Finite Group Databasebeta

← #28308156

Subgroups — #28308156

Order 13568.

706 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 84 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#20289960C212yesmaximal
2#20289642C214yesmaximal, complemented
3C2×#16810484C211yesmaximal, characteristic
4#16811593—22no—
5#16811466—22no—
6#16811466C2214yes—
7C24×Dic53C411yescharacteristic
8C22×C22⋊Dic53C2211yescharacteristic
9#16810484—22no—
10#16810484C412yes—
11#16810484C2212yes—
12C2×#13133124—22no—
13C22×C22⋊C4×C53C411yescharacteristic
14C2×#11719940—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53—21no—
19C23×Dic53—24no—
20C23×Dic53—42no—
21C23×Dic53—42no—
22C23×Dic53C2×C412yes—
23C23×Dic53D412yes—
24C2×C22⋊Dic53—42no—
25C2×C22⋊Dic53C2×C411yescharacteristic
26C2×C22⋊Dic53D412yes—
27C2×C22⋊Dic53C2311yescharacteristic
28#13133124—24no—
29#13133124—24no—
30C25×C53C2×C411yescharacteristic
31C23×C4×C53—21no—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—42no—
34C2×C22⋊C4×C53C2×C412yes—
35C24×C53—22no—
36C24×C53—22no—
37C24×C53—22no—
38C24×C53—44no—
39C24×C53C22⋊C411yescharacteristic
40C24×C53C22⋊C411yescharacteristic
41C24×C53C22×C411yescharacteristic
42C22×C4×C53—22no—
43C22×C4×C53—42no—
44C22×C4×C53—44no—
45C22×Dic53—24no—
46C22×Dic53—24no—
47C22×Dic53—22no—
48C22×Dic53—22no—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—416no—
52C22×Dic53—42no—
53C22×Dic53—42no—
54C22×Dic53—42no—
55C22×Dic53—42no—
56C22×Dic53—42no—
57C22×Dic53—42no—
58C22×Dic53—44no—
59C22×Dic53C22⋊C412yes—
60C22×Dic53M1614yescomplemented
61C22×Dic53C2×D412yes—
62#11719940—84no—
63C2×#88324—44no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C22⋊Dic53—41no—
67C22⋊Dic53—42no—
68C22⋊Dic53—41no—
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—44no—
88C2×C4×C53—42no—
89C2×C4×C53—84no—
90C2×Dic53—24no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—48no—
94C2×Dic53—48no—
95C2×Dic53—44no—
96C2×Dic53—44no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—44no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—82no—
103#88324—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—82no—
124Dic53—42no—
125Dic53—42no—
126Dic53—44no—
127C2×#381—531no—
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×C4—531no—
138C22×C22⋊C4—531no—
139C22×C22⋊C4#615711yescharacteristic
140#407—1062no—
141#381—532no—
142#381—532no—
143#381—1062no—
144C2×#95—1062no—
145#323—534no—
146#323—1062no—
147C53#1150011yescharacteristic, complemented, Sylow
148C25C2×#615711yescharacteristic
149C23×C4—21no—
150C23×C4—532no—
151C23×C4—532no—
152C23×C4—1064no—
153C23×C4—1061no—
154C23×C4—1061no—
155C23×C4—2122no—
156C23×C4—2122no—
157C2×M16—2122no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—2122no—
164C2×C22⋊C4C2×#615712yes—
165C22⋊C8—1064no—
166C22⋊C8—2122no—
167#95—1064no—
168#95—1064no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—44no—
173C24C22⋊#615711yescharacteristic
174C24C22⋊#615711yescharacteristic
175C24C22×#615711yescharacteristic
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—532no—
180C22×C4—532no—
181C22×C4—534no—
182C22×C4—1062no—
183C22×C4—1064no—
184C22×C4—1064no—
185C22×C4—1062no—
186C22×C4—1062no—
187C22×C4—1062no—
188C22×C4—2124no—
189C22×C4—21216no—
190C22×C4—2122no—
191C22×C4—2122no—
192C22×C4—2122no—
193C22×C4—2122no—
194C22×C4—2122no—
195C22×C4—2122no—
196M16—4244no—
197C2×C8—2124no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—2121no—
201C22⋊C4—2122no—
202C22⋊C4—2121no—
203C23—24no—
204C23—21no—
205C23—21no—
206C23—28no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—42no—
212C23—42no—
213C23—48no—
214C23—48no—
215C23—44no—
216C23—44no—
217C23#1313327612yes—
218C23#1313327712yes—
219C23C2×C22⋊#615711yescharacteristic
220C23D53⋊M1612yes—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—84no—
224C2×C4—1064no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—4242no—
228C2×C4—2128no—
229C2×C4—2128no—
230C2×C4—2124no—
231C2×C4—2124no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2124no—
237C8—4244no—
238C22—24no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—416no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22#1681152614yes—
253C22#1681159511yescharacteristic
254C22C2×#1313327611yescharacteristic
255C22C2×#1313327711yescharacteristic
256C4—82no—
257C4—2122no—
258C4—2122no—
259C4—2124no—
260C2—21no—
261C2—24no—
262C2—21no—
263C2—42no—
264C2—42no—
265C2#2028996211yescharacteristic
266C2C2×#1681152612yes—
subgroupG/H
centerC22C2×#13133276
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#11500
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289642C2semidirect
60. C22×Dic53M16semidirect
147. C53#11500semidirect

Maximal subgroups

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

subgroupindex
#202899602
#202896422
C2×#168104842
#1150053