Finite Group Databasebeta

← #28303430

Subgroups — #28303430

Order 13568.

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

normal subgroups only

This group has 538 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#20290610C212yesmaximal
2#20289614C214yesmaximal
3#20277310C211yesmaximal, characteristic
4#16811564C2214yes—
5#16811501—22no—
6#16811501—22no—
7C22×#11719921C2211yescharacteristic
8#16810514—22no—
9#16810512—22no—
10#16810481C412yescomplemented
11#16810478C2212yes—
12C4×C22⋊Dic53C411yescharacteristic, complemented
13C4×C22⋊C4×C53C411yescharacteristic, complemented
14C2×C22⋊#6157—24no—
15C2×C22⋊#6157—24no—
16C2×C22⋊#6157—24no—
17C24×D53C2311yescharacteristic
18C23×Dic53C2×C411yescharacteristic
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—42no—
35C23×D53—81no—
36C23×D53—81no—
37C23×D53C2×D412yes—
38C23×D53#4114yes—
39C22×#6157—44no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—81no—
43C22×C4×C53C22⋊C411yescharacteristic
44C22×C4×C53C22⋊C411yescharacteristic
45C22×Dic53—22no—
46C22×Dic53—22no—
47C22×Dic53—81no—
48C22×Dic53C22⋊C412yes—
49C22⋊#6157—48no—
50C22⋊#6157—84no—
51C22⋊#6157—84no—
52C22⋊C4×C53—42no—
53C42×C53—81no—
54C22⋊Dic53—42no—
55#11719921—24no—
56#11719921—24no—
57#11719921—82no—
58#11719921—82no—
59#11719919—82no—
60C4×Dic53—81no—
61C23×C53—22no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53C2×C22⋊C411yescharacteristic
65C23×C53#11712yes—
66C22×D53—24no—
67C22×D53—44no—
68C22×D53—44no—
69C22×D53—44no—
70C22×D53—42no—
71C22×D53—42no—
72C22×D53—88no—
73C22×D53#12314yes—
74C2×#6157—84no—
75C2×#6157—84no—
76C2×#6157—84no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—82no—
85C2×Dic53—22no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—82no—
91#8750—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—22no—
95C22×C53—22no—
96C22×C53—22no—
97C22×C53—44no—
98C22×C53#38111yescharacteristic
99C22×C53#41012yes—
100C2×D53—24no—
101C2×D53—24no—
102C2×D53—24no—
103C2×D53—84no—
104C2×D53—88no—
105#6157—164no—
106C4×C53—41no—
107C4×C53—41no—
108C4×C53—82no—
109Dic53—42no—
110Dic53—82no—
111#2312—531no—
112#1819—532no—
113#1304—534no—
114C2×C53—22no—
115C2×C53—42no—
116C2×C53#182011yescharacteristic
117C2×C53#182811yescharacteristic
118C2×C53#182811yescharacteristic
119D53—44no—
120C22×C22⋊C4—531no—
121#388—1062no—
122#388—1062no—
123#380—532no—
124#379—532no—
125#379—534no—
126C4×C22⋊C4—531no—
127C4×C22⋊C4#615711yescharacteristic, complemented
128#342—1062no—
129#342—1062no—
130C53#875011yescharacteristic, complemented, Sylow
131C25—531no—
132C23×C4—531no—
133C23×C4C2×#615711yescharacteristic
134C2×C4⋊C4—1062no—
135C2×C22⋊C4—532no—
136C2×C22⋊C4—532no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4—1064no—
139C2×C22⋊C4—1062no—
140C2×C22⋊C4—1062no—
141C2×C22⋊C4—1064no—
142C2×C22⋊C4—1064no—
143C2×C22⋊C4—1064no—
144C2×C22⋊C4C2×#615712yes—
145C2×C42—21no—
146C2×C42—1061no—
147#95—21no—
148#95—1062no—
149#95—1061no—
150C24—532no—
151C24—534no—
152C24—4241no—
153C24—4241no—
154C24—2122no—
155C24C22×#615711yescharacteristic
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—81no—
159C22×C4—532no—
160C22×C4—1062no—
161C22×C4—1062no—
162C22×C4—4241no—
163C22×C4—2124no—
164C22×C4C22⋊#615711yescharacteristic
165C22×C4C22⋊#615711yescharacteristic
166C4⋊C4—4242no—
167C22⋊C4—42no—
168C22⋊C4—1064no—
169C22⋊C4—1064no—
170C22⋊C4—4242no—
171C22⋊C4—4242no—
172C22⋊C4—4244no—
173C22⋊C4—4244no—
174C22⋊C4—2128no—
175C22⋊C4—2122no—
176C42—81no—
177C42—4241no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—534no—
182C23—1064no—
183C23—4248no—
184C23—2122no—
185C23—2122no—
186C23—2124no—
187C23—2124no—
188C23—2124no—
189C23C2×C22⋊#615711yescharacteristic
190C23#1313329612yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—1062no—
200C2×C4—4244no—
201C2×C4—4244no—
202C2×C4—4244no—
203C2×C4—4242no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C22—21no—
209C22—21no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—44no—
214C22—1064no—
215C22—1064no—
216C22—1064no—
217C22—4244no—
218C22—4248no—
219C22#1681156611yescharacteristic
220C22#1681160212yes—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—8484no—
225C4—4242no—
226C4—2122no—
227C2—22no—
228C2—42no—
229C2—2124no—
230C2#2029061211yescharacteristic
231C2#2029062611yescharacteristic
232C2#2029062711yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC4×C22⋊C4×C53C4
socleC22×C53#381
prime radical, cosocleC24×D53C23
hypercenterC4×C22⋊C4#6157
hypocenterC53#8750
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
10. #16810481C4semidirect
12. C4×C22⋊Dic53C4semidirect
13. C4×C22⋊C4×C53C4semidirect
127. C4×C22⋊C4#6157semidirect
130. C53#8750semidirect

Maximal subgroups

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

subgroupindex
#202906102
#202896142
#202773102
#875053