Finite Group Databasebeta

← #27281629

Subgroups — #27281629

Order 13568.

534 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285089C211yesmaximal, characteristic
2#20273429C211yesmaximal, characteristic
3#20273265C212yesmaximal, complemented
4#20272579C212yesmaximal, complemented
5C53×#2380C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810987—22no—
8#16810969C414yescomplemented
9#16810680C2212yes—
10#16810628—22no—
11C4×#11719917C2211yescharacteristic
12#16810367—22no—
13#16810359—24no—
14C2×#127×C53C2211yescharacteristic
15C53×#386—22no—
16C53×#386—24no—
17C53×#383C2212yes—
18C2×#11719917—21no—
19#13133217C2×C412yescomplemented
20#13133209—21no—
21#13133196—22no—
22C4×Dic106—22no—
23#13133176—21no—
24C4⋊#88322—22no—
25#13133138—28no—
26C22⋊#88322—22no—
27#13133128—21no—
28#13133128—22no—
29C4×#88322C2×C412yescomplemented
30C22×D4×C53—22no—
31C22×D4×C53D412yes—
32C23×C4×C53—22no—
33#127×C53—22no—
34#127×C53C2×C412yes—
35#127×C53D412yes—
36C2×C22⋊C4×C53—24no—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53C2311yescharacteristic
39#95×C53—22no—
40#95×C53—22no—
41#95×C53D414yes—
42C24×C53—22no—
43C2×D4×C53—24no—
44C2×D4×C53—24no—
45C2×D4×C53—22no—
46C2×D4×C53—42no—
47C2×D4×C53—42no—
48C2×D4×C53—82no—
49C2×D4×C53—82no—
50C2×D4×C53C22⋊C414yes—
51C22×C4×C53—24no—
52C22×C4×C53—84no—
53C22×C4×C53—82no—
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D412yes—
56C2×Dic106—22no—
57C22⋊C4×C53—84no—
58C22⋊C4×C53—82no—
59C42×C53C22×C411yescharacteristic
60C42×C53C2×D411yescharacteristic
61C42×C53#4112yes—
62C22⋊Dic53—41no—
63C4⋊Dic53—22no—
64C4⋊Dic53—22no—
65#11719919—42no—
66C4×Dic53—41no—
67#11719917—42no—
68C2×#88322—22no—
69C2×#88322—42no—
70C23×C53—24no—
71C23×C53—44no—
72C23×C53—44no—
73C23×C53—42no—
74C23×C53—42no—
75C23×C53#12011yescharacteristic
76D4×C53—44no—
77D4×C53—82no—
78D4×C53—84no—
79D4×C53—84no—
80C2×C4×C53—22no—
81C2×C4×C53—42no—
82C2×C4×C53—48no—
83C2×C4×C53—44no—
84C2×C4×C53—88no—
85C2×C4×C53C2×C22⋊C411yescharacteristic
86C2×C4×C53C4×D412yes—
87C2×C4×C53#12112yes—
88C2×C4×C53#12311yescharacteristic
89C2×Dic53—42no—
90Dic106—82no—
91#88322—42no—
92#88322—82no—
93#9425—531nomaximal, Sylow
94C22×C53—22no—
95C22×C53—44no—
96C22×C53—48no—
97C22×C53—82no—
98C22×C53—84no—
99C22×C53—82no—
100C22×C53#38611yescharacteristic
101C22×C53#45512yes—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—84no—
107Dic53—82no—
108#3138—531no—
109#2380D5311yescharacteristic
110#1555—531no—
111#1515—532no—
112#1288—532no—
113C2×C53—22no—
114C2×C53—84no—
115C2×C53#188511yescharacteristic
116C2×C53#190412yes—
117#533—531no—
118C2×#127C2×D5311yescharacteristic
119#490—1062no—
120#486—534no—
121#432—532no—
122#418—1062no—
123C4×M16—531no—
124#386—22no—
125#386—24no—
126#383C2×D5312yes—
127#331—1062no—
128#328—1064no—
129C53#942511yescharacteristic, complemented, Sylow
130C22×D4—22no—
131C22×D4C22⋊D5312yes—
132C23×C4—22no—
133C2×M16—1061no—
134C4⋊Q8—532no—
135#127—22no—
136#127C4×D5312yes—
137#127D21212yes—
138#125—1061no—
139C22⋊Q8—1062no—
140C4×Q8—1062no—
141#117—1061no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—1062no—
146#102—1068no—
147C22⋊C8—1062no—
148#97—1061no—
149#97—1062no—
150C4×C8—532no—
151#95—22no—
152#95—22no—
153#95C22⋊D5314yes—
154C24—22no—
155C2×Q8—1062no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4#1171992114yes—
164C22×C4—24no—
165C22×C4—84no—
166C22×C4—82no—
167C22×C4C2×C22⋊D5311yescharacteristic
168C22×C4C2×C22⋊D5312yes—
169M16—2122no—
170C2×C8—1062no—
171C2×C8—2122no—
172C4⋊C4—1062no—
173C4⋊C4—1062no—
174C4⋊C4—2122no—
175C22⋊C4—84no—
176C22⋊C4—82no—
177C22⋊C4—2121no—
178C42—2121no—
179C42C2×C4×D5311yescharacteristic
180C42C2×D21211yescharacteristic
181C42#1171994512yes—
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1313318611yescharacteristic
188Q8—4242no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—48no—
196C2×C4—44no—
197C2×C4—88no—
198C2×C4—2122no—
199C2×C4C2×#1171992111yescharacteristic
200C2×C4C4×C22⋊D5312yes—
201C2×C4#1313319112yes—
202C2×C4#1313320011yescharacteristic
203C8—4242no—
204C8—2122no—
205C22—22no—
206C22—48no—
207C22—44no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1681050211yescharacteristic
212C22#1681080112yes—
213C4—22no—
214C4—22no—
215C4—21no—
216C4—21no—
217C4—84no—
218C4—4242no—
219C2—22no—
220C2—84no—
221C2#2027510311yescharacteristic
222C2#2027520212yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2380C2
socleC22×C53#386
prime radicalC2×#127×C53C22
cosocleC2×C42×C53C23
hypercenter#2380D53
hypocenterC53#9425
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2380

Complements

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

NH ≅ G/Ntype
3. #20273265C2semidirect
4. #20272579C2semidirect
8. #16810969C4semidirect
19. #13133217C2×C4semidirect
29. C4×#88322C2×C4semidirect
129. C53#9425semidirect

Maximal subgroups

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

subgroupindex
#202850892
#202734292
#202732652
#202725792
C53×#23802
#942553