Finite Group Databasebeta

← #28298582

Subgroups — #28298582

Order 13568.

286 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 62 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#20290592C211yesmaximal, characteristic
3#20274752C211yesmaximal, characteristic
4C2×#13133292C412yescomplemented
5C2×#13133286—21no—
6#16811565—21no—
7#16811518—21no—
8C22⋊#11719910—21no—
9#16811497C412yes—
10C22×C8×D53C2211yescharacteristic
11C2×#13133152—21no—
12#16810586—21no—
13#16810388C411yescharacteristic, complemented
14C53×#341C411yescharacteristic, complemented
15C2×C4⋊#6157—22no—
16C2×C22⋊#6157—41no—
17#13133292—42no—
18C2×#11719910—22no—
19#13133286—42no—
20(C2×#6157)⋊C4—41no—
21C22×C4×D53D411yescharacteristic
22C2×#11719917—21no—
23C22×#88322D411yescharacteristic
24C2×C8×D53—42no—
25C2×C8×D53C2×C411yescharacteristic
26C2×C8×D53C2×C411yescharacteristic
27C2×C8×D53D411yescharacteristic
28C2×C8×D53Q811yescharacteristic
29#13133152—42no—
30#13133129—41no—
31#13133129—41no—
32C2×M16×C53—21no—
33C22×C8×C53C2×C411yescharacteristic
34C23×D53SD1611yescharacteristic
35C22×#6157—42no—
36C22×C4×C53C22⋊C411yescharacteristic
37C22×Dic53Q1611yescharacteristic
38C2×C4×D53—41no—
39C2×C4×D53—41no—
40C2×C4×D53C22⋊C411yescharacteristic
41C2×C4×D53C4⋊C411yescharacteristic
42C2×C4×D53D811yescharacteristic
43C2×C4×D53SD1611yescharacteristic
44C22⋊#6157—82no—
45C4⋊#6157—42no—
46C4⋊#6157—42no—
47M16×C53—41no—
48M16×C53—41no—
49C2×C8×C53—41no—
50C2×C8×C53—41no—
51C2×C8×C53C4211yescharacteristic
52C2×C8×C53C4⋊C411yescharacteristic
53#11719917—41no—
54#11719917—41no—
55C2×#88322—22no—
56C2×#88322—41no—
57C2×#88322C22⋊C411yescharacteristic
58C2×#88322C4⋊C411yescharacteristic
59C8×D53—22no—
60C8×D53—22no—
61C8×D53—41no—
62C8×D53—41no—
63#11719910—42no—
64C23×C53#10311yescharacteristic
65C22×D53—21no—
66C22×D53—21no—
67C22×D53—41no—
68C22×D53—41no—
69C22×D53#10211yescharacteristic
70C22×D53#10611yescharacteristic
71C2×#6157—82no—
72C2×#6157—82no—
73C2×#6157—82no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53#9511yescharacteristic
77C2×C4×C53#10211yescharacteristic
78C2×Dic53—41no—
79C2×Dic53#10311yescharacteristic
80C2×Dic53#10711yescharacteristic
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—21no—
84C4×D53—21no—
85C4×D53—41no—
86C4×D53—41no—
87C8×C53—21no—
88C8×C53—21no—
89C8×C53—81no—
90#88322—21no—
91#88322—81no—
92#88322#9912yes—
93#7235—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53#34011yescharacteristic
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53#36111yescharacteristic
106C2×D53#36211yescharacteristic
107#6157—162no—
108C4×C53—21no—
109C4×C53H(3,ℤ/4)11yescharacteristic
110C4×C53#34211yescharacteristic
111Dic53—21no—
112Dic53—21no—
113#2112—531no—
114#1810—531no—
115#1808—531no—
116C2×C53—41no—
117C2×C53#128911yescharacteristic
118C2×C53#138711yescharacteristic
119C2×C53#138911yescharacteristic
120D53—21no—
121D53—21no—
122D53—41no—
123C23×C8—531no—
124C2×M32—532no—
125C2×#108—1061no—
126C2×#107—1061no—
127#407—1061no—
128#380—1061no—
129C22⋊C16—1061no—
130C22⋊C16—1061no—
131#341—531no—
132#341#615711yescharacteristic, complemented
133#340—532no—
134C53#723511yescharacteristic, complemented, Sylow
135C23×C4—531no—
136C2×M16—21no—
137C2×M16—1061no—
138C22×C8—531no—
139C22×C8—531no—
140C22×C8—531no—
141C22×C8—531no—
142C22×C8—531no—
143C22×C8—2122no—
144C22×C8C2×#615711yescharacteristic
145C2×C4⋊C4—1062no—
146C2×C22⋊C4—2121no—
147M32—2122no—
148C2×C16—1062no—
149#108—2122no—
150#107—2122no—
151C22⋊C8—2121no—
152C22⋊C8—2121no—
153#95—2121no—
154C24—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—2121no—
161C22×C4—2121no—
162C22×C4—2122no—
163C22×C4C22⋊#615711yescharacteristic
164M16—41no—
165M16—41no—
166M16—2121no—
167M16—2121no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—531no—
171C2×C8—531no—
172C2×C8—1062no—
173C2×C8—1062no—
174C2×C8—1062no—
175C2×C8—2121no—
176C2×C8—2121no—
177C2×C8—2121no—
178C2×C8C4×#615711yescharacteristic
179C2×C8C4⋊#615711yescharacteristic
180C4⋊C4—2122no—
181C4⋊C4—2122no—
182C22⋊C4—4242no—
183C16—2122no—
184C23—531no—
185C23—531no—
186C23—1061no—
187C23—1061no—
188C23—2121no—
189C23—2121no—
190C23#1313328011yescharacteristic
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—531no—
194C2×C4—531no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—4242no—
200C2×C4—4242no—
201C2×C4—4242no—
202C2×C4—2121no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4(C2×#6157)⋊C411yescharacteristic
206C2×C4#1313327911yescharacteristic
207C8—21no—
208C8—21no—
209C8—81no—
210C8—532no—
211C8—1061no—
212C8—4241no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—531no—
217C22—531no—
218C22—1061no—
219C22—1061no—
220C22—1061no—
221C22—1061no—
222C22—1061no—
223C22—2122no—
224C22—2122no—
225C22#1681149611yescharacteristic
226C4—21no—
227C4—8482no—
228C4—1061no—
229C4—1061no—
230C4#1681149011yescharacteristic
231C4#1681150011yescharacteristic
232C2—41no—
233C2—1061no—
234C2—1061no—
235C2—2121no—
236C2#2028957711yescharacteristic
237C2#2028979111yescharacteristic
238C2#2028979511yescharacteristic
subgroupG/H
centerC22#16811496
commutatorC2×C8×C53C42
FrattiniC22×C8C2×#6157
FittingC53×#341C4
socleC22×C53#340
prime radicalC23×D53SD16
cosocleC22×C8×D53C22
hypercenter#341#6157
hypocenterC53#7235
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #341

Complements

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

NH ≅ G/Ntype
4. C2×#13133292C4semidirect
13. #16810388C4semidirect
14. C53×#341C4semidirect
132. #341#6157semidirect
134. C53#7235semidirect

Maximal subgroups

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

subgroupindex
#202909432
#202905922
#202747522
#723553