Finite Group Databasebeta

← #28309806

Subgroups — #28309806

Order 13568.

280 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 64 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#20291088C211yesmaximal, characteristic, complemented
2#20291058C211yesmaximal, characteristic, complemented
3#20289735C212yesmaximal, complemented
4#20289726C212yesmaximal, complemented
5#20285344C211yesmaximal, characteristic
6C2×#13133291—21no—
7#16811546—21no—
8#16811546C2212yes—
9#16811546C2212yes—
10#16811519—21no—
11#16811515—21no—
12#16811228C411yescharacteristic
13#16811022C2211yescharacteristic
14#16810997C2211yescharacteristic
15#16810991C411yescharacteristic
16#16810989C411yescharacteristic
17#16810871—21no—
18#16810854—21no—
19#16810721—21no—
20C4×#11719924—21no—
21#16810579C2211yescharacteristic
22C53×#541C411yescharacteristic
23#13133291—22no—
24#13133291—22no—
25#13133291—41no—
26C2×#11719910—41no—
27C2×#11719924—21no—
28C2×#11719924—21no—
29C2×#11719917D411yescharacteristic
30C22×#88322D411yescharacteristic
31#13133214C2×C411yescharacteristic
32#13133209C2×C411yescharacteristic
33#13133202—41no—
34#13133191—21no—
35#13133190—41no—
36C4×D212—21no—
37#13133150—21no—
38#13133148—21no—
39#13133141—21no—
40#13133141—21no—
41#13133137—21no—
42#13133137—21no—
43#13133128C2×C411yescharacteristic
44#13133128D411yescharacteristic
45C4×#88322D411yescharacteristic
46C4×#88322C2311yescharacteristic
47C4⋊Q8×C53C2×C411yescharacteristic
48#126×C53—41no—
49#125×C53—41no—
50#125×C53C2×C411yescharacteristic
51C22⋊Q8×C53—21no—
52C4×Q8×C53—21no—
53#117×C53C2×C411yescharacteristic
54C2×Q8×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C2×C22⋊D53—41no—
57C2×D212—21no—
58C2×D212—41no—
59C2×C4×D53—41no—
60C4⋊C4×C53—21no—
61C4⋊C4×C53—21no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—21no—
67C22⋊C4×C53—41no—
68C42×C53—41no—
69C42×C53C22⋊C411yescharacteristic
70C42×C53C22×C411yescharacteristic
71#11719924—42no—
72#11719924—41no—
73#11719924—41no—
74#11719921—41no—
75#11719921—41no—
76C4⋊Dic53—21no—
77#11719919—41no—
78#11719917—22no—
79C2×#88322—21no—
80C2×#88322—21no—
81C2×#88322C22⋊C411yescharacteristic
82C2×#88322C22⋊C411yescharacteristic
83C2×#88322C2×D411yescharacteristic
84C2×#88322C2×D411yescharacteristic
85#11719910—42no—
86#11719910—42no—
87C23×C53#10011yescharacteristic
88C22×D53—41no—
89Q8×C53—41no—
90Q8×C53—41no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53#10011yescharacteristic
98C2×C4×C53C2×C22⋊C411yescharacteristic
99C22⋊D53—161no—
100C2×Dic53—41no—
101D212—41no—
102D212—41no—
103D212—161no—
104C4×D53—161no—
105#88322—21no—
106#88322—21no—
107#88322#13512yes—
108#88322AGL(1,ℤ/8)11yescharacteristic
109#88322#13711yescharacteristic
110#11991—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—41no—
113C22×C53C2×#10011yescharacteristic
114C2×D53—81no—
115C2×D53—81no—
116C2×D53—81no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53—41no—
120C4×C53—81no—
121C4×C53—81no—
122C4×C53#41611yescharacteristic
123C4×C53#41711yescharacteristic
124Dic53—81no—
125#3174—531no—
126#2166—531no—
127#2153—531no—
128#1366—532no—
129#1359—532no—
130C2×C53—21no—
131C2×C53—41no—
132C2×C53#155111yescharacteristic
133D53—161no—
134#541#615711yescharacteristic
135#540—531no—
136C2×M32—1061no—
137#499—531no—
138#492—531no—
139#491—531no—
140#490—531no—
141#468—1061no—
142#465—1061no—
143#440—1061no—
144C4×SD16—1061no—
145#405—531no—
146#364—532no—
147#364—532no—
148#364—1061no—
149#349—1061no—
150#347—1061no—
151C53#1199111yescharacteristic, complemented, Sylow
152C2×SD16—1061no—
153C2×SD16—1061no—
154C2×M16—531no—
155C22×C8—531no—
156C4⋊Q8C2×#615711yescharacteristic
157#127—531no—
158#126—41no—
159#125—41no—
160#125—531no—
161#125C2×#615711yescharacteristic
162#123—2121no—
163C22⋊Q8—21no—
164#121—1061no—
165#121—2121no—
166C4×Q8—21no—
167C4×D4—1061no—
168#117C2×#615711yescharacteristic
169M32—1062no—
170M32—1062no—
171M32—2121no—
172C2×C16—2121no—
173#107—1061no—
174#106—1061no—
175#103—1061no—
176#103—1061no—
177#102—1061no—
178#102—1061no—
179#97—531no—
180#97—531no—
181C4×C8—531no—
182C4×C8—531no—
183C2×Q8—21no—
184C2×D4—1061no—
185C2×D4—2121no—
186C2×D4—2121no—
187C22×C4—2121no—
188C22×C4C22⋊#615711yescharacteristic
189SD16—2122no—
190SD16—2121no—
191SD16—2121no—
192M16—1062no—
193C2×C8—531no—
194C2×C8—531no—
195C2×C8—531no—
196C2×C8—531no—
197C2×C8—1061no—
198C2×C8—1061no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—1061no—
206C4⋊C4—2121no—
207C22⋊C4—21no—
208C22⋊C4—41no—
209C22⋊C4—2121no—
210C22⋊C4—2121no—
211C42—41no—
212C42C22⋊#615711yescharacteristic
213C42C22×#615711yescharacteristic
214C16—2122no—
215C16—2122no—
216C23—2121no—
217C23#1313327711yescharacteristic
218Q8—41no—
219Q8—41no—
220D4—8481no—
221D4—8481no—
222D4—2121no—
223D4—2121no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—8481no—
231C2×C4—2121no—
232C2×C4#1313327711yescharacteristic
233C2×C4C2×C22⋊#615711yescharacteristic
234C8—532no—
235C8—531no—
236C8—531no—
237C8—1061no—
238C8—1061no—
239C22—21no—
240C22—41no—
241C22—4241no—
242C22—4241no—
243C22—4241no—
244C22C2×#1313327711yescharacteristic
245C4—21no—
246C4—41no—
247C4—41no—
248C4—81no—
249C4—81no—
250C4—4241no—
251C4#1681161311yescharacteristic
252C4#1681161411yescharacteristic
253C2—21no—
254C2—41no—
255C2—8481no—
256C2#2029023411yescharacteristic
subgroupG/H
centerC2#20290234
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#541C4
socleC2×C53#1551
prime radical#16811228C4
cosocleC4×#88322C23
hypercenter#541#6157
hypocenterC53#11991
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #20291088C2semidirect
2. #20291058C2semidirect
3. #20289735C2semidirect
4. #20289726C2semidirect
151. C53#11991semidirect

Maximal subgroups

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

subgroupindex
#202910882
#202910582
#202897352
#202897262
#202853442
#1199153