Finite Group Databasebeta

← #28308193

Subgroups — #28308193

Order 13568.

490 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 84 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#20289963C212yesmaximal
2#20289656C214yesmaximal, complemented
3C2×#16810488C211yesmaximal, characteristic
4#16811593—22no—
5#16811468—22no—
6#16811468C2214yes—
7C22×C4⋊Dic53C2211yescharacteristic
8C22×C4×Dic53C411yescharacteristic
9#16810488—22no—
10#16810488C412yes—
11#16810488C2212yes—
12C2×#13133124—22no—
13C22×C4⋊C4×C53C411yescharacteristic
14C2×#11719940—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—41no—
20C2×C4⋊Dic53—41no—
21C2×C4⋊Dic53C2×C411yescharacteristic
22C2×C4⋊Dic53D412yes—
23C2×C4⋊Dic53C2311yescharacteristic
24C2×C4×Dic53—24no—
25C2×C4×Dic53—21no—
26C2×C4×Dic53C2×C412yes—
27#13133124—24no—
28#13133124—24no—
29C23×C4×C53—21no—
30C23×C4×C53C2×C411yescharacteristic
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—41no—
33C2×C4⋊C4×C53—41no—
34C2×C4⋊C4×C53C2×C412yes—
35C24×C53C22⋊C411yescharacteristic
36C22×C4×C53—21no—
37C22×C4×C53—21no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—44no—
42C22×C4×C53—41no—
43C22×C4×C53—41no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C22×C411yescharacteristic
46C22×Dic53—24no—
47C22×Dic53—22no—
48C22×Dic53—22no—
49C22×Dic53—22no—
50C22×Dic53—42no—
51C22×Dic53—42no—
52C22×Dic53—44no—
53C22×Dic53C22⋊C412yes—
54C22×Dic53C2×D412yes—
55#11719940—84no—
56C2×#88324—44no—
57C4⋊C4×C53—42no—
58C4⋊C4×C53—42no—
59C4⋊Dic53—41no—
60C4⋊Dic53—42no—
61C4⋊Dic53—41no—
62C4×Dic53—24no—
63C4×Dic53—22no—
64C4×Dic53M1614yescomplemented
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53#9911yescharacteristic
70C23×C53#9911yescharacteristic
71C23×C53C2×C22⋊C411yescharacteristic
72C2×C4×C53—24no—
73C2×C4×C53—24no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—44no—
79C2×C4×C53—42no—
80C2×C4×C53—82no—
81C2×C4×C53—82no—
82C2×C4×C53#10112yes—
83C2×C4×C53C2×M1612yes—
84C2×Dic53—24no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—48no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—44no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—82no—
94#88324—84no—
95#11511—531nomaximal, Sylow
96C22×C53—24no—
97C22×C53—24no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53#35412yes—
105C22×C53#35512yes—
106C22×C53#40711yescharacteristic
107C22×C53C2×#9911yescharacteristic
108C22×C53C2×#10111yescharacteristic
109C4×C53—22no—
110C4×C53—22no—
111C4×C53—82no—
112Dic53—44no—
113Dic53—42no—
114Dic53—42no—
115C2×#382—531no—
116#1463—532no—
117#1324—534no—
118C2×C53—21no—
119C2×C53—22no—
120C2×C53—22no—
121C2×C53—21no—
122C2×C53#146311yescharacteristic
123C2×C53C2×#35411yescharacteristic
124C2×C53C2×#35511yescharacteristic
125C22×C4⋊C4—531no—
126C22×C4⋊C4#615711yescharacteristic
127C22×C42—531no—
128#407—1062no—
129#382—532no—
130#382—532no—
131#382—1062no—
132C2×#95—1062no—
133#324—534no—
134#324—1062no—
135C53#1151111yescharacteristic, complemented, Sylow
136C23×C4—21no—
137C23×C4—532no—
138C23×C4—1061no—
139C23×C4C2×#615711yescharacteristic
140C2×M16—2122no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—531no—
146C2×C4⋊C4—531no—
147C2×C4⋊C4—2121no—
148C2×C4⋊C4—2121no—
149C2×C4⋊C4C2×#615712yes—
150C2×C42—532no—
151C2×C42—1064no—
152C2×C42—1061no—
153C22⋊C8—1064no—
154C22⋊C8—2122no—
155#95—1064no—
156#95—1064no—
157C24C22⋊#615711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—44no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—532no—
167C22×C4—532no—
168C22×C4—1062no—
169C22×C4—1064no—
170C22×C4—1062no—
171C22×C4—1062no—
172C22×C4—2124no—
173C22×C4—2122no—
174C22×C4—2122no—
175C22×C4C22⋊#615711yescharacteristic
176C22×C4C22×#615711yescharacteristic
177M16—4244no—
178C2×C8—2124no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—2122no—
182C4⋊C4—2121no—
183C4⋊C4—2121no—
184C42—534no—
185C42—1064no—
186C42—1062no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23#1313327511yescharacteristic
192C23#1313327511yescharacteristic
193C23C2×C22⋊#615711yescharacteristic
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—1064no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—4242no—
208C2×C4—2128no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2124no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4#1313327812yes—
215C2×C4D53⋊M1612yes—
216C8—4244no—
217C22—24no—
218C22—24no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22#1681152912yes—
226C22#1681153112yes—
227C22#1681159511yescharacteristic
228C22C2×#1313327511yescharacteristic
229C22C2×#1313327811yescharacteristic
230C4—22no—
231C4—22no—
232C4—82no—
233C4—2124no—
234C4—2122no—
235C4—2122no—
236C2—21no—
237C2—22no—
238C2—22no—
239C2—21no—
240C2#2028996511yescharacteristic
241C2C2×#1681152911yescharacteristic
242C2C2×#1681153111yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11511
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289656C2semidirect
64. C4×Dic53M16semidirect
135. C53#11511semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202896562
C2×#168104882
#1151153