Finite Group Databasebeta

← #28298576

Subgroups — #28298576

Order 13568.

308 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 66 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#20274739C211yesmaximal, characteristic
4C2×#13133292C412yescomplemented
5C2×#13133285—21no—
6#16811565—21no—
7#16811518—21no—
8C22⋊#11719910—21no—
9#16811497C412yescomplemented
10C22×C8×D53C2211yescharacteristic
11C2×#13133148—21no—
12#16810480—21no—
13#16810386C411yescharacteristic, complemented
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
28C2×C4⋊Dic53—21no—
29C2×#11719921—41no—
30#13133148—42no—
31#13133124—41no—
32C22×C8×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53—21no—
34C23×D53D811yescharacteristic
35C22×#6157—42no—
36C22×C4×C53—41no—
37C22×C4×C53C22⋊C411yescharacteristic
38C22×Dic53—41no—
39C22×Dic53SD1611yescharacteristic
40C2×C4×D53—41no—
41C2×C4×D53—41no—
42C2×C4×D53C22⋊C411yescharacteristic
43C2×C4×D53C4⋊C411yescharacteristic
44C2×C4×D53SD1611yescharacteristic
45C2×C4×D53Q1611yescharacteristic
46C22⋊#6157—81no—
47C22⋊#6157—81no—
48C4⋊#6157—42no—
49C4⋊#6157—42no—
50C2×C8×C53—41no—
51C2×C8×C53C4211yescharacteristic
52C2×C8×C53C4⋊C411yescharacteristic
53C4⋊C4×C53—41no—
54C4⋊C4×C53—41no—
55#11719921—82no—
56C4⋊Dic53—41no—
57C4⋊Dic53—41no—
58C2×#88322—22no—
59C2×#88322C22⋊C411yescharacteristic
60C2×#88322C4⋊C411yescharacteristic
61C8×D53—22no—
62C8×D53—22no—
63C8×D53—41no—
64C8×D53—41no—
65#11719910—42no—
66C23×C53#10211yescharacteristic
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—22no—
70C22×D53—41no—
71C22×D53#10211yescharacteristic
72C22×D53#10711yescharacteristic
73C2×#6157—82no—
74C2×#6157—82no—
75C2×#6157—82no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—81no—
79C2×C4×C53—81no—
80C2×C4×C53—81no—
81C2×C4×C53#9511yescharacteristic
82C2×C4×C53#10311yescharacteristic
83C2×Dic53—41no—
84C2×Dic53—81no—
85C2×Dic53—81no—
86C2×Dic53—81no—
87C2×Dic53#10311yescharacteristic
88C2×Dic53#10611yescharacteristic
89C4×D53—21no—
90C4×D53—21no—
91C4×D53—21no—
92C4×D53—21no—
93C4×D53—41no—
94C4×D53—41no—
95C8×C53—21no—
96C8×C53—21no—
97#88322—21no—
98#88322#9912yes—
99#7233—531nomaximal, Sylow
100C22×C53—21no—
101C22×C53—21no—
102C22×C53#34011yescharacteristic
103C22×C53#36011yescharacteristic
104C22×C53#36011yescharacteristic
105C2×D53—21no—
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—21no—
110C2×D53—42no—
111C2×D53—42no—
112C2×D53#36012yes—
113#6157—162no—
114C4×C53—21no—
115C4×C53—161no—
116C4×C53H(3,ℤ/4)11yescharacteristic
117C4×C53#34211yescharacteristic
118Dic53—21no—
119Dic53—21no—
120Dic53—161no—
121#2112—531no—
122#1807—531no—
123#1807—531no—
124C2×C53—21no—
125C2×C53—21no—
126C2×C53#128911yescharacteristic
127C2×C53#138611yescharacteristic
128C2×C53#138611yescharacteristic
129D53—22no—
130D53—41no—
131C23×C8—531no—
132C2×M32—532no—
133C2×#106—1061no—
134C2×#106—1061no—
135#380—1061no—
136#380—1061no—
137C22⋊C16—1061no—
138C22⋊C16—1061no—
139#340—532no—
140#340—531no—
141#340#615711yescharacteristic, complemented
142C53#723311yescharacteristic, complemented, Sylow
143C23×C4—531no—
144C22×C8—531no—
145C22×C8—531no—
146C22×C8—531no—
147C22×C8—531no—
148C22×C8—531no—
149C22×C8—2122no—
150C22×C8C2×#615711yescharacteristic
151C2×C4⋊C4—21no—
152C2×C4⋊C4—1062no—
153C2×C4⋊C4—1061no—
154C2×C22⋊C4—2121no—
155C2×C22⋊C4—2121no—
156M32—2122no—
157C2×C16—1062no—
158#106—2122no—
159#106—2122no—
160#95—2121no—
161#95—2121no—
162C24—531no—
163C22×C4—41no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—2121no—
170C22×C4—2121no—
171C22×C4—2122no—
172C22×C4—2121no—
173C22×C4C22⋊#615711yescharacteristic
174C2×C8—41no—
175C2×C8—531no—
176C2×C8—531no—
177C2×C8—1062no—
178C2×C8—1062no—
179C2×C8—1062no—
180C2×C8—2121no—
181C2×C8—2121no—
182C2×C8C4×#615711yescharacteristic
183C2×C8C4⋊#615711yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—2121no—
187C4⋊C4—2121no—
188C4⋊C4—2122no—
189C4⋊C4—2122no—
190C22⋊C4—4242no—
191C22⋊C4—4241no—
192C22⋊C4—4241no—
193C16—2122no—
194C23—531no—
195C23—531no—
196C23—1062no—
197C23—1061no—
198C23—1061no—
199C23—2121no—
200C23#1313327911yescharacteristic
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—531no—
207C2×C4—531no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—4241no—
213C2×C4—4241no—
214C2×C4—4242no—
215C2×C4—4242no—
216C2×C4—4242no—
217C2×C4—4241no—
218C2×C4—2121no—
219C2×C4—2121no—
220C2×C4—2121no—
221C2×C4(C2×#6157)⋊C411yescharacteristic
222C2×C4#1313328011yescharacteristic
223C8—21no—
224C8—21no—
225C8—532no—
226C8—1061no—
227C22—21no—
228C22—21no—
229C22—532no—
230C22—1061no—
231C22—1061no—
232C22—1061no—
233C22—1061no—
234C22—1061no—
235C22—2122no—
236C22—2122no—
237C22#1681149611yescharacteristic
238C22#1681153911yescharacteristic
239C22#1681154011yescharacteristic
240C4—21no—
241C4—161no—
242C4—8481no—
243C4—8482no—
244C4—1061no—
245C4—1061no—
246C4#1681149011yescharacteristic
247C4#1681150011yescharacteristic
248C2—21no—
249C2—21no—
250C2—1062no—
251C2—2121no—
252C2#2028957711yescharacteristic
253C2#2028978611yescharacteristic
254C2#2028978811yescharacteristic
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#7233
LCSG ▷ C2×C8×C53 ▷ 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
4. C2×#13133292C4semidirect
9. #16811497C4semidirect
13. #16810386C4semidirect
14. C53×#340C4semidirect
141. #340#6157semidirect
142. C53#7233semidirect

Maximal subgroups

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

subgroupindex
#202909432
#202905912
#202747392
#723353