Finite Group Databasebeta

← #28303360

Subgroups — #28303360

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—44no—
25C2×#11719919—44no—
26C2×C4×Dic53—24no—
27C2×C4×Dic53C2×C411yescharacteristic
28C2×C4×Dic53D412yes—
29C2×C4×Dic53C2311yescharacteristic
30C23×C4×C53—21no—
31C23×C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—42no—
34C2×C4⋊C4×C53C2×C412yes—
35C24×C53C22⋊C411yescharacteristic
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—42no—
41C22×C4×C53—44no—
42C22×C4×C53C22⋊C411yescharacteristic
43C22×C4×C53C22×C411yescharacteristic
44C22×Dic53—24no—
45C22×Dic53—24no—
46C22×Dic53—22no—
47C22×Dic53—42no—
48C22×Dic53—44no—
49C22×Dic53C22⋊C412yes—
50C22×Dic53D814yes—
51C22×Dic53SD1614yes—
52C22×Dic53C2×D412yes—
53C2×#88324—44no—
54C2×#88324—44no—
55C2×#88324—44no—
56C4⋊C4×C53—42no—
57C4⋊C4×C53—42no—
58C4⋊Dic53—42no—
59C4⋊Dic53—42no—
60#11719919—48no—
61#11719919—44no—
62#11719919—44no—
63C4×Dic53—22no—
64C4×Dic53—24no—
65C4×Dic53#4114yes—
66C23×C53—21no—
67C23×C53—21no—
68C23×C53—22no—
69C23×C53#10012yes—
70C23×C53#10214yes—
71C23×C53C2×C22⋊C411yescharacteristic
72C2×C4×C53—28no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—42no—
78C2×C4×C53—44no—
79C2×C4×C53—84no—
80C2×C4×C53#10112yes—
81C2×C4×C53#11712yes—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—28no—
85C2×Dic53—28no—
86C2×Dic53—42no—
87C2×Dic53—44no—
88C2×Dic53—84no—
89C2×Dic53#10214yes—
90C2×Dic53#12314yes—
91C2×Dic53C2×D812yes—
92C2×Dic53C2×SD1612yes—
93C2×Dic53#13714yes—
94#88324—84no—
95#8723—531nomaximal, Sylow
96C22×C53—24no—
97C22×C53—22no—
98C22×C53—22no—
99C22×C53—24no—
100C22×C53#33114yes—
101C22×C53#38111yescharacteristic
102C22×C53C2×#10011yescharacteristic
103C22×C53C2×#10111yescharacteristic
104C22×C53C2×#10212yes—
105C22×C53#41912yes—
106C4×C53—22no—
107C4×C53—22no—
108C4×C53—82no—
109Dic53—24no—
110Dic53—82no—
111Dic53#48014yes—
112Dic53#48314yes—
113C2×#384—531no—
114#1797—532no—
115#1296—534no—
116C2×C53—24no—
117C2×C53C2×#33111yescharacteristic
118C2×C53#154612yes—
119C2×C53#155811yescharacteristic
120C2×C53#182111yescharacteristic
121C2×C53#182312yes—
122C22×C4⋊C4—531no—
123C22×C4⋊C4—1062no—
124C22×C4⋊C4#615711yescharacteristic
125C22×C42—531no—
126C2×C4⋊C8—534no—
127C2×C22⋊C8—1062no—
128#384—532no—
129#384—532no—
130#384—1062no—
131#336—1062no—
132C53#872311yescharacteristic, complemented, Sylow
133C23×C4—21no—
134C23×C4—532no—
135C23×C4—1061no—
136C23×C4C2×#615711yescharacteristic
137C22×C8—1064no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—1064no—
142C2×C4⋊C4—1064no—
143C2×C4⋊C4—1061no—
144C2×C4⋊C4—2122no—
145C2×C4⋊C4—2124no—
146C2×C4⋊C4—2124no—
147C2×C4⋊C4C2×#615712yes—
148C2×C42—531no—
149C2×C42—532no—
150C2×C42—531no—
151C2×C42—1064no—
152C4⋊C8—1068no—
153C22⋊C8—2124no—
154C24C22⋊#615711yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—532no—
162C22×C4—534no—
163C22×C4—534no—
164C22×C4—532no—
165C22×C4—1064no—
166C22×C4—1064no—
167C22×C4—1062no—
168C22×C4—2122no—
169C22×C4—2124no—
170C22×C4C22⋊#615711yescharacteristic
171C22×C4C22×#615711yescharacteristic
172C2×C8—2124no—
173C2×C8—2124no—
174C2×C8—2124no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—2122no—
178C4⋊C4—2122no—
179C4⋊C4—2124no—
180C4⋊C4—2124no—
181C4⋊C4—2128no—
182C42—534no—
183C42—1064no—
184C42—1062no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23#1313327712yes—
189C23#1313327914yes—
190C23C2×C22⋊#615711yescharacteristic
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—84no—
199C2×C4—534no—
200C2×C4—532no—
201C2×C4—532no—
202C2×C4—534no—
203C2×C4—534no—
204C2×C4—1068no—
205C2×C4—1068no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—4244no—
209C2×C4—2122no—
210C2×C4—2124no—
211C2×C4#1313327812yes—
212C2×C4#1313329612yes—
213C8—4244no—
214C22—22no—
215C22—24no—
216C22—24no—
217C22—22no—
218C22#1681148114yes—
219C22#1681156611yescharacteristic
220C22C2×#1313327711yescharacteristic
221C22C2×#1313327811yescharacteristic
222C22C2×#1313327912yes—
223C22#1681161612yes—
224C4—22no—
225C4—22no—
226C4—82no—
227C4—534no—
228C4—534no—
229C4—1064no—
230C4—4242no—
231C2—24no—
232C2C2×#1681148111yescharacteristic
233C2#2029022412yes—
234C2#2029024811yescharacteristic
235C2#2029061411yescharacteristic
236C2#2029061812yes—
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#8723
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
132. C53#8723semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202895932
C2×#168104922
#872353