Finite Group Databasebeta

← #27281627

Subgroups — #27281627

Order 13568.

534 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 95 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#20277892C211yesmaximal, characteristic
3#20273267C212yesmaximal, complemented
4#20272581C212yesmaximal, complemented
5C53×#1555C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810987—22no—
8#16810969C414yescomplemented
9#16810628—22no—
10C4×#11719917C2211yescharacteristic
11#16810503—24no—
12#16810503—22no—
13#16810491C2212yes—
14#16810361—24no—
15C2×#127×C53C2211yescharacteristic
16C53×#432C2212yes—
17C53×#331—22no—
18C23×Dic53—22no—
19C2×#11719917—21no—
20#13133217C2×C412yescomplemented
21#13133209—21no—
22#13133196—22no—
23C4×Dic106—22no—
24#13133176—21no—
25C2×C22⋊Dic53—24no—
26C2×C22⋊Dic53—22no—
27#13133138—28no—
28#13133128—21no—
29C4×#88322C2×C412yescomplemented
30#13133124—22no—
31#13133124—22no—
32#13133124D414yes—
33C22×D4×C53—22no—
34C22×D4×C53D412yes—
35#127×C53—22no—
36#127×C53C2×C412yes—
37#127×C53D412yes—
38C2×C42×C53C2311yescharacteristic
39C4⋊C8×C53—22no—
40C22⋊C8×C53—22no—
41#97×C53—22no—
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×C53C2×D411yescharacteristic
52C22×C4×C53C2×D412yes—
53C22×Dic53—24no—
54C22×Dic53—84no—
55C22×Dic53—82no—
56C2×Dic106—22no—
57C2×C8×C53—42no—
58C42×C53C22×C411yescharacteristic
59C42×C53C2×D411yescharacteristic
60C42×C53#4112yes—
61C22⋊Dic53—41no—
62C22⋊Dic53—84no—
63C22⋊Dic53—82no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66#11719919—42no—
67C4×Dic53—41no—
68#11719917—42no—
69C2×#88322—22no—
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×C53C2×C22⋊C411yescharacteristic
83C2×C4×C53C4×D412yes—
84C2×C4×C53#12112yes—
85C2×C4×C53#12311yescharacteristic
86C2×Dic53—48no—
87C2×Dic53—42no—
88C2×Dic53—44no—
89C2×Dic53—88no—
90Dic106—82no—
91C8×C53—82no—
92#88322—42no—
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—
106Dic53—84no—
107Dic53—82no—
108#3138—531no—
109#2380—531no—
110#1555D5311yescharacteristic
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#432C2×D5312yes—
122#418—1062no—
123C4×M16—531no—
124#386—1064no—
125#386—1062no—
126#383—532no—
127#331—22no—
128#328—1064no—
129C53#942511yescharacteristic, complemented, Sylow
130C22×D4—22no—
131C22×D4D21212yes—
132C23×C4—1062no—
133C2×M16—1061no—
134C4⋊Q8—532no—
135#127—22no—
136#127C4×D5312yes—
137#127C22⋊D5312yes—
138#125—1061no—
139C22⋊Q8—1062no—
140C4×Q8—1062no—
141#117—1061no—
142C2×C22⋊C4—1064no—
143C2×C22⋊C4—1062no—
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—22no—
146#102—1068no—
147C22⋊C8—22no—
148#97—22no—
149#97—1061no—
150C4×C8—532no—
151#95—534no—
152#95—1062no—
153#95—1062no—
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—1064no—
165C22×C4—4244no—
166C22×C4—4242no—
167C22×C4C2×D21211yescharacteristic
168C22×C4D4×D5312yes—
169M16—2122no—
170C2×C8—42no—
171C2×C8—1062no—
172C4⋊C4—1062no—
173C4⋊C4—1062no—
174C4⋊C4—2122no—
175C22⋊C4—4244no—
176C22⋊C4—4242no—
177C22⋊C4—2121no—
178C42—2121no—
179C42C2×C4×D5311yescharacteristic
180C42#1171994712yes—
181C42C2×C22⋊D5311yescharacteristic
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1313318711yescharacteristic
188Q8—4242no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—4248no—
196C2×C4—2128no—
197C2×C4—2122no—
198C2×C4—2124no—
199C2×C4C2×#1171992111yescharacteristic
200C2×C4#1313317712yes—
201C2×C4#1313318912yes—
202C2×C4#1313320111yescharacteristic
203C8—82no—
204C8—2122no—
205C22—22no—
206C22—48no—
207C22—44no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1681050011yescharacteristic
212C22#1681080312yes—
213C4—22no—
214C4—22no—
215C4—21no—
216C4—21no—
217C4—4244no—
218C4—4242no—
219C2—22no—
220C2—84no—
221C2#2027510111yescharacteristic
222C2#2027520012yes—
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1555C2
socleC22×C53#386
prime radicalC2×#127×C53C22
cosocleC2×C42×C53C23
hypercenter#1555D53
hypocenterC53#9425
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
3. #20273267C2semidirect
4. #20272581C2semidirect
8. #16810969C4semidirect
20. #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
#202778922
#202732672
#202725812
C53×#15552
#942553