Finite Group Databasebeta

← #27531007

Subgroups — #27531007

Order 13568.

1140 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 175 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#20282997C212yesmaximal, complemented
2#20276138C213yesmaximal, complemented
3#20275898C216yesmaximal, complemented
4#20275713C213yesmaximal, complemented
5C53×#1779C211yesmaximal, characteristic, complemented
6C22×D424C2211yescharacteristic, complemented
7C2×#13133191C2216yes—
8C2×#13133186—22no—
9C2×#13133151C2213yescomplemented
10C2×#13133139C2216yescomplemented
11#16810502—26no—
12#16810483C2216yes—
13#16810386C2213yescomplemented
14#16810386C2213yescomplemented
15C24×C4×C53C2211yescharacteristic
16C2×C22⋊C8×C53C2213yes—
17C2×C22⋊C8×C53C2213yes—
18C22×D212C2312yes—
19C22×C22⋊D53—26no—
20C2×D424—43no—
21#13133191—46no—
22#13133191—46no—
23#13133186—48no—
24C2×C4⋊Dic53C2316yes—
25C2×C22⋊Dic53—26no—
26C2×#11719921—26no—
27#13133151—43no—
28#13133139—46no—
29#13133124—26no—
30C25×C53D411yescharacteristic
31C23×C4×C53—43no—
32C23×C4×C53—43no—
33C23×C4×C53D413yes—
34C23×C4×C53C2313yes—
35C22×C8×C53C2313yes—
36C22×C8×C53C2311yescharacteristic
37C22⋊C8×C53—43no—
38C22⋊C8×C53—43no—
39C24×C53—43no—
40C24×C53—43no—
41C24×C53D814yescomplemented
42C24×C53C2×D413yes—
43C23×D53—22no—
44C22×C4×C53—23no—
45C22×C4×C53—23no—
46C22×C4×C53—26no—
47C22×C4×C53—26no—
48C22×C4×C53—46no—
49C22×C4×C53—43no—
50C22×C4×C53—412no—
51C22×C4×C53—44no—
52C22×C4×C53C2×D413yes—
53C22×C4×C53C2411yescharacteristic
54C2×C22⋊D53—412no—
55C2×C22⋊D53—412no—
56C2×C22⋊D53—412no—
57C22×Dic53—26no—
58C2×D212—26no—
59C2×D212—26no—
60C2×C8×C53—26no—
61C2×C8×C53—23no—
62C2×C8×C53—23no—
63C22⋊Dic53—412no—
64#11719921—412no—
65C4⋊Dic53—26no—
66C4⋊Dic53—26no—
67D424—44no—
68C23×C53—26no—
69C23×C53—23no—
70C23×C53—26no—
71C23×C53—23no—
72C23×C53—412no—
73C23×C53—44no—
74C23×C53—412no—
75C23×C53C2×D816yes—
76C23×C53C22×D411yescharacteristic
77C22×D53—46no—
78C22×D53—42no—
79C22×D53—46no—
80C2×C4×C53—23no—
81C2×C4×C53—26no—
82C2×C4×C53—412no—
83C2×C4×C53—46no—
84C2×C4×C53—43no—
85C2×C4×C53—43no—
86C2×C4×C53H(5,𝔽2)13yes—
87C2×C4×C53H(5,𝔽2)13yes—
88C22⋊D53—824no—
89C2×Dic53—46no—
90C2×Dic53—46no—
91C2×Dic53—46no—
92D212—46no—
93D212—42no—
94C8×C53—43no—
95C8×C53—41no—
96#32706—531nomaximal, Sylow
97C22×C53—212no—
98C22×C53—26no—
99C22×C53—416no—
100C22×C53—46no—
101C22×C53—43no—
102C22×C53—43no—
103C22×C53#53413yes—
104C22×C53C22×D811yescharacteristic
105C22×C53#57513yes—
106C2×D53—86no—
107C2×D53—86no—
108C2×D53—82no—
109C4×C53—43no—
110C4×C53#56013yes—
111C4×C53H(3,𝔽4)11yescharacteristic
112Dic53—86no—
113#2847—532no—
114#2075—533no—
115#2033—536no—
116#1999—533no—
117#1779D5311yescharacteristic, complemented
118C2×C53—46no—
119C2×C53#281211yescharacteristic
120C2×C53#318613yes—
121C2×C53#319713yes—
122D53—162no—
123C24×C4C2×D5311yescharacteristic
124C22×D8—531no—
125C2×#121—536no—
126C2×#120—1062no—
127C2×#107—533no—
128C2×#102—536no—
129C2×C22⋊C8C2×D5313yes—
130C2×C22⋊C8C2×D5313yes—
131#386—1066no—
132#380—536no—
133#340—533no—
134#340—533no—
135C53#3270611yescharacteristic, complemented, Sylow
136C25D21211yescharacteristic
137C22×D4—532no—
138C22×D4—1066no—
139C23×C4—43no—
140C23×C4—43no—
141C23×C4D21213yes—
142C23×C4C22×D5313yes—
143C2×D8—2123no—
144C22×C8C22×D5311yescharacteristic
145C22×C8C22×D5313yes—
146#121—2126no—
147#121—2126no—
148#120—2128no—
149C2×C4⋊C4—536no—
150C2×C22⋊C4—1066no—
151C2×C22⋊C4—1066no—
152#107—2123no—
153#102—2126no—
154C22⋊C8—43no—
155C22⋊C8—43no—
156#95—1066no—
157C24—43no—
158C24—43no—
159C24—1062no—
160C24D42414yescomplemented
161C24C2×D21213yes—
162C2×D4—1066no—
163C2×D4—1066no—
164C2×D4—21212no—
165C2×D4—21212no—
166C2×D4—21212no—
167C22×C4—23no—
168C22×C4—23no—
169C22×C4—26no—
170C22×C4—26no—
171C22×C4—43no—
172C22×C4—46no—
173C22×C4—412no—
174C22×C4—44no—
175C22×C4—1066no—
176C22×C4C2×D21213yes—
177C22×C4C23×D5311yescharacteristic
178D8—2124no—
179C2×C8—26no—
180C2×C8—23no—
181C2×C8—23no—
182C4⋊C4—1066no—
183C4⋊C4—1066no—
184C22⋊C4—21212no—
185C22⋊C4—21212no—
186C23—26no—
187C23—23no—
188C23—26no—
189C23—23no—
190C23—412no—
191C23—44no—
192C23—412no—
193C23—2122no—
194C23—2126no—
195C23—2126no—
196C23C2×D42416yes—
197C23C22×D21211yescharacteristic
198D4—42424no—
199D4—2126no—
200D4—2122no—
201C2×C4—26no—
202C2×C4—23no—
203C2×C4—412no—
204C2×C4—46no—
205C2×C4—43no—
206C2×C4—43no—
207C2×C4—2126no—
208C2×C4—2126no—
209C2×C4—2126no—
210C2×C4#1313326213yes—
211C2×C4#1313326213yes—
212C8—43no—
213C8—41no—
214C22—212no—
215C22—26no—
216C22—416no—
217C22—43no—
218C22—43no—
219C22—46no—
220C22—4246no—
221C22—4242no—
222C22—4246no—
223C22#1681118513yes—
224C22C22×D42411yescharacteristic
225C22#1681142313yes—
226C4—43no—
227C4—4246no—
228C4#1681136213yes—
229C4#1681136611yescharacteristic
230C2—46no—
231C2—8482no—
232C2#2028271211yescharacteristic
233C2#2028546613yes—
234C2#2028555713yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1779C2
socleC23×C53C22×D4
hypercenter#1779D53
hypocenterC53#32706
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
1. #20282997C2semidirect
2. #20276138C2semidirect
3. #20275898C2semidirect
4. #20275713C2semidirect
5. C53×#1779C2semidirect
6. C22×D424C22semidirect
9. C2×#13133151C22semidirect
10. C2×#13133139C22semidirect
13. #16810386C22semidirect
14. #16810386C22semidirect
41. C24×C53D8semidirect
117. #1779D53semidirect
135. C53#32706semidirect
160. C24D424semidirect

Maximal subgroups

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

subgroupindex
#202829972
#202761382
#202758982
#202757132
C53×#17792
#3270653