Finite Group Databasebeta

← #27272339

Subgroups — #27272339

Order 13568.

590 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 131 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#20274409C212yesmaximal
2#20272538C214yesmaximal, complemented
3C53×#2296C211yesmaximal, characteristic
4C22×#11719917—22no—
5#16810378—22no—
6#16810378C2214yes—
7C2×#117×C53C411yescharacteristic
8C2×#117×C53C411yescharacteristic
9C22×C42×C53C2211yescharacteristic
10C53×#389C412yes—
11C53×#389C2212yes—
12C53×#385—22no—
13C4×C4⋊C4×C53—22no—
14C2×#11719917—28no—
15C2×#11719917—44no—
16C2×#11719917—44no—
17C22×#88322—24no—
18C23×C4×C53C2×C411yescharacteristic
19C23×C4×C53D412yes—
20#117×C53—24no—
21#117×C53—42no—
22C2×C4⋊C4×C53—22no—
23C2×C4⋊C4×C53—22no—
24C2×C4⋊C4×C53C2×C412yes—
25C2×C4⋊C4×C53C2×C412yes—
26C2×C22⋊C4×C53—21no—
27C2×C22⋊C4×C53—21no—
28C2×C42×C53—21no—
29C2×C42×C53—21no—
30C2×C42×C53—42no—
31C2×C42×C53C2×C411yescharacteristic
32C2×C42×C53D412yes—
33C2×C42×C53C2311yescharacteristic
34#95×C53—22no—
35#95×C53—22no—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—22no—
38C22×C4×C53—24no—
39C22×C4×C53—22no—
40C22×C4×C53—24no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22⋊C412yes—
48C22×C4×C53C22×C411yescharacteristic
49C22×C4×C53C2×D412yes—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—44no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—42no—
55C4⋊C4×C53M1614yescomplemented
56C22⋊C4×C53—24no—
57C22⋊C4×C53—42no—
58C42×C53—24no—
59C42×C53—41no—
60C42×C53—42no—
61C42×C53—41no—
62C42×C53—42no—
63#11719917—416no—
64C2×#88322—28no—
65C2×#88322—44no—
66C2×#88322—44no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53#9912yes—
72C23×C53C2×C22⋊C411yescharacteristic
73C2×C4×C53—28no—
74C2×C4×C53—28no—
75C2×C4×C53—28no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—24no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53#10112yes—
91C2×C4×C53#10414yes—
92C2×C4×C53C2×M1612yes—
93C2×C4×C53AGL(1,ℤ/8)12yes—
94C2×C4×C53#13712yes—
95#88322—48no—
96#7632—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53—24no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53#40711yescharacteristic
103C22×C53C2×#9911yescharacteristic
104C22×C53C2×#10111yescharacteristic
105C22×C53#41712yes—
106C22×C53C2×#10412yes—
107C4×C53—22no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—24no—
111C4×C53—22no—
112C4×C53—44no—
113C4×C53—82no—
114#2296D5311yescharacteristic
115#1728—532no—
116#1271—534no—
117C2×C53—42no—
118C2×C53#146311yescharacteristic
119C2×C53#148912yes—
120C2×C53#150412yes—
121C2×C53#152611yescharacteristic
122C2×C53#156111yescharacteristic
123C22×M16—1062no—
124C2×#117Dic5311yescharacteristic
125C2×#117Dic5311yescharacteristic
126C22×C42C2×D5311yescharacteristic
127#389Dic5312yes—
128#389C2×D5312yes—
129#385—22no—
130C4×C4⋊C4—22no—
131#336—534no—
132#336—1062no—
133C53#763211yescharacteristic, complemented, Sylow
134C23×C4C2×Dic5311yescharacteristic
135C23×C4C22⋊D5312yes—
136C2×M16—1068no—
137C2×M16—2124no—
138C2×M16—2124no—
139C22×C8—1064no—
140#117—24no—
141#117—42no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C2×Dic5312yes—
145C2×C4⋊C4C2×Dic5312yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—42no—
151C2×C42C2×Dic5311yescharacteristic
152C2×C42C22⋊D5312yes—
153C2×C42C22×D5311yescharacteristic
154#95—22no—
155#95—22no—
156C24C22⋊Dic5311yescharacteristic
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C22⋊Dic5311yescharacteristic
167C22×C4C22⋊Dic5312yes—
168C22×C4C22×Dic5311yescharacteristic
169C22×C4C2×C22⋊D5312yes—
170M16—21216no—
171C2×C8—1068no—
172C2×C8—2124no—
173C2×C8—2124no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4#1171991714yescomplemented
180C22⋊C4—24no—
181C22⋊C4—42no—
182C42—24no—
183C42—41no—
184C42—42no—
185C42—41no—
186C42—42no—
187C23—21no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23#1313313212yes—
192C23C2×C22⋊Dic5311yescharacteristic
193C2×C4—28no—
194C2×C4—28no—
195C2×C4—28no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4#1313313612yes—
211C2×C4#1313314414yes—
212C2×C4C2×#1171991712yes—
213C2×C4#1313324212yes—
214C2×C4#1313324712yes—
215C8—2128no—
216C22—24no—
217C22—22no—
218C22—42no—
219C22—41no—
220C22—41no—
221C22#1681058711yescharacteristic
222C22C2×#1313313211yescharacteristic
223C22C2×#1313313611yescharacteristic
224C22#1681062412yes—
225C22C2×#1313314412yes—
226C4—22no—
227C4—24no—
228C4—22no—
229C4—22no—
230C4—22no—
231C4—44no—
232C4—82no—
233C2—42no—
234C2#2027304511yescharacteristic
235C2#2027315412yes—
236C2#2027321912yes—
237C2#2027331011yescharacteristic
238C2#2027345311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2296C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2296D53
hypocenterC53#7632
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #20272538C2semidirect
55. C4⋊C4×C53M16semidirect
133. C53#7632semidirect
179. C4⋊C4#11719917semidirect

Maximal subgroups

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

subgroupindex
#202744092
#202725382
C53×#22962
#763253