Finite Group Databasebeta

← #28302875

Subgroups — #28302875

Order 13568.

1154 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 108 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#20290549C212yesmaximal, complemented
2#20289962C214yesmaximal, complemented
3C22×#13133205C211yesmaximal, characteristic
4C2×#13133278—24no—
5#16811593C2214yes—
6#16811467—28no—
7C23×Dic106C2211yescharacteristic
8C23×D212C411yescharacteristic
9C2×#13133205—24no—
10C2×#13133205—28no—
11C22×#11719921C412yes—
12C22×#11719921C2212yes—
13C22×C42×C53C411yescharacteristic
14C2×#11719940—24no—
15#13133278—48no—
16C22⋊#88324—28no—
17C24×D53C2×C412yes—
18C22×Dic106—24no—
19C22×Dic106—44no—
20C22×Dic106D414yes—
21C22×D212—24no—
22C22×D212—22no—
23C22×D212—44no—
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2311yescharacteristic
26#13133205—232no—
27C2×#11719921—24no—
28C2×#11719921—24no—
29C2×#11719921—24no—
30C2×#11719921—24no—
31C2×#11719921—24no—
32C2×#11719921D418yes—
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53C2×C412yes—
35C2×C42×C53—24no—
36C2×C42×C53—22no—
37C24×C53C22×C411yescharacteristic
38C23×D53—22no—
39C23×D53—22no—
40C23×D53—44no—
41C23×D53—44no—
42C23×D53—44no—
43C23×D53C22⋊C414yes—
44C22×C4×C53—22no—
45C22×C4×C53—24no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—44no—
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C22⋊C414yes—
51C22×Dic53—22no—
52C22×Dic53—42no—
53C22×Dic53—42no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C22×Dic53C2×D414yes—
57C2×D212—28no—
58C2×D212—416no—
59C2×D212—44no—
60C2×D212—44no—
61C2×Dic106—28no—
62C2×Dic106—28no—
63C2×Dic106—44no—
64C2×Dic106—416no—
65#11719940—48no—
66C2×#88324—44no—
67C42×C53—28no—
68#11719921—216no—
69#11719921—216no—
70C23×C53—22no—
71C23×C53—24no—
72C23×C53C2×C22⋊C411yescharacteristic
73C23×C53C2×C22⋊C412yes—
74C22×D53—24no—
75C22×D53—28no—
76C22×D53—44no—
77C22×D53—44no—
78C22×D53—44no—
79C22×D53—44no—
80C22×D53—44no—
81C22×D53—44no—
82C22×D53—44no—
83C22×D53—44no—
84C22×D53—816no—
85C2×C4×C53—28no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—24no—
89C2×C4×C53—44no—
90C2×C4×C53—44no—
91C2×C4×C53—44no—
92C2×C4×C53#9914yes—
93C2×Dic53—24no—
94C2×Dic53—22no—
95C2×Dic53—24no—
96C2×Dic53—44no—
97C2×Dic53—44no—
98C2×Dic53#12014yes—
99D212—416no—
100Dic106—416no—
101#88324—84no—
102#8552—531nomaximal, Sylow
103C22×C53—28no—
104C22×C53—22no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53#37911yescharacteristic
108C22×C53C2×#9912yes—
109C22×C53#41314yes—
110C2×D53—44no—
111C2×D53—44no—
112C2×D53—44no—
113C2×D53—88no—
114C2×D53—88no—
115C2×D53—88no—
116C4×C53—24no—
117C4×C53—44no—
118Dic53—24no—
119Dic53—42no—
120C22×#124—531no—
121#1784—532no—
122#1462—534no—
123C2×C53—22no—
124C2×C53—24no—
125C2×C53#178111yescharacteristic
126C2×C53#178512yes—
127D53—84no—
128C23×Q8—531no—
129C23×D4—531no—
130C2×#124—1068no—
131C2×#124—1064no—
132C22×C22⋊C4—532no—
133C22×C22⋊C4—532no—
134C22×C42#615711yescharacteristic
135C2×#101—1064no—
136#407—534no—
137#323—1068no—
138C53#855211yescharacteristic, complemented, Sylow
139C25—532no—
140C22×Q8—534no—
141C22×Q8—1064no—
142C22×Q8—2124no—
143C22×D4—1064no—
144C22×D4—1062no—
145C22×D4—2124no—
146C23×C4—531no—
147C23×C4—531no—
148C23×C4C2×#615711yescharacteristic
149C23×C4C2×#615712yes—
150C2×M16—1064no—
151#124—10632no—
152C2×C22⋊C4—538no—
153C2×C22⋊C4—1064no—
154C2×C22⋊C4—1064no—
155C2×C22⋊C4—1064no—
156C2×C22⋊C4—1064no—
157C2×C22⋊C4—1064no—
158C2×C42—24no—
159C2×C42—22no—
160#101—2128no—
161C22⋊C8—1068no—
162C24—534no—
163C24—1062no—
164C24—1062no—
165C24—2124no—
166C24—2124no—
167C24—2124no—
168C24C22×#615711yescharacteristic
169C2×Q8—1068no—
170C2×Q8—1068no—
171C2×Q8—2124no—
172C2×Q8—21216no—
173C2×D4—1068no—
174C2×D4—21216no—
175C2×D4—2124no—
176C2×D4—2124no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—44no—
182C22×C4—532no—
183C22×C4—534no—
184C22×C4—532no—
185C22×C4—1062no—
186C22×C4—2122no—
187C22×C4—2122no—
188C22×C4C22⋊#615712yes—
189C22×C4C22⋊#615714yes—
190M16—2128no—
191C2×C8—2124no—
192C22⋊C4—10616no—
193C22⋊C4—10616no—
194C42—28no—
195C23—22no—
196C23—24no—
197C23—1068no—
198C23—1064no—
199C23—42416no—
200C23—2124no—
201C23—2124no—
202C23—2124no—
203C23—2124no—
204C23—2124no—
205C23—2124no—
206C23—2124no—
207C23—2124no—
208C23C2×C22⋊#615711yescharacteristic
209C23C2×C22⋊#615712yes—
210Q8—21216no—
211D4—21216no—
212C2×C4—28no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—534no—
220C2×C4—1064no—
221C2×C4—1062no—
222C2×C4—1064no—
223C2×C4—2124no—
224C2×C4—2124no—
225C2×C4#1313327614yes—
226C8—4244no—
227C22—28no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—4248no—
232C22—4248no—
233C22—4248no—
234C22—2124no—
235C22—2124no—
236C22—2124no—
237C22#1681156411yescharacteristic
238C22C2×#1313327612yes—
239C22#1681160914yes—
240C4—24no—
241C4—44no—
242C4—1064no—
243C4—2122no—
244C2—22no—
245C2—24no—
246C2—4244no—
247C2#2029054311yescharacteristic
248C2#2029055112yes—
subgroupG/H
centerC22#16811564
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C42×C53C4
socleC22×C53#379
prime radicalC23×D212C4
cosocleC23×Dic53C23
hypercenterC22×C42#6157
hypocenterC53#8552
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. #20290549C2semidirect
2. #20289962C2semidirect
138. C53#8552semidirect

Maximal subgroups

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

subgroupindex
#202905492
#202899622
C22×#131332052
#855253