Finite Group Databasebeta

← #27293452

Subgroups — #27293452

Order 13568.

706 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 107 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#20273042C212yesmaximal
2#20272645C214yesmaximal, complemented
3C2×C53×#381C211yesmaximal, characteristic
4#16810587—22no—
5#16810347—22no—
6#16810347C2214yes—
7C24×C4×C53C411yescharacteristic
8C22×C22⋊C4×C53C411yescharacteristic
9C22×C22⋊C4×C53C2211yescharacteristic
10C53×#381—22no—
11C53×#381C412yes—
12C53×#381C2212yes—
13C2×#95×C53—22no—
14C2×#11719917—42no—
15C22⋊#88322—24no—
16C22⋊#88322—42no—
17C25×C53C2×C411yescharacteristic
18C23×C4×C53—21no—
19C23×C4×C53—21no—
20C23×C4×C53—24no—
21C23×C4×C53—21no—
22C23×C4×C53—42no—
23C23×C4×C53—42no—
24C23×C4×C53C2×C412yes—
25C23×C4×C53D412yes—
26C2×C22⋊C4×C53—21no—
27C2×C22⋊C4×C53—42no—
28C2×C22⋊C4×C53—42no—
29C2×C22⋊C4×C53C2×C412yes—
30C2×C22⋊C4×C53C2×C411yescharacteristic
31C2×C22⋊C4×C53D412yes—
32C2×C22⋊C4×C53C2311yescharacteristic
33#95×C53—24no—
34#95×C53—24no—
35C24×C53—22no—
36C24×C53—22no—
37C24×C53—22no—
38C24×C53—44no—
39C24×C53C22⋊C411yescharacteristic
40C24×C53C22⋊C411yescharacteristic
41C24×C53C22×C411yescharacteristic
42C22×C4×C53—24no—
43C22×C4×C53—24no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—416no—
50C22×C4×C53—42no—
51C22×C4×C53—42no—
52C22×C4×C53—42no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53—44no—
57C22×C4×C53—42no—
58C22×C4×C53—44no—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53M1614yescomplemented
61C22×C4×C53C2×D412yes—
62C22⋊C4×C53—41no—
63C22⋊C4×C53—42no—
64C22⋊C4×C53—41no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67#11719917—84no—
68C2×#88322—44no—
69C23×C53—24no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—28no—
73C23×C53—21no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53—48no—
80C23×C53—48no—
81C23×C53—44no—
82C23×C53—44no—
83C23×C53#9912yes—
84C23×C53#10012yes—
85C23×C53C2×C22⋊C411yescharacteristic
86C23×C53C2×M1612yes—
87C2×C4×C53—24no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—48no—
91C2×C4×C53—48no—
92C2×C4×C53—44no—
93C2×C4×C53—44no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—44no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—44no—
100C2×C4×C53—42no—
101C2×C4×C53—82no—
102C2×C4×C53—84no—
103#88322—84no—
104#11500—531nomaximal, Sylow
105C22×C53—24no—
106C22×C53—24no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—22no—
110C22×C53—22no—
111C22×C53—22no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—416no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53#35214yes—
120C22×C53#40711yescharacteristic
121C22×C53C2×#9911yescharacteristic
122C22×C53C2×#10011yescharacteristic
123C4×C53—42no—
124C4×C53—42no—
125C4×C53—44no—
126C4×C53—82no—
127C2×#381D5311yescharacteristic
128#1462—532no—
129#1317—534no—
130C2×C53—21no—
131C2×C53—24no—
132C2×C53—21no—
133C2×C53—42no—
134C2×C53—42no—
135C2×C53#146211yescharacteristic
136C2×C53C2×#35212yes—
137C24×C4Dic5311yescharacteristic
138C22×C22⋊C4Dic5311yescharacteristic
139C22×C22⋊C4C2×D5311yescharacteristic
140#407—1062no—
141#381—22no—
142#381Dic5312yes—
143#381C2×D5312yes—
144C2×#95—22no—
145#323—534no—
146#323—1062no—
147C53#1150011yescharacteristic, complemented, Sylow
148C25C2×Dic5311yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×Dic5312yes—
156C23×C4C22⋊D5312yes—
157C2×M16—2122no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4C2×Dic5312yes—
162C2×C22⋊C4C2×Dic5311yescharacteristic
163C2×C22⋊C4C22⋊D5312yes—
164C2×C22⋊C4C22×D5311yescharacteristic
165C22⋊C8—1064no—
166C22⋊C8—2122no—
167#95—24no—
168#95—24no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—44no—
173C24C22⋊Dic5311yescharacteristic
174C24C22⋊Dic5311yescharacteristic
175C24C22×Dic5311yescharacteristic
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—416no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4#1171991714yescomplemented
194C22×C4C22⋊Dic5312yes—
195C22×C4C2×C22⋊D5312yes—
196M16—4244no—
197C2×C8—2124no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C23—21no—
204C23—21no—
205C23—24no—
206C23—28no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—42no—
212C23—42no—
213C23—48no—
214C23—48no—
215C23—44no—
216C23—44no—
217C23#1313313212yes—
218C23#1313313412yes—
219C23C2×C22⋊Dic5311yescharacteristic
220C23C2×#1171991712yes—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—48no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—84no—
237C8—4244no—
238C22—24no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—416no—
250C22—42no—
251C22—42no—
252C22#1681041314yes—
253C22#1681058711yescharacteristic
254C22C2×#1313313211yescharacteristic
255C22C2×#1313313411yescharacteristic
256C4—42no—
257C4—42no—
258C4—44no—
259C4—82no—
260C2—21no—
261C2—21no—
262C2—24no—
263C2—42no—
264C2—42no—
265C2#2027304211yescharacteristic
266C2C2×#1681041312yes—
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC2×C53×#381C2
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenterC2×#381D53
hypocenterC53#11500
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. #20272645C2semidirect
60. C22×C4×C53M16semidirect
147. C53#11500semidirect
193. C22×C4#11719917semidirect

Maximal subgroups

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

subgroupindex
#202730422
#202726452
C2×C53×#3812
#1150053