Finite Group Databasebeta

← #28298668

Subgroups — #28298668

Order 13568.

352 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 70 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#20290943C211yesmaximal, characteristic
2#20290591C211yesmaximal, characteristic
3#20274430C211yesmaximal, characteristic
4C2×#13133292—21no—
5C2×#13133285C412yescomplemented
6#16811565—21no—
7#16811518C412yescomplemented
8C22⋊#11719910—21no—
9#16811497—21no—
10C22×C8×D53C2211yescharacteristic
11C2×#13133175—21no—
12#16810386C411yescharacteristic, complemented
13#16810385—21no—
14C53×#340C411yescharacteristic, complemented
15C2×C4⋊#6157—22no—
16C2×C22⋊#6157—41no—
17#13133292—42no—
18C2×#11719910—22no—
19#13133285—42no—
20(C2×#6157)⋊C4—41no—
21C22×C4×D53D411yescharacteristic
22C22×#88322D411yescharacteristic
23C2×C8×D53—42no—
24C2×C8×D53C2×C411yescharacteristic
25C2×C8×D53C2×C411yescharacteristic
26C2×C8×D53D411yescharacteristic
27C2×C8×D53Q811yescharacteristic
28#13133175—24no—
29#13133175—42no—
30C2×C4⋊Dic53—21no—
31C2×#11719921—41no—
32C2×C4×Dic53—41no—
33C22×C8×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53—21no—
35C23×D53D811yescharacteristic
36C22×#6157—42no—
37C22×C4×C53—41no—
38C22×C4×C53C22⋊C411yescharacteristic
39C22×Dic53—41no—
40C22×Dic53SD1611yescharacteristic
41C2×C4×D53—22no—
42C2×C4×D53—22no—
43C2×C4×D53C22⋊C411yescharacteristic
44C2×C4×D53C4⋊C411yescharacteristic
45C2×C4×D53SD1611yescharacteristic
46C2×C4×D53Q1611yescharacteristic
47C22⋊#6157—81no—
48C22⋊#6157—81no—
49C4⋊#6157—42no—
50C4⋊#6157—42no—
51C2×C8×C53—41no—
52C2×C8×C53C4211yescharacteristic
53C2×C8×C53C4⋊C411yescharacteristic
54C4⋊C4×C53—22no—
55C4⋊C4×C53—41no—
56#11719921—44no—
57C4⋊Dic53—22no—
58C4⋊Dic53—41no—
59C4×Dic53—44no—
60C2×#88322—41no—
61C2×#88322C22⋊C411yescharacteristic
62C2×#88322C4⋊C411yescharacteristic
63C8×D53—22no—
64C8×D53—22no—
65C8×D53—41no—
66C8×D53—41no—
67#11719910—42no—
68C23×C53#10211yescharacteristic
69C22×D53—21no—
70C22×D53—21no—
71C22×D53—22no—
72C22×D53—41no—
73C22×D53#10211yescharacteristic
74C22×D53#10711yescharacteristic
75C2×#6157—82no—
76C2×#6157—82no—
77C2×#6157—82no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—42no—
81C2×C4×C53—81no—
82C2×C4×C53—81no—
83C2×C4×C53#9511yescharacteristic
84C2×C4×C53#10311yescharacteristic
85C2×Dic53—22no—
86C2×Dic53—42no—
87C2×Dic53—81no—
88C2×Dic53—81no—
89C2×Dic53#10311yescharacteristic
90C2×Dic53#10611yescharacteristic
91C4×D53—22no—
92C4×D53—21no—
93C4×D53—21no—
94C4×D53—22no—
95C4×D53#9912yes—
96C4×D53#10412yes—
97C8×C53—21no—
98C8×C53—21no—
99#88322—21no—
100#88322—21no—
101#7262—531nomaximal, Sylow
102C22×C53—21no—
103C22×C53—21no—
104C22×C53#34011yescharacteristic
105C22×C53#36011yescharacteristic
106C22×C53#36011yescharacteristic
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—21no—
110C2×D53—21no—
111C2×D53—21no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53#32712yes—
115#6157—162no—
116C4×C53—21no—
117C4×C53—82no—
118C4×C53#33911yescharacteristic
119C4×C53#34211yescharacteristic
120Dic53—21no—
121Dic53—82no—
122Dic53#32812yes—
123#2112—531no—
124#1807—531no—
125#1735—531no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53#129411yescharacteristic
129C2×C53#138411yescharacteristic
130C2×C53#138611yescharacteristic
131D53—22no—
132D53—41no—
133C23×C8—531no—
134C2×#117—1061no—
135C2×M32—1061no—
136C2×#106—532no—
137#380—1061no—
138C22⋊C16—532no—
139C22⋊C16—1061no—
140#340—531no—
141#340—1061no—
142#340—1061no—
143#340#615711yescharacteristic, complemented
144C53#726211yescharacteristic, complemented, Sylow
145C23×C4—531no—
146C22×C8—531no—
147C22×C8—531no—
148C22×C8—531no—
149C22×C8—531no—
150C22×C8—531no—
151C22×C8—2122no—
152C22×C8C2×#615711yescharacteristic
153#117—1064no—
154#117—2122no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—1061no—
157C2×C4⋊C4—1062no—
158C2×C22⋊C4—2121no—
159C2×C22⋊C4—2121no—
160C2×C42—2121no—
161M32—2122no—
162C2×C16—1062no—
163#106—2122no—
164#95—2121no—
165C24—531no—
166C22×C4—41no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—531no—
171C22×C4—531no—
172C22×C4—1062no—
173C22×C4—1062no—
174C22×C4—2122no—
175C22×C4—2121no—
176C22×C4C22⋊#615711yescharacteristic
177C2×C8—41no—
178C2×C8—531no—
179C2×C8—531no—
180C2×C8—1062no—
181C2×C8—1062no—
182C2×C8—2121no—
183C2×C8—2121no—
184C2×C8—2121no—
185C2×C8C4×#615711yescharacteristic
186C2×C8C4⋊#615711yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—1062no—
190C4⋊C4—2121no—
191C4⋊C4—2122no—
192C4⋊C4—2122no—
193C22⋊C4—4241no—
194C22⋊C4—4241no—
195C22⋊C4—2124no—
196C42—2124no—
197C16—2122no—
198C23—531no—
199C23—531no—
200C23—1062no—
201C23—1061no—
202C23—1061no—
203C23—2121no—
204C23#1313327911yescharacteristic
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—531no—
211C2×C4—532no—
212C2×C4—532no—
213C2×C4—531no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1061no—
218C2×C4—1061no—
219C2×C4—4242no—
220C2×C4—4242no—
221C2×C4—4242no—
222C2×C4—4241no—
223C2×C4—4241no—
224C2×C4—2122no—
225C2×C4(C2×#6157)⋊C411yescharacteristic
226C2×C4#1313328011yescharacteristic
227C8—21no—
228C8—21no—
229C8—1061no—
230C8—1061no—
231C22—21no—
232C22—21no—
233C22—532no—
234C22—1061no—
235C22—1061no—
236C22—1061no—
237C22—1061no—
238C22—1061no—
239C22—2122no—
240C22—2122no—
241C22#1681149611yescharacteristic
242C22#1681153911yescharacteristic
243C22#1681154011yescharacteristic
244C4—21no—
245C4—82no—
246C4—532no—
247C4—8482no—
248C4—1061no—
249C4—4242no—
250C4#1681149411yescharacteristic
251C4#1681150011yescharacteristic
252C2—21no—
253C2—21no—
254C2—1062no—
255C2—2121no—
256C2#2028958911yescharacteristic
257C2#2028978211yescharacteristic
258C2#2028978611yescharacteristic
subgroupG/H
centerC22#16811496
commutatorC2×C8×C53C42
FrattiniC22×C8C2×#6157
FittingC53×#340C4
socleC22×C53#340
prime radicalC23×D53D8
cosocleC22×C8×D53C22
hypercenter#340#6157
hypocenterC53#7262
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #340

Complements

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

NH ≅ G/Ntype
5. C2×#13133285C4semidirect
7. #16811518C4semidirect
12. #16810386C4semidirect
14. C53×#340C4semidirect
143. #340#6157semidirect
144. C53#7262semidirect

Maximal subgroups

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

subgroupindex
#202909432
#202905912
#202744302
#726253