Finite Group Databasebeta

← #28303370

Subgroups — #28303370

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#20289593C212yesmaximal, complemented
3#20289593C212yesmaximal, complemented
4#20277118C211yesmaximal, characteristic
5C2×#13133284C2214yes—
6C2×C22⋊#88324—22no—
7#16811489—22no—
8C2×#13133176C411yescharacteristic
9C22×C4×Dic53C2211yescharacteristic
10#16810492C2211yescharacteristic
11#16810492C2211yescharacteristic
12#16810488C411yescharacteristic
13#16810488C411yescharacteristic
14#16810481—22no—
15C4×#11719919—22no—
16C2×#117×C53C411yescharacteristic
17C22×#88324—24no—
18#13133284—28no—
19C22⋊#88324—44no—
20C23×Dic53D412yes—
21#13133176—42no—
22C2×C4⋊Dic53C2×C411yescharacteristic
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×#11719919—22no—
25C2×#11719919—22no—
26C2×C22⋊Dic53—21no—
27C2×C4×Dic53—24no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53D412yes—
31C2×C4×Dic53C2311yescharacteristic
32#13133124—22no—
33#13133124—22no—
34C23×C4×C53C2×C411yescharacteristic
35#117×C53—42no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C22⋊C4×C53—21no—
39C2×C42×C53—21no—
40C24×C53C22⋊C411yescharacteristic
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—24no—
49C22×Dic53—24no—
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C2×#88324—44no—
57C2×#88324—44no—
58C2×#88324—44no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—41no—
61C22⋊C4×C53—42no—
62C42×C53—42no—
63C22⋊Dic53—42no—
64C4⋊Dic53—41no—
65C4⋊Dic53—41no—
66#11719919—44no—
67#11719919—42no—
68#11719919—42no—
69C4×Dic53—22no—
70C4×Dic53—24no—
71C4×Dic53—42no—
72C4×Dic53#4114yes—
73C23×C53—21no—
74C23×C53—21no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53#10112yes—
78C23×C53C2×C22⋊C411yescharacteristic
79C2×C4×C53—28no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53#10012yes—
89C2×C4×C53#11712yes—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—44no—
94C2×Dic53—44no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53#12314yes—
100C2×Dic53#13514yes—
101C2×Dic53AGL(1,ℤ/8)12yes—
102C2×Dic53#13712yes—
103#88324—84no—
104#8728—531nomaximal, Sylow
105C22×C53—24no—
106C22×C53—22no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—42no—
110C22×C53#38111yescharacteristic
111C22×C53C2×#10011yescharacteristic
112C22×C53C2×#10111yescharacteristic
113C22×C53#41612yes—
114C22×C53#41712yes—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—81no—
118C4×C53—81no—
119Dic53—42no—
120Dic53—81no—
121Dic53—81no—
122Dic53#48214yes—
123Dic53#48514yes—
124#2285—531no—
125#1797—532no—
126#1296—532no—
127#1296—532no—
128C2×C53—42no—
129C2×C53#154311yescharacteristic
130C2×C53#155212yes—
131C2×C53#156011yescharacteristic
132C2×C53#182111yescharacteristic
133C2×C53#182512yes—
134C2×#117—531no—
135C2×#117#615711yescharacteristic
136C22×C42—531no—
137C2×C4⋊C8—534no—
138C2×C22⋊C8—1062no—
139#384—531no—
140#384—531no—
141#382—531no—
142#382—531no—
143#380—1062no—
144C4×C4⋊C4—1062no—
145#336—1062no—
146C53#872811yescharacteristic, complemented, Sylow
147C23×C4—532no—
148C23×C4C2×#615711yescharacteristic
149C22×C8—1064no—
150#117—42no—
151#117—2122no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4C2×#615711yescharacteristic
157C2×C4⋊C4C2×#615711yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—1061no—
160C2×C42—21no—
161C2×C42—531no—
162C2×C42—532no—
163C2×C42—531no—
164C2×C42—1064no—
165C2×C42—1061no—
166C4⋊C8—1068no—
167C22⋊C8—2124no—
168#95—1062no—
169#95—1062no—
170C24C22⋊#615711yescharacteristic
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—532no—
177C22×C4—532no—
178C22×C4—1062no—
179C22×C4—1064no—
180C22×C4—1064no—
181C22×C4—1062no—
182C22×C4—1061no—
183C22×C4—1061no—
184C22×C4C22⋊#615711yescharacteristic
185C22×C4C22×#615711yescharacteristic
186C2×C8—2124no—
187C2×C8—2124no—
188C2×C8—2124no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—2121no—
192C4⋊C4—2121no—
193C4⋊C4—2124no—
194C4⋊C4—2122no—
195C4⋊C4—2122no—
196C22⋊C4—42no—
197C22⋊C4—2122no—
198C42—42no—
199C42—534no—
200C42—1064no—
201C42—1062no—
202C42—2122no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23#1313327812yes—
208C23C2×C22⋊#615711yescharacteristic
209C2×C4—28no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—534no—
219C2×C4—534no—
220C2×C4—532no—
221C2×C4—532no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—2124no—
226C2×C4—2124no—
227C2×C4—2121no—
228C2×C4—2121no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4#1313327712yes—
232C2×C4#1313329612yes—
233C8—4244no—
234C22—22no—
235C22—24no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22#1681156611yescharacteristic
240C22C2×#1313327711yescharacteristic
241C22C2×#1313327811yescharacteristic
242C22#1681161312yes—
243C22#1681161412yes—
244C4—22no—
245C4—22no—
246C4—81no—
247C4—81no—
248C4—534no—
249C4—534no—
250C4—4241no—
251C4—4241no—
252C4—2122no—
253C2—42no—
254C2#2029021811yescharacteristic
255C2#2029023612yes—
256C2#2029025211yescharacteristic
257C2#2029061411yescharacteristic
258C2#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#8728
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
3. #20289593C2semidirect
146. C53#8728semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202895932
#202771182
#872853