24no—
57C22×C4×C53—48no—
58C22×C4×C53—44no—
59C22×C4×C53C2×D412yes—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D412yes—
62C22×C4×C53#4114yes—
63C22×C4×C53C2411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C4⋊C4×C53—28no—
67C4⋊C4×C53—48no—
68C4⋊C4×C53—44no—
69C4⋊C4×C53—48no—
70C42×C53—24no—
71C42×C53—22no—
72C42×C53#4114yes—
73C22⋊Dic53—44no—
74C4⋊Dic53—24no—
75C4⋊Dic53—24no—
76#11719919—44no—
77#11719919—44no—
78C2×#88322—24no—
79C2×#88322—24no—
80C2×#88322—24no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—22no—
84C23×C53#12412yes—
85C23×C53#13714yes—
86C23×C53C22×D411yescharacteristic
87C2×C4×C53—216no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—28no—
95C2×C4×C53—44no—
96C2×C4×C53—48no—
97C2×C4×C53—88no—
98C2×C4×C53#12314yes—
99C2×C4×C53#12412yes—
100C2×C4×C53AGL(1,ℤ/8)14yes—
101C2×C4×C53C2×#4112yes—
102C2×C4×C53C2×#4112yes—
103C2×C4×C532−1+412yes—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110#88322—44no—
111#27965—531nomaximal, Sylow
112C22×C53—24no—
113C22×C53—22no—
114C22×C53—24no—
115C22×C53—22no—
116C22×C53#49114yes—
117C22×C53C2×#12311yescharacteristic
118C22×C53C2×#12411yescharacteristic
119C22×C53#53611yescharacteristic
120C22×C53C2×AGL(1,ℤ/8)12yes—
121C22×C53C2×#13712yes—
122C4×C53—24no—
123C4×C53—24no—
124C4×C53—22no—
125C4×C53—84no—
126C4×C53#55214yes—
127Dic53—82no—
128Dic53—82no—
129#2851—531no—
130C2×#384D5311yescharacteristic
131#2099—534no—
132#1992—532no—
133#1836—532no—
134#1824—534no—
135#1749—531no—
136C2×C53—24no—
137C2×C53#250311yescharacteristic
138C2×C53#309212yes—
139C2×C53C2×#49111yescharacteristic
140C2×C53#313811yescharacteristic
141C2×C53#323912yes—
142C2×#125—531no—
143C22×C4⋊C4—22no—
144C22×C4⋊C4C2×D5312yes—
145C22×C42C2×D5311yescharacteristic
146C2×C22⋊C8—532no—
147C2×#97—532no—
148#389—532no—
149#387—1062no—
150#385—1062no—
151#384—22no—
152#384C2×D5312yes—
153#384C2×D5312yes—
154#382—531no—
155#380—532no—
156#340—538no—
157#340—538no—
158#336—532no—
159C53#2796511yescharacteristic, complemented, Sylow
160C23×C4—22no—
161C23×C4C22⋊D5312yes—
162C23×C4C22×D5311yescharacteristic
163C22×C8—534no—
164#125—1064no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4C22×D5314yes—
175C2×C22⋊C4—1062no—
176C2×C42—24no—
177C2×C42C22⋊D5312yes—
178C2×C42C22×D5311yescharacteristic
179C2×C42C22×D5311yescharacteristic
180C22⋊C8—1064no—
181#97—1064no—
182#95—1062no—
183#95—1062no—
184#95—1062no—
185C24C2×C22⋊D5311yescharacteristic
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—48no—
193C22×C4—44no—
194C22×C4—1062no—
195C22×C4—1062no—
196C22×C4#1171994714yes—
197C22×C4C2×C22⋊D5312yes—
198C22×C4C2×C22⋊D5311yescharacteristic
199C22×C4C2×C22⋊D5312yes—
200C22×C4C23×D5311yescharacteristic
201C2×C8—1064no—
202C2×C8—1064no—
203C2×C8—1064no—
204C4⋊C4—28no—
205C4⋊C4—48no—
206C4⋊C4—44no—
207C4⋊C4—48no—
208C4⋊C4—1064no—
209C4⋊C4—1064no—
210C4⋊C4—2124no—
211C4⋊C4—2124no—
212C22⋊C4—2124no—
213C42—24no—
214C42—22no—
215C42#1171994514yes—
216C23—21no—
217C23—22no—
218C23—21no—
219C23#1313320412yes—
220C23#1313324814yes—
221C23C22×C22⋊D5311yescharacteristic
222C2×C4—216no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—28no—
230C2×C4—44no—
231C2×C4—48no—
232C2×C4—88no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4#1313320014yes—
240C2×C4#1313320412yes—
241C2×C4#1313324314yes—
242C2×C4C2×#1171994712yes—
243C2×C4C2×#1171994512yes—
244C2×C4#1313326512yes—
245C8—2124no—
246C22—22no—
247C22—22no—
248C22—24no—
249C22—24no—
250C22#1681099114yes—
251C22C2×#1313320011yescharacteristic
252C22C2×#1313320411yescharacteristic
253C22#1681119711yescharacteristic
254C22C2×#1313324312yes—
255C22C2×#1313324812yes—
256C4—22no—
257C4—24no—
258C4—24no—
259C4—84no—
260C4—4242no—
261C4—4242no—
262C4#1681130814yes—
263C2—24no—
264C2#2027919611yescharacteristic
265C2#2028463812yes—
266C2C2×#1681099111yescharacteristic
267C2#2028508911yescharacteristic
268C2#2028594512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#384C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#384D53
hypocenterC53#27965
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. #20276272C2semidirect
3. #20275683C2semidirect
4. #20274850C2semidirect
159. C53#27965semidirect

Maximal subgroups

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

subgroupindex
#202830252
#202762722
#202756832
#202748502
#202748052
#202744772
C2×C53×#3842
#2796553