Finite Group Databasebeta

← #28303364

Subgroups — #28303364

Order 13568.

630 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 140 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#20290571C212yesmaximal
2#20289593C214yesmaximal, complemented
3C2×#16810492C211yesmaximal, characteristic
4C2×#13133284C2214yes—
5C2×C22⋊#88324—22no—
6#16811489—22no—
7C22×C4⋊Dic53C411yescharacteristic
8C22×#11719919—22no—
9C22×C4×Dic53C2211yescharacteristic
10#16810492—22no—
11#16810492C412yes—
12#16810492C2212yes—
13C22×C4⋊C4×C53C411yescharacteristic
14C22×#88324—24no—
15#13133284—28no—
16C22⋊#88324—44no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—42no—
21C2×C4⋊Dic53C2×C412yes—
22C2×#11719919—24no—
23C2×#11719919—24no—
24C2×#11719919—48no—
25C2×C4×Dic53—24no—
26C2×C4×Dic53C2×C411yescharacteristic
27C2×C4×Dic53D412yes—
28C2×C4×Dic53C2311yescharacteristic
29C23×C4×C53—21no—
30C23×C4×C53C2×C411yescharacteristic
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—42no—
33C2×C4⋊C4×C53C2×C412yes—
34C24×C53C22⋊C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—44no—
40C22×C4×C53—42no—
41C22×C4×C53C22⋊C411yescharacteristic
42C22×C4×C53C22×C411yescharacteristic
43C22×Dic53—24no—
44C22×Dic53—24no—
45C22×Dic53—22no—
46C22×Dic53—44no—
47C22×Dic53—42no—
48C22×Dic53C22⋊C412yes—
49C22×Dic53SD1614yes—
50C22×Dic53Q1614yes—
51C22×Dic53C2×D412yes—
52C2×#88324—44no—
53C2×#88324—44no—
54C2×#88324—44no—
55C4⋊C4×C53—42no—
56C4⋊C4×C53—42no—
57C4⋊Dic53—42no—
58C4⋊Dic53—42no—
59#11719919—48no—
60#11719919—48no—
61C4×Dic53—22no—
62C4×Dic53—24no—
63C4×Dic53#4114yes—
64C23×C53—21no—
65C23×C53—21no—
66C23×C53—22no—
67C23×C53#10012yes—
68C23×C53#10314yes—
69C23×C53C2×C22⋊C411yescharacteristic
70C2×C4×C53—28no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—42no—
76C2×C4×C53—44no—
77C2×C4×C53—84no—
78C2×C4×C53#10112yes—
79C2×C4×C53#11712yes—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—28no—
83C2×Dic53—28no—
84C2×Dic53—42no—
85C2×Dic53—44no—
86C2×Dic53—84no—
87C2×Dic53#10314yes—
88C2×Dic53#12314yes—
89C2×Dic53C2×SD1612yes—
90C2×Dic53C2×Q1612yes—
91C2×Dic53AGL(1,ℤ/8)14yes—
92#88324—84no—
93#8725—531nomaximal, Sylow
94C22×C53—24no—
95C22×C53—22no—
96C22×C53—22no—
97C22×C53—24no—
98C22×C53#33314yes—
99C22×C53#38111yescharacteristic
100C22×C53C2×#10011yescharacteristic
101C22×C53C2×#10111yescharacteristic
102C22×C53C2×#10312yes—
103C22×C53#41812yes—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—82no—
107Dic53—24no—
108Dic53—82no—
109Dic53#48114yes—
110Dic53#48414yes—
111C2×#384—531no—
112#1797—532no—
113#1296—534no—
114C2×C53—24no—
115C2×C53C2×#33311yescharacteristic
116C2×C53#154912yes—
117C2×C53#155511yescharacteristic
118C2×C53#182111yescharacteristic
119C2×C53#182412yes—
120C22×C4⋊C4—531no—
121C22×C4⋊C4—1062no—
122C22×C4⋊C4#615711yescharacteristic
123C22×C42—531no—
124C2×C4⋊C8—534no—
125C2×C22⋊C8—1062no—
126#384—532no—
127#384—532no—
128#384—1062no—
129#336—1062no—
130C53#872511yescharacteristic, complemented, Sylow
131C23×C4—21no—
132C23×C4—532no—
133C23×C4—1061no—
134C23×C4C2×#615711yescharacteristic
135C22×C8—1064no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—1064no—
140C2×C4⋊C4—1064no—
141C2×C4⋊C4—1061no—
142C2×C4⋊C4—2122no—
143C2×C4⋊C4—2128no—
144C2×C4⋊C4C2×#615712yes—
145C2×C42—531no—
146C2×C42—532no—
147C2×C42—531no—
148C2×C42—1064no—
149C4⋊C8—1068no—
150C22⋊C8—2124no—
151C24C22⋊#615711yescharacteristic
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—532no—
159C22×C4—534no—
160C22×C4—534no—
161C22×C4—532no—
162C22×C4—1064no—
163C22×C4—1064no—
164C22×C4—1062no—
165C22×C4—2124no—
166C22×C4—2122no—
167C22×C4C22⋊#615711yescharacteristic
168C22×C4C22×#615711yescharacteristic
169C2×C8—2124no—
170C2×C8—2124no—
171C2×C8—2124no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—2122no—
175C4⋊C4—2122no—
176C4⋊C4—2128no—
177C4⋊C4—2128no—
178C42—534no—
179C42—1064no—
180C42—1062no—
181C23—21no—
182C23—21no—
183C23—22no—
184C23#1313327712yes—
185C23#1313328014yes—
186C23C2×C22⋊#615711yescharacteristic
187C2×C4—28no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—84no—
195C2×C4—534no—
196C2×C4—532no—
197C2×C4—532no—
198C2×C4—534no—
199C2×C4—534no—
200C2×C4—1068no—
201C2×C4—1068no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—4244no—
205C2×C4—2122no—
206C2×C4—2124no—
207C2×C4#1313327812yes—
208C2×C4#1313329612yes—
209C8—4244no—
210C22—22no—
211C22—24no—
212C22—24no—
213C22—22no—
214C22#1681148314yes—
215C22#1681156611yescharacteristic
216C22C2×#1313327711yescharacteristic
217C22C2×#1313327811yescharacteristic
218C22C2×#1313328012yes—
219C22#1681161512yes—
220C4—22no—
221C4—22no—
222C4—82no—
223C4—534no—
224C4—534no—
225C4—1064no—
226C4—4242no—
227C2—24no—
228C2C2×#1681148311yescharacteristic
229C2#2029023012yes—
230C2#2029024211yescharacteristic
231C2#2029061411yescharacteristic
232C2#2029061912yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8725
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289593C2semidirect
130. C53#8725semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202895932
C2×#168104922
#872553