Finite Group Databasebeta

← #28303451

Subgroups — #28303451

Order 13568.

474 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289962C212yesmaximal
2#20289616C214yesmaximal, complemented
3#20277310C211yesmaximal, characteristic
4#16811593C2214yes—
5#16811467—22no—
6#16811467—22no—
7C22×#11719921C2211yescharacteristic
8#16810514—22no—
9#16810512—22no—
10#16810481C2212yes—
11#16810478C412yes—
12C4×C22⋊Dic53C411yescharacteristic
13C4×C22⋊C4×C53C411yescharacteristic
14C2×#11719940—24no—
15C22⋊#88324—24no—
16C22⋊#88324—24no—
17C24×D53C2×C411yescharacteristic
18C23×Dic53C2311yescharacteristic
19C2×#11719919—22no—
20C2×C22⋊Dic53C2×C412yes—
21C2×#11719921—22no—
22C2×#11719921—22no—
23C2×#11719921—24no—
24C2×#11719921D412yes—
25C2×#11719921D412yes—
26C2×C4×Dic53—21no—
27#13133124—21no—
28#13133124—22no—
29C23×C4×C53C2×C411yescharacteristic
30C2×C22⋊C4×C53C2×C412yes—
31C2×C42×C53—21no—
32#95×C53—21no—
33C24×C53C22×C411yescharacteristic
34C23×D53—22no—
35C23×D53—42no—
36C23×D53—81no—
37C23×D53—81no—
38C23×D53C22⋊C412yes—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—81no—
42C22×C4×C53C22⋊C411yescharacteristic
43C22×C4×C53C22⋊C411yescharacteristic
44C22×Dic53—22no—
45C22×Dic53—81no—
46C22×Dic53C2×D412yes—
47C22×Dic53#4114yes—
48#11719940—84no—
49C2×#88324—44no—
50C22⋊C4×C53—42no—
51C42×C53—81no—
52C22⋊Dic53—42no—
53#11719921—24no—
54#11719921—24no—
55#11719921—82no—
56#11719921—82no—
57#11719919—82no—
58C4×Dic53—81no—
59C23×C53—22no—
60C23×C53—41no—
61C23×C53—41no—
62C23×C53C2×C22⋊C411yescharacteristic
63C23×C53#11712yes—
64C22×D53—22no—
65C22×D53—24no—
66C22×D53—44no—
67C22×D53—42no—
68C22×D53—42no—
69C22×D53—82no—
70C22×D53—82no—
71C22×D53—88no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—82no—
80C2×Dic53—44no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53#12314yes—
86#88324—84no—
87#8759—531nomaximal, Sylow
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—22no—
91C22×C53—22no—
92C22×C53—22no—
93C22×C53—44no—
94C22×C53#38111yescharacteristic
95C22×C53#41312yes—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—88no—
100C2×D53—84no—
101C4×C53—41no—
102C4×C53—41no—
103C4×C53—82no—
104Dic53—24no—
105Dic53—82no—
106#2312—531no—
107#1462—532no—
108#1305—534no—
109C2×C53—22no—
110C2×C53—42no—
111C2×C53#182211yescharacteristic
112C2×C53#182811yescharacteristic
113C2×C53#182911yescharacteristic
114D53—82no—
115C22×C22⋊C4—531no—
116#407—534no—
117#388—1062no—
118#388—1062no—
119#380—532no—
120#379—532no—
121C4×C22⋊C4—531no—
122C4×C22⋊C4#615711yescharacteristic
123#323—1062no—
124#323—1062no—
125C53#875911yescharacteristic, complemented, Sylow
126C25—531no—
127C23×C4—531no—
128C23×C4C2×#615711yescharacteristic
129C2×M16—1064no—
130C2×C4⋊C4—1062no—
131C2×C22⋊C4—532no—
132C2×C22⋊C4—532no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—1064no—
135C2×C22⋊C4—1062no—
136C2×C22⋊C4—1062no—
137C2×C22⋊C4C2×#615712yes—
138C2×C42—21no—
139C2×C42—1061no—
140C22⋊C8—1064no—
141C22⋊C8—1064no—
142#95—21no—
143#95—1062no—
144#95—1061no—
145C24—532no—
146C24—1062no—
147C24—4241no—
148C24—4241no—
149C24—2122no—
150C24C22×#615711yescharacteristic
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—81no—
154C22×C4—532no—
155C22×C4—534no—
156C22×C4—1062no—
157C22×C4—4241no—
158C22×C4C22⋊#615711yescharacteristic
159C22×C4C22⋊#615711yescharacteristic
160M16—4244no—
161C2×C8—2124no—
162C4⋊C4—4242no—
163C22⋊C4—42no—
164C22⋊C4—1064no—
165C22⋊C4—1064no—
166C22⋊C4—4242no—
167C22⋊C4—4242no—
168C22⋊C4—2122no—
169C42—81no—
170C42—4241no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—1064no—
175C23—1062no—
176C23—4242no—
177C23—4242no—
178C23—4248no—
179C23—2122no—
180C23—2122no—
181C23—2124no—
182C23C2×C22⋊#615711yescharacteristic
183C23#1313329612yes—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—82no—
192C2×C4—534no—
193C2×C4—2122no—
194C2×C4—2124no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2122no—
198C8—4244no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—44no—
205C22—4244no—
206C22—4248no—
207C22—2122no—
208C22—2122no—
209C22—2122no—
210C22#1681156611yescharacteristic
211C22#1681160912yes—
212C4—41no—
213C4—41no—
214C4—82no—
215C4—1064no—
216C4—4242no—
217C2—22no—
218C2—42no—
219C2—4242no—
220C2#2029061611yescharacteristic
221C2#2029062711yescharacteristic
222C2#2029062811yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC4×C22⋊C4×C53C4
socleC22×C53#381
prime radicalC24×D53C2×C4
cosocleC23×Dic53C23
hypercenterC4×C22⋊C4#6157
hypocenterC53#8759
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289616C2semidirect
125. C53#8759semidirect

Maximal subgroups

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

subgroupindex
#202899622
#202896162
#202773102
#875953