Finite Group Databasebeta

← #28309796

Subgroups — #28309796

Order 13568.

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

normal subgroups only

This group has 292 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#20289723C212yesmaximal, complemented
5#20285359C211yesmaximal, characteristic
6C2×#13133291—21no—
7#16811546—21no—
8#16811546C2212yes—
9#16811546C2212yes—
10#16811519—21no—
11#16811515—21no—
12#16811228C411yescharacteristic
13#16811022C2211yescharacteristic
14#16810999C2211yescharacteristic
15#16810989C411yescharacteristic
16#16810987C411yescharacteristic
17#16810874—21no—
18#16810857—21no—
19#16810733—21no—
20C4×#11719922—21no—
21#16810579C2211yescharacteristic
22C53×#540C411yescharacteristic
23#13133291—22no—
24#13133291—22no—
25#13133291—41no—
26C2×#11719910—41no—
27C2×#11719922—21no—
28C2×#11719922—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#13133138—21no—
40#13133138—21no—
41#13133137—21no—
42#13133137—21no—
43#13133128C2×C411yescharacteristic
44#13133128D411yescharacteristic
45C4×#88322D411yescharacteristic
46C4×#88322C2311yescharacteristic
47#127×C53C2×C411yescharacteristic
48#125×C53C2×C411yescharacteristic
49#123×C53—41no—
50#121×C53—21no—
51#121×C53—41no—
52C4×D4×C53—21no—
53#117×C53C2×C411yescharacteristic
54C2×D4×C53—21no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53C22⋊C411yescharacteristic
59C2×C22⋊D53—41no—
60C2×D212—21no—
61C2×D212—41no—
62C2×C4×D53—41no—
63C4⋊C4×C53—21no—
64C4⋊C4×C53—21no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—21no—
67C22⋊C4×C53—41no—
68C22⋊C4×C53—41no—
69C42×C53C22⋊C411yescharacteristic
70C42×C53C22×C411yescharacteristic
71#11719922—42no—
72#11719922—41no—
73#11719922—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—41no—
88C23×C53#10011yescharacteristic
89C22×D53—41no—
90D4×C53—41no—
91D4×C53—41no—
92D4×C53—161no—
93D4×C53—161no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—41no—
99C2×C4×C53—161no—
100C2×C4×C53#10011yescharacteristic
101C2×C4×C53C2×C22⋊C411yescharacteristic
102C22⋊D53—161no—
103C2×Dic53—41no—
104D212—41no—
105D212—41no—
106D212—161no—
107C4×D53—161no—
108#88322—21no—
109#88322—21no—
110#88322#13512yes—
111#88322AGL(1,ℤ/8)11yescharacteristic
112#88322#13711yescharacteristic
113#11989—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—41no—
116C22×C53—81no—
117C22×C53—81no—
118C22×C53—81no—
119C22×C53C2×#10011yescharacteristic
120C2×D53—81no—
121C2×D53—81no—
122C2×D53—81no—
123C4×C53—21no—
124C4×C53—41no—
125C4×C53—41no—
126C4×C53—81no—
127C4×C53#41611yescharacteristic
128C4×C53#41711yescharacteristic
129Dic53—81no—
130#3176—531no—
131#2166—531no—
132#2153—531no—
133#1366—532no—
134#1357—532no—
135C2×C53—21no—
136C2×C53—41no—
137C2×C53—161no—
138C2×C53#155111yescharacteristic
139D53—161no—
140#540—531no—
141#540#615711yescharacteristic
142C2×M32—1061no—
143#499—531no—
144#493—531no—
145#490—531no—
146#490—531no—
147#469—1061no—
148#466—1061no—
149#442—1061no—
150C4×D8—1061no—
151#405—531no—
152#364—532no—
153#364—532no—
154#364—1061no—
155#349—1061no—
156#347—1061no—
157C53#1198911yescharacteristic, complemented, Sylow
158C2×D8—1061no—
159C2×D8—1061no—
160C2×M16—531no—
161C22×C8—531no—
162#127—531no—
163#127C2×#615711yescharacteristic
164#125—531no—
165#125C2×#615711yescharacteristic
166#123—41no—
167#123—2121no—
168#121—21no—
169#121—41no—
170#121—1061no—
171#121—2121no—
172C4×D4—21no—
173C4×D4—1061no—
174#117C2×#615711yescharacteristic
175M32—1062no—
176M32—1062no—
177M32—2121no—
178C2×C16—2121no—
179#107—1061no—
180#106—1061no—
181#102—1061no—
182#102—1061no—
183#102—1061no—
184#102—1061no—
185#97—531no—
186#97—531no—
187C4×C8—531no—
188C4×C8—531no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—1061no—
193C2×D4—2121no—
194C2×D4—2121no—
195C22×C4—41no—
196C22×C4—2121no—
197C22×C4C22⋊#615711yescharacteristic
198D8—2122no—
199D8—2121no—
200D8—2121no—
201M16—1062no—
202C2×C8—531no—
203C2×C8—531no—
204C2×C8—531no—
205C2×C8—531no—
206C2×C8—1061no—
207C2×C8—1061no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—41no—
211C4⋊C4—1061no—
212C4⋊C4—2121no—
213C22⋊C4—21no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—2121no—
217C22⋊C4—2121no—
218C42C22⋊#615711yescharacteristic
219C42C22×#615711yescharacteristic
220C16—2122no—
221C16—2122no—
222C23—41no—
223C23—2121no—
224C23#1313327711yescharacteristic
225D4—41no—
226D4—41no—
227D4—161no—
228D4—161no—
229D4—8481no—
230D4—8481no—
231D4—2121no—
232D4—2121no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—161no—
239C2×C4—8481no—
240C2×C4—2121no—
241C2×C4#1313327711yescharacteristic
242C2×C4C2×C22⋊#615711yescharacteristic
243C8—532no—
244C8—531no—
245C8—531no—
246C8—1061no—
247C8—1061no—
248C22—21no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22—4241no—
254C22—4241no—
255C22—4241no—
256C22C2×#1313327711yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—81no—
261C4—4241no—
262C4#1681161311yescharacteristic
263C4#1681161411yescharacteristic
264C2—21no—
265C2—41no—
266C2—161no—
267C2—8481no—
268C2#2029023411yescharacteristic
subgroupG/H
centerC2#20290234
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#540C4
socleC2×C53#1551
prime radical#20285359C2
cosocleC4×#88322C23
hypercenter#540#6157
hypocenterC53#11989
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20291088C2semidirect
2. #20291058C2semidirect
3. #20289735C2semidirect
4. #20289723C2semidirect
157. C53#11989semidirect

Maximal subgroups

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

subgroupindex
#202910882
#202910582
#202897352
#202897232
#202853592
#1198953