Finite Group Databasebeta

← #28307971

Subgroups — #28307971

Order 13568.

454 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×#16811468C212yesmaximal
2#20289658C212yesmaximal, complemented
3#20289656C212yesmaximal, complemented
4#20277335C211yesmaximal, characteristic
5C2×C22⋊#88324—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×#13133176C411yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10#16810521—22no—
11#16810495—22no—
12#16810492C411yescharacteristic
13#16810492C2211yescharacteristic
14#16810488C411yescharacteristic
15#16810488C2211yescharacteristic
16C2×#117×C53C411yescharacteristic
17C22×#88324—42no—
18C22⋊#88324—24no—
19C22⋊#88324—42no—
20C23×Dic53D412yes—
21#13133176—22no—
22#13133176C814yescomplemented
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—42no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53C2311yescharacteristic
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33#13133124—22no—
34#13133124—22no—
35C23×C4×C53C2×C411yescharacteristic
36#117×C53—42no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53—21no—
40C2×C42×C53—21no—
41C24×C53C22⋊C411yescharacteristic
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C2×C812yes—
48C22×C4×C53M1612yes—
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53—22no—
54C22×Dic53—82no—
55C22×Dic53C22⋊C412yes—
56C22×Dic53C2×D412yes—
57C2×#88324—44no—
58C2×#88324—84no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—41no—
61C22⋊C4×C53—42no—
62C42×C53—42no—
63C22⋊Dic53—24no—
64C4⋊Dic53—24no—
65C4⋊Dic53—41no—
66C4⋊Dic53—42no—
67C4⋊Dic53—41no—
68#11719919—82no—
69#11719919—82no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53C2×C812yescomplemented
73C4×Dic53C2×C812yescomplemented
74C23×C53—41no—
75C23×C53—41no—
76C23×C53C22⋊C812yes—
77C23×C53#9912yes—
78C23×C53#10112yes—
79C23×C53C2×C22⋊C411yescharacteristic
80C2×C4×C53—24no—
81C2×C4×C53—24no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53C22⋊C812yes—
91C2×C4×C53C22×C811yescharacteristic
92C2×C4×C53C2×M1611yescharacteristic
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—44no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—82no—
105#88324—84no—
106#11448—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—22no—
109C22×C53—21no—
110C22×C53—22no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53#32414yes—
114C22×C53C2×C22⋊C811yescharacteristic
115C22×C53C2×#9911yescharacteristic
116C22×C53C2×#10111yescharacteristic
117C4×C53—22no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—81no—
121C4×C53—81no—
122Dic53—42no—
123Dic53—42no—
124Dic53—42no—
125Dic53—42no—
126#2317—531no—
127C2×#324—532no—
128#1325—532no—
129#1324—532no—
130C2×C53—21no—
131C2×C53—24no—
132C2×C53—21no—
133C2×C53C2×#32411yescharacteristic
134C2×C53#212711yescharacteristic
135C2×C53#213311yescharacteristic
136C2×#117—531no—
137C2×#117#615711yescharacteristic
138C22×C4⋊C4—531no—
139C2×C22⋊C8—1062no—
140#389—1062no—
141#385—1062no—
142#384—531no—
143#384—531no—
144#382—531no—
145#382—531no—
146#324—534no—
147#324—1062no—
148C53#1144811yescharacteristic, complemented, Sylow
149C23×C4—532no—
150C23×C4C2×#615711yescharacteristic
151C22×C8—2122no—
152#117—42no—
153#117—534no—
154#117—1062no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—531no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—2122no—
162C2×C4⋊C4C2×#615711yescharacteristic
163C2×C4⋊C4C2×#615711yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—1061no—
166C2×C42—21no—
167C2×C42—531no—
168C2×C42—531no—
169C22⋊C8—1064no—
170C22⋊C8—2122no—
171#95—1062no—
172#95—1062no—
173C24C22⋊#615711yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—532no—
179C22×C4—532no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4—1061no—
183C22×C4—1061no—
184C22×C4—4242no—
185C22×C4D53⋊C812yes—
186C22×C4#1171993612yes—
187C22×C4C22⋊#615711yescharacteristic
188C22×C4C22×#615711yescharacteristic
189C2×C8—4244no—
190C2×C8—2124no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—1064no—
194C4⋊C4—4242no—
195C4⋊C4—4242no—
196C4⋊C4—2122no—
197C4⋊C4—2121no—
198C4⋊C4—2121no—
199C22⋊C4—42no—
200C22⋊C4—1064no—
201C42—42no—
202C42—532no—
203C42—532no—
204C42—1061no—
205C42—1061no—
206C23—41no—
207C23—41no—
208C23#1313327412yes—
209C23#1313327512yes—
210C23#1313327812yes—
211C23C2×C22⋊#615711yescharacteristic
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—4242no—
227C2×C4—2121no—
228C2×C4—2121no—
229C2×C4—2124no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4#1313327412yes—
235C2×C4C2×D53⋊C811yescharacteristic
236C2×C4C2×#1171993611yescharacteristic
237C8—4244no—
238C22—21no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22#1681146914yes—
245C22C2×#1313327411yescharacteristic
246C22C2×#1313327511yescharacteristic
247C22C2×#1313327811yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—81no—
252C4—81no—
253C4—2122no—
254C4—2122no—
255C4—2122no—
256C4—2122no—
257C2—21no—
258C2—21no—
259C2—24no—
260C2C2×#1681146911yescharacteristic
261C2#2029099011yescharacteristic
262C2#2029101011yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11448
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289658C2semidirect
3. #20289656C2semidirect
22. #13133176C8semidirect
72. C4×Dic53C2×C8semidirect
73. C4×Dic53C2×C8semidirect
148. C53#11448semidirect

Maximal subgroups

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

subgroupindex
C2×#168114682
#202896582
#202896562
#202773352
#1144853