Finite Group Databasebeta

← #28303347

Subgroups — #28303347

Order 13568.

554 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 112 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#20290441C212yesmaximal
2#20289591C214yesmaximal, complemented
3C2×#16810488C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5C2×#13133272C2214yes—
6#16811489—22no—
7C22×C4⋊Dic53C411yescharacteristic
8C22×C4×Dic53C2211yescharacteristic
9#16810488—22no—
10#16810488C412yes—
11#16810488C2212yes—
12C2×#13133124—22no—
13C22×C4⋊C4×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—28no—
16#13133272—28no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—42no—
21C2×C4⋊Dic53C2×C412yes—
22C2×C4×Dic53—24no—
23C2×C4×Dic53C2×C411yescharacteristic
24C2×C4×Dic53D412yes—
25C2×C4×Dic53C2311yescharacteristic
26#13133124—24no—
27#13133124—24no—
28C23×C4×C53—21no—
29C23×C4×C53C2×C411yescharacteristic
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—42no—
32C2×C4⋊C4×C53C2×C412yes—
33C24×C53C22⋊C411yescharacteristic
34C22×C4×C53—22no—
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—44no—
39C22×C4×C53—42no—
40C22×C4×C53C22⋊C411yescharacteristic
41C22×C4×C53C22×C411yescharacteristic
42C22×Dic53—24no—
43C22×Dic53—24no—
44C22×Dic53—22no—
45C22×Dic53—42no—
46C22×Dic53—42no—
47C22×Dic53—44no—
48C22×Dic53C22⋊C412yes—
49C22×Dic53C2×D412yes—
50C2×#88324—28no—
51C2×#88324—44no—
52C2×#88324—44no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—42no—
55C4⋊Dic53—42no—
56C4⋊Dic53—42no—
57C4×Dic53—22no—
58C4×Dic53—24no—
59C4×Dic53#4114yes—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—22no—
63C23×C53#10012yes—
64C23×C53#10414yes—
65C23×C53C2×C22⋊C411yescharacteristic
66C2×C4×C53—28no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—42no—
72C2×C4×C53—44no—
73C2×C4×C53—84no—
74C2×C4×C53#10112yes—
75C2×C4×C53#11712yes—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—48no—
82C2×Dic53—42no—
83C2×Dic53—44no—
84C2×Dic53—84no—
85C2×Dic53#12314yes—
86#88324—48no—
87#8718—531nomaximal, Sylow
88C22×C53—22no—
89C22×C53—24no—
90C22×C53—24no—
91C22×C53—22no—
92C22×C53#32914yes—
93C22×C53#38111yescharacteristic
94C22×C53C2×#10011yescharacteristic
95C22×C53C2×#10111yescharacteristic
96C22×C53C2×#10412yes—
97C22×C53#42112yes—
98C4×C53—22no—
99C4×C53—22no—
100C4×C53—82no—
101Dic53—24no—
102Dic53—42no—
103Dic53—82no—
104C2×#382—531no—
105#1749—532no—
106#1295—534no—
107C2×C53—24no—
108C2×C53C2×#32911yescharacteristic
109C2×C53#152812yes—
110C2×C53#153311yescharacteristic
111C2×C53#182111yescharacteristic
112C2×C53#182612yes—
113C22×C4⋊C4—531no—
114C22×C4⋊C4#615711yescharacteristic
115C22×C42—531no—
116C2×C22⋊C8—1062no—
117C2×#97—534no—
118#382—532no—
119#382—532no—
120#382—1062no—
121C2×#95—1062no—
122#336—1062no—
123C53#871811yescharacteristic, complemented, Sylow
124C23×C4—21no—
125C23×C4—532no—
126C23×C4—1061no—
127C23×C4C2×#615711yescharacteristic
128C22×C8—1064no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—1061no—
133C2×C4⋊C4—2122no—
134C2×C4⋊C4C2×#615712yes—
135C2×C42—531no—
136C2×C42—532no—
137C2×C42—531no—
138C2×C42—1064no—
139C22⋊C8—1068no—
140#97—1068no—
141#95—1064no—
142#95—1064no—
143C24C22⋊#615711yescharacteristic
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4—1064no—
153C22×C4—1064no—
154C22×C4—1062no—
155C22×C4—2122no—
156C22×C4—2122no—
157C22×C4—2124no—
158C22×C4C22⋊#615711yescharacteristic
159C22×C4C22×#615711yescharacteristic
160C2×C8—1068no—
161C2×C8—2124no—
162C2×C8—2124no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—2122no—
166C4⋊C4—2122no—
167C42—534no—
168C42—1064no—
169C42—1062no—
170C23—21no—
171C23—21no—
172C23—22no—
173C23#1313327712yes—
174C23#1313328214yes—
175C23C2×C22⋊#615711yescharacteristic
176C2×C4—28no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—84no—
184C2×C4—534no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—4244no—
191C2×C4—2128no—
192C2×C4—2122no—
193C2×C4—2124no—
194C2×C4#1313327812yes—
195C2×C4#1313329612yes—
196C8—2128no—
197C22—22no—
198C22—24no—
199C22—24no—
200C22—22no—
201C22#1681147914yes—
202C22#1681156611yescharacteristic
203C22C2×#1313327711yescharacteristic
204C22C2×#1313327811yescharacteristic
205C22C2×#1313328212yes—
206C22#1681162112yes—
207C4—22no—
208C4—22no—
209C4—82no—
210C4—1064no—
211C4—4242no—
212C4—2122no—
213C2—24no—
214C2C2×#1681147911yescharacteristic
215C2#2029017912yes—
216C2#2029019411yescharacteristic
217C2#2029061411yescharacteristic
218C2#2029062212yes—
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#8718
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. #20289591C2semidirect
123. C53#8718semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895912
C2×#168104882
#871853