Finite Group Databasebeta

← #28303361

Subgroups — #28303361

Order 13568.

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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#20290572C212yesmaximal
2#20289593C214yesmaximal, complemented
3#384×D53C211yesmaximal, characteristic
4C2×#13133284C2214yes—
5C2×#13133274—22no—
6#16811489—22no—
7C2×C4⋊C4×D53—22no—
8C2×C4⋊C4×D53C412yes—
9C2×C42×D53C2211yescharacteristic
10#16810493C411yescharacteristic
11#16810492—22no—
12#16810492C2212yes—
13C53×#384C411yescharacteristic
14C22×#88324—24no—
15#13133284—28no—
16#13133274—44no—
17C22×C4×D53—22no—
18C22×C4×D53C2×C411yescharacteristic
19C22×C4×D53D412yes—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53C2×C412yes—
22C2×#11719919—22no—
23C2×#11719919—24no—
24C4⋊C4×D53—44no—
25C4⋊C4×D53—44no—
26C4⋊C4×D53—44no—
27C2×C4×Dic53D412yes—
28C2×C4×Dic53C2311yescharacteristic
29C42×D53—24no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53C2×C412yes—
32C2×C42×C53C2×C411yescharacteristic
33C23×D53C22⋊C411yescharacteristic
34C22×C4×C53—22no—
35C22×C4×C53C22⋊C412yes—
36C22×C4×C53C22×C411yescharacteristic
37C22×Dic53—22no—
38C22×Dic53C22⋊C411yescharacteristic
39C22×Dic53C2×D412yes—
40C2×C4×D53—22no—
41C2×C4×D53—22no—
42C2×C4×D53—22no—
43C2×C4×D53—24no—
44C2×C4×D53—24no—
45C2×C4×D53—44no—
46C2×C4×D53—48no—
47C2×C4×D53D814yes—
48C2×C4×D53SD1614yes—
49C2×#88324—44no—
50C2×#88324—44no—
51C2×#88324—44no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—44no—
54C42×C53—22no—
55C4⋊Dic53—44no—
56C4⋊Dic53—42no—
57#11719919—44no—
58#11719919—44no—
59#11719919—44no—
60C4×Dic53—24no—
61C4×Dic53#4114yes—
62C23×C53C2×C22⋊C411yescharacteristic
63C22×D53—21no—
64C22×D53—21no—
65C22×D53—22no—
66C22×D53#10012yes—
67C22×D53#10214yes—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—42no—
73C2×C4×C53—44no—
74C2×C4×C53#10214yes—
75C2×C4×C53#11712yes—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—42no—
79C2×Dic53—44no—
80C2×Dic53#10112yes—
81C2×Dic53#12314yes—
82C2×Dic53C2×D812yes—
83C2×Dic53C2×SD1612yes—
84C2×Dic53#13714yes—
85C4×D53—28no—
86C4×D53—28no—
87C4×D53—28no—
88C4×D53—88no—
89#88324—84no—
90#8723—531nomaximal, Sylow
91C22×C53#38111yescharacteristic
92C22×C53C2×#10011yescharacteristic
93C22×C53C2×#10111yescharacteristic
94C22×C53C2×#10212yes—
95C22×C53#41912yes—
96C2×D53—24no—
97C2×D53—22no—
98C2×D53—22no—
99C2×D53—24no—
100C2×D53#33114yes—
101C4×C53—22no—
102C4×C53—24no—
103C4×C53—82no—
104Dic53—22no—
105Dic53—82no—
106Dic53#48014yes—
107Dic53#48314yes—
108C2×#384—531no—
109#1797—532no—
110#1296—534no—
111C2×C53C2×#33111yescharacteristic
112C2×C53#154612yes—
113C2×C53#155811yescharacteristic
114C2×C53#182111yescharacteristic
115C2×C53#182312yes—
116D53—24no—
117C22×C4⋊C4—532no—
118C22×C4⋊C4—1062no—
119C22×C42—531no—
120C2×C4⋊C8—534no—
121C2×C22⋊C8—1062no—
122#384—532no—
123#384—531no—
124#384—1062no—
125#384#615711yescharacteristic
126#336—1062no—
127C53#872311yescharacteristic, complemented, Sylow
128C23×C4—532no—
129C23×C4—531no—
130C23×C4—1062no—
131C22×C8—1064no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4—1064no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4—2124no—
138C2×C4⋊C4—2124no—
139C2×C4⋊C4—2124no—
140C2×C4⋊C4C2×#615712yes—
141C2×C42—531no—
142C2×C42—532no—
143C2×C42—1064no—
144C2×C42C2×#615711yescharacteristic
145C4⋊C8—1068no—
146C22⋊C8—2124no—
147C24—531no—
148C22×C4—22no—
149C22×C4—532no—
150C22×C4—531no—
151C22×C4—534no—
152C22×C4—534no—
153C22×C4—1064no—
154C22×C4—1062no—
155C22×C4—1064no—
156C22×C4—1062no—
157C22×C4—1062no—
158C22×C4—1062no—
159C22×C4—2128no—
160C22×C4—2124no—
161C22×C4C22⋊#615712yes—
162C22×C4C22×#615711yescharacteristic
163C2×C8—2124no—
164C2×C8—2124no—
165C2×C8—2124no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—2122no—
169C4⋊C4—2124no—
170C4⋊C4—2124no—
171C4⋊C4—2124no—
172C4⋊C4—2124no—
173C42—22no—
174C42—534no—
175C42—1064no—
176C23—532no—
177C23—534no—
178C23—1061no—
179C23—1061no—
180C23—1062no—
181C23C2×C22⋊#615711yescharacteristic
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—534no—
189C2×C4—532no—
190C2×C4—532no—
191C2×C4—534no—
192C2×C4—532no—
193C2×C4—1068no—
194C2×C4—1068no—
195C2×C4—1068no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—4248no—
199C2×C4—2122no—
200C2×C4—2124no—
201C2×C4#1313327914yes—
202C2×C4#1313329612yes—
203C8—4244no—
204C22—534no—
205C22—1062no—
206C22—1064no—
207C22—1064no—
208C22—1062no—
209C22#1681156611yescharacteristic
210C22D53⋊#10011yescharacteristic
211C22D53⋊#10111yescharacteristic
212C22C2×#1313327912yes—
213C22#1681161612yes—
214C4—22no—
215C4—24no—
216C4—82no—
217C4—534no—
218C4—534no—
219C4—1062no—
220C4—4242no—
221C2—1064no—
222C2D53⋊#33111yescharacteristic
223C2#2029022512yes—
224C2#2029024911yescharacteristic
225C2#2029061511yescharacteristic
226C2#2029061812yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#8723
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #20289593C2semidirect
127. C53#8723semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202895932
#384×D532
#872353