Finite Group Databasebeta

← #28303372

Subgroups — #28303372

Order 13568.

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

normal subgroups only

This group has 514 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#20290571C212yesmaximal
2#20289593C214yesmaximal, complemented
3#20277118C211yesmaximal, characteristic
4C2×#13133284C2214yes—
5C2×C22⋊#88324—22no—
6#16811489—22no—
7C2×#13133176C411yescharacteristic
8C22×C4×Dic53C2211yescharacteristic
9#16810492C2212yes—
10#16810488C412yes—
11#16810481—22no—
12C4×#11719919—22no—
13C2×#117×C53C411yescharacteristic
14C22×#88324—24no—
15#13133284—28no—
16C22⋊#88324—44no—
17C23×Dic53D412yes—
18#13133176—42no—
19C2×C4⋊Dic53C2×C412yes—
20C2×#11719919—22no—
21C2×#11719919—22no—
22C2×C22⋊Dic53—21no—
23C2×C4×Dic53—24no—
24C2×C4×Dic53—21no—
25C2×C4×Dic53C2×C411yescharacteristic
26C2×C4×Dic53D412yes—
27C2×C4×Dic53C2311yescharacteristic
28#13133124—24no—
29C23×C4×C53C2×C411yescharacteristic
30#117×C53—42no—
31C2×C4⋊C4×C53C2×C412yes—
32C2×C22⋊C4×C53—21no—
33C2×C42×C53—21no—
34C24×C53C22⋊C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53C22⋊C411yescharacteristic
40C22×C4×C53C22×C411yescharacteristic
41C22×Dic53—24no—
42C22×Dic53—24no—
43C22×Dic53—22no—
44C22×Dic53—22no—
45C22×Dic53—22no—
46C22×Dic53C22⋊C412yes—
47C22×Dic53C2×D412yes—
48C2×#88324—44no—
49C2×#88324—44no—
50C2×#88324—44no—
51C4⋊C4×C53—42no—
52C22⋊C4×C53—42no—
53C42×C53—42no—
54C22⋊Dic53—42no—
55C4⋊Dic53—42no—
56#11719919—44no—
57#11719919—42no—
58#11719919—42no—
59C4×Dic53—22no—
60C4×Dic53—24no—
61C4×Dic53—42no—
62C4×Dic53#4114yes—
63C23×C53—21no—
64C23×C53—21no—
65C23×C53—22no—
66C23×C53—22no—
67C23×C53#10012yes—
68C23×C53C2×C22⋊C411yescharacteristic
69C2×C4×C53—28no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53#10112yes—
78C2×C4×C53#11712yes—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—44no—
83C2×Dic53—44no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53#12314yes—
88C2×Dic53#13514yes—
89C2×Dic53AGL(1,ℤ/8)12yes—
90C2×Dic53#13712yes—
91#88324—84no—
92#8729—531nomaximal, Sylow
93C22×C53—24no—
94C22×C53—22no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—42no—
98C22×C53#38111yescharacteristic
99C22×C53C2×#10011yescharacteristic
100C22×C53C2×#10111yescharacteristic
101C22×C53#41612yes—
102C22×C53#41712yes—
103C4×C53—22no—
104C4×C53—22no—
105C4×C53—82no—
106Dic53—42no—
107Dic53—82no—
108Dic53#48214yes—
109Dic53#48514yes—
110#2285—531no—
111#1797—532no—
112#1296—534no—
113C2×C53—42no—
114C2×C53#154411yescharacteristic
115C2×C53#155112yes—
116C2×C53#156111yescharacteristic
117C2×C53#182111yescharacteristic
118C2×C53#182512yes—
119C2×#117—531no—
120C2×#117#615711yescharacteristic
121C22×C42—531no—
122C2×C4⋊C8—534no—
123C2×C22⋊C8—1062no—
124#384—532no—
125#382—532no—
126#380—1062no—
127C4×C4⋊C4—1062no—
128#336—1062no—
129C53#872911yescharacteristic, complemented, Sylow
130C23×C4—532no—
131C23×C4C2×#615711yescharacteristic
132C22×C8—1064no—
133#117—42no—
134#117—2122no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4—1062no—
138C2×C4⋊C4C2×#615712yes—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—1061no—
141C2×C42—21no—
142C2×C42—531no—
143C2×C42—532no—
144C2×C42—531no—
145C2×C42—1064no—
146C2×C42—1061no—
147C4⋊C8—1068no—
148C22⋊C8—2124no—
149#95—1064no—
150C24C22⋊#615711yescharacteristic
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—532no—
156C22×C4—532no—
157C22×C4—1062no—
158C22×C4—1064no—
159C22×C4—1064no—
160C22×C4—1062no—
161C22×C4—1062no—
162C22×C4C22⋊#615711yescharacteristic
163C22×C4C22×#615711yescharacteristic
164C2×C8—2124no—
165C2×C8—2124no—
166C2×C8—2124no—
167C4⋊C4—42no—
168C4⋊C4—2122no—
169C4⋊C4—2124no—
170C4⋊C4—2122no—
171C4⋊C4—2122no—
172C22⋊C4—42no—
173C22⋊C4—2122no—
174C42—42no—
175C42—534no—
176C42—1064no—
177C42—1062no—
178C42—2122no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23#1313327712yes—
184C23C2×C22⋊#615711yescharacteristic
185C2×C4—28no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—534no—
194C2×C4—534no—
195C2×C4—532no—
196C2×C4—532no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—2124no—
201C2×C4—2124no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4#1313327812yes—
206C2×C4#1313329612yes—
207C8—4244no—
208C22—22no—
209C22—24no—
210C22—42no—
211C22—41no—
212C22—41no—
213C22#1681156611yescharacteristic
214C22C2×#1313327711yescharacteristic
215C22C2×#1313327811yescharacteristic
216C22#1681161312yes—
217C22#1681161412yes—
218C4—22no—
219C4—22no—
220C4—82no—
221C4—534no—
222C4—534no—
223C4—4242no—
224C4—2122no—
225C2—42no—
226C2#2029022011yescharacteristic
227C2#2029023412yes—
228C2#2029025411yescharacteristic
229C2#2029061411yescharacteristic
230C2#2029062012yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8729
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289593C2semidirect
129. C53#8729semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202895932
#202771182
#872953