Finite Group Databasebeta

← #27281630

Subgroups — #27281630

Order 13568.

534 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277892C211yesmaximal, characteristic
2#20273429C211yesmaximal, characteristic
3#20273266C212yesmaximal, complemented
4#20272580C212yesmaximal, complemented
5C53×#3138C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810503—24no—
8#16810503—22no—
9#16810491C2212yes—
10#16810367—22no—
11#16810360—24no—
12C53×#533C2211yescharacteristic
13C2×#127×C53C2211yescharacteristic
14C53×#490—22no—
15C53×#486C414yescomplemented
16C53×#418—22no—
17C4×M16×C53C2211yescharacteristic
18C23×Dic53—22no—
19C2×C22⋊Dic53—24no—
20C2×C22⋊Dic53—22no—
21C4⋊#88322—22no—
22C22⋊#88322—22no—
23#13133128—22no—
24#13133124—22no—
25#13133124—22no—
26#13133124D414yes—
27C22×D4×C53—22no—
28C22×D4×C53D412yes—
29C2×M16×C53—21no—
30C4⋊Q8×C53C2×C412yescomplemented
31#127×C53—22no—
32#127×C53C2×C412yes—
33#127×C53D412yes—
34#125×C53—21no—
35C22⋊Q8×C53—22no—
36C4×Q8×C53—22no—
37#117×C53—21no—
38C2×C42×C53C2311yescharacteristic
39#102×C53—28no—
40#97×C53—21no—
41C4×C8×C53C2×C412yescomplemented
42C24×C53—22no—
43C2×Q8×C53—22no—
44C2×D4×C53—24no—
45C2×D4×C53—24no—
46C2×D4×C53—22no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C2×D4×C53—82no—
50C2×D4×C53—82no—
51C2×D4×C53C22⋊C414yes—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D412yes—
54C22×Dic53—24no—
55C22×Dic53—84no—
56C22×Dic53—82no—
57M16×C53—42no—
58C2×C8×C53—22no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—42no—
62C22⋊C4×C53—41no—
63C42×C53—41no—
64C42×C53C22×C411yescharacteristic
65C42×C53C2×D411yescharacteristic
66C42×C53#4112yes—
67C22⋊Dic53—84no—
68C22⋊Dic53—82no—
69C2×#88322—42no—
70C23×C53—24no—
71C23×C53—44no—
72C23×C53—44no—
73C23×C53—42no—
74C23×C53—42no—
75C23×C53#12011yescharacteristic
76Q8×C53—82no—
77D4×C53—44no—
78D4×C53—82no—
79D4×C53—84no—
80D4×C53—84no—
81C2×C4×C53—22no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53C2×C22⋊C411yescharacteristic
85C2×C4×C53C4×D412yes—
86C2×C4×C53#12112yes—
87C2×C4×C53#12311yescharacteristic
88C2×Dic53—48no—
89C2×Dic53—44no—
90C2×Dic53—88no—
91C8×C53—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—82no—
107Dic53—84no—
108#3138D5311yescharacteristic
109#2380—531no—
110#1555—531no—
111#1515—532no—
112#1288—532no—
113C2×C53—22no—
114C2×C53—84no—
115C2×C53#188511yescharacteristic
116C2×C53#190412yes—
117#533C2×D5311yescharacteristic
118C2×#127C2×D5311yescharacteristic
119#490—22no—
120#486Dic5314yescomplemented
121#432—532no—
122#418—22no—
123C4×M16C2×D5311yescharacteristic
124#386—1064no—
125#386—1062no—
126#383—532no—
127#331—1062no—
128#328—1064no—
129C53#942511yescharacteristic, complemented, Sylow
130C22×D4—22no—
131C22×D4C22⋊D5312yes—
132C23×C4—1062no—
133C2×M16—21no—
134C4⋊Q8C2×Dic5312yescomplemented
135#127—22no—
136#127C2×Dic5312yes—
137#127C22⋊D5312yes—
138#125—21no—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117—21no—
142C2×C22⋊C4—1064no—
143C2×C22⋊C4—1062no—
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—1062no—
146#102—28no—
147C22⋊C8—1062no—
148#97—21no—
149#97—1062no—
150C4×C8C2×Dic5312yescomplemented
151#95—534no—
152#95—1062no—
153#95—1062no—
154C24—22no—
155C2×Q8—22no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4C22⋊Dic5314yes—
164C22×C4—1064no—
165C22×C4—4244no—
166C22×C4—4242no—
167C22×C4D4×D5312yes—
168C22×C4C2×C22⋊D5311yescharacteristic
169M16—42no—
170C2×C8—22no—
171C2×C8—2122no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C22⋊C4—41no—
176C22⋊C4—4244no—
177C22⋊C4—4242no—
178C42—41no—
179C42#1171994712yes—
180C42C22×Dic5311yescharacteristic
181C42C2×C22⋊D5311yescharacteristic
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1313318511yescharacteristic
188Q8—82no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—4248no—
197C2×C4—2128no—
198C2×C4—2124no—
199C2×C4C2×C22⋊Dic5311yescharacteristic
200C2×C4D4×Dic5312yes—
201C2×C4#1313318912yes—
202C2×C4#1313319811yescharacteristic
203C8—42no—
204C8—4242no—
205C22—22no—
206C22—48no—
207C22—44no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1681050311yescharacteristic
212C22#1681080012yes—
213C4—22no—
214C4—22no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—4244no—
219C2—22no—
220C2—84no—
221C2#2027510411yescharacteristic
222C2#2027520412yes—
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3138C2
socleC22×C53#386
prime radicalC2×#127×C53C22
cosocleC2×C42×C53C23
hypercenter#3138D53
hypocenterC53#9425
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
3. #20273266C2semidirect
4. #20272580C2semidirect
15. C53×#486C4semidirect
30. C4⋊Q8×C53C2×C4semidirect
41. C4×C8×C53C2×C4semidirect
120. #486Dic53semidirect
129. C53#9425semidirect
134. C4⋊Q8C2×Dic53semidirect
150. C4×C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202778922
#202734292
#202732662
#202725802
C53×#31382
#942553