Finite Group Databasebeta

← C4⋊(A4⋊#157) #13180651

Subgroups — C4⋊(A4⋊#157) #13180651

Order 1728.

374 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 36 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×A4⋊#157C212yesmaximal
2C4×A4×C3⋊D3C211yesmaximal, characteristic
3#10593812—31nomaximal
4C2×A4×C3⋊D3C2211yescharacteristic
5A4⋊#157—22no—
6C32×C4×A4C411yescharacteristic, complemented
7C4⋊(C3⋊#157)—41nomaximal
8A4×C3⋊Dic3C411yescharacteristic, complemented
9C22×C4×C3⋊D3D311yescharacteristic
10C2×C22⋊#157—32no—
11C2×C4⋊#157—31no—
12C2×C4⋊#157—31no—
13C4×D3×A4—62no—
14#63781—32no—
15C2×C32×A4C2×C411yescharacteristic
16C2×C3⋊#157—42no—
17A4×C3⋊D3D411yescharacteristic
18A4×C3⋊D3Q811yescharacteristic
19C3×C4×C3⋊D3—41no—
20C4⋊SL(2,ℤ/4)—91nomaximal
21C23×C3⋊D3C2×D311yescharacteristic
22C22×#157—32no—
23C22×#157—32no—
24C2×D3×A4—62no—
25C22×C32×C4Dic311yescharacteristic, complemented
26C22×C3⋊Dic3Dic311yescharacteristic, complemented
27C2×C4×C3⋊D3—31no—
28C2×C4×C3⋊D3—31no—
29C2×C4×C3⋊D3—31no—
30C2×C4×C3⋊D3—31no—
31C3×C4×A4—22no—
32C3×C4×A4—42no—
33C22⋊#157—62no—
34C22⋊#157—62no—
35C4⋊#157—61no—
36C4⋊#157—61no—
37C4⋊#157—61no—
38C4⋊#157—61no—
39Dic3×A4—62no—
40C2×C3×C3⋊D3—41no—
41C32×A4C4⋊C411yescharacteristic, complemented
42C3⋊#157—82no—
43C33×C4—41no—
44C3×C3⋊Dic3—41no—
45C22×C4×D3—62no—
46C2×C4×A4—91no—
47C2×SL(2,ℤ/4)—92no—
48C23×C32C2×Dic311yescharacteristic
49C22×C3⋊D3—31no—
50C22×C3⋊D3—31no—
51C22×C3⋊D3—31no—
52C22×C3⋊D3—31no—
53C22×C3⋊D3Dic611yescharacteristic
54C22×C3⋊D3D1211yescharacteristic
55C2×C3×A4—22no—
56C2×C3×A4—42no—
57C2×#157—62no—
58C2×#157—62no—
59C2×#157—62no—
60C2×#157—62no—
61C2×#157—62no—
62C2×#157—62no—
63D3×A4—62no—
64D3×A4—62no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C3⋊Dic3—31no—
68C2×C3⋊Dic3—31no—
69C4×C3⋊D3—31no—
70C4×C3⋊D3—31no—
71C4×C3⋊D3—31no—
72C4×C3⋊D3—61no—
73C4×C3⋊D3S411yescharacteristic
74C3×C4×D3—242no—
75#380—271noSylow
76C2×C33—41no—
77C3×C3⋊D3—41no—
78C3×C3⋊D3—41no—
79C23×D3—62no—
80C22×A4—91no—
81C22×C3×C4—22no—
82C22×Dic3—62no—
83C2×C4×D3—182no—
84C2×C4×D3—182no—
85C2×C4×D3—182no—
86C2×C4×D3—182no—
87C4×A4—42no—
88C4×A4—91no—
89C4×A4#15711yescharacteristic, complemented
90SL(2,ℤ/4)—182no—
91C4⋊Dic3—361no—
92C22×C32—31no—
93C22×C32—31no—
94C22×C32C4⋊Dic311yescharacteristic, complemented
95C2×C3⋊D3—31no—
96C2×C3⋊D3—31no—
97C2×C3⋊D3—31no—
98C2×C3⋊D3—31no—
99C2×C3⋊D3—31no—
100C2×C3⋊D3—31no—
101C2×C3⋊D3—31no—
102C2×C3⋊D3—61no—
103C2×C3⋊D3C2×S411yescharacteristic
104C2×C3×D3—242no—
105C3×A4—22no—
106C3×A4—42no—
107#157—122no—
108#157—122no—
109C32×C4—31no—
110C32×C4—82no—
111C32×C4—162no—
112C32×C4SL(2,ℤ/4)11yescharacteristic, complemented
113C3⋊Dic3—31no—
114C3⋊Dic3SL(2,ℤ/4)11yescharacteristic, complemented
115C3×Dic3—242no—
116C23×C4—91no—
117C2×C4⋊C4—271no—
118C2×C4⋊C4—271no—
119C2×C22⋊C4—272no—
120#95—272no—
121C33—41noSylow
122C23×C3—22no—
123C22×D3—62no—
124C22×D3—62no—
125C22×D3—182no—
126C22×D3—182no—
127C22×D3—182no—
128C22×D3—182no—
129C2×A4—42no—
130C2×A4—91no—
131C2×A4—91no—
132C2×A4C2×#15711yescharacteristic
133C2×C3×C4—62no—
134C2×C3×C4—62no—
135C2×C3×C4—361no—
136C2×Dic3—182no—
137C2×Dic3—182no—
138C2×Dic3—362no—
139C4×D3—62no—
140C4×D3—182no—
141C4×D3—182no—
142C4×D3—182no—
143C4×D3—184no—
144C2×C32—31no—
145C2×C32—31no—
146C2×C32—82no—
147C2×C32—162no—
148C2×C32C2×SL(2,ℤ/4)11yescharacteristic
149C3⋊D3—31no—
150C3⋊D3—31no—
151C3⋊D3A4⋊Q811yescharacteristic
152C3⋊D3C4⋊S411yescharacteristic
153C3×D3—242no—
154C3×D3—242no—
155C24—91no—
156C22×C4—91no—
157C22×C4—271no—
158C22×C4—271no—
159C22×C4—271no—
160C22×C4—271no—
161C22×C4—272no—
162C22×C4—272no—
163C22×C4C3⋊#15711yescharacteristic, complemented
164C4⋊C4—541no—
165C4⋊C4—541no—
166C4⋊C4—541no—
167C4⋊C4—541no—
168C22⋊C4—542no—
169C22⋊C4—542no—
170C22×C3—22no—
171C22×C3—62no—
172C22×C3—62no—
173C22×C3—361no—
174C2×D3—62no—
175C2×D3—182no—
176C2×D3—182no—
177C2×D3—182no—
178C2×D3—182no—
179C2×D3—182no—
180C2×D3—182no—
181C2×D3—182no—
182C2×D3—184no—
183A4—42no—
184A4C4⋊#15711yescharacteristic, complemented
185C3×C4—22no—
186C3×C4—41no—
187C3×C4—62no—
188C3×C4—162no—
189C3×C4—361no—
190Dic3—62no—
191Dic3—182no—
192Dic3—722no—
193C32—82no—
194C32—162no—
195C32C4⋊SL(2,ℤ/4)11yescharacteristic, complemented
196C23—91no—
197C23—91no—
198C23—271no—
199C23—271no—
200C23—271no—
201C23—271no—
202C23C2×C3⋊#15711yescharacteristic
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—91no—
206C2×C4—271no—
207C2×C4—271no—
208C2×C4—271no—
209C2×C4—271no—
210C2×C4—271no—
211C2×C4—541no—
212C2×C4—542no—
213C2×C4—542no—
214C2×C4—542no—
215C2×C4—542no—
216C2×C4—542no—
217C2×C4—542no—
218C2×C3—22no—
219C2×C3—41no—
220C2×C3—62no—
221C2×C3—62no—
222C2×C3—162no—
223C2×C3—361no—
224C2×C3—361no—
225D3—62no—
226D3—62no—
227D3—182no—
228D3—182no—
229C22—31no—
230C22—31no—
231C22—91no—
232C22—271no—
233C22—271no—
234C22—271no—
235C22—271no—
236C22—271no—
237C22—271no—
238C22—271no—
239C22—541no—
240C22C4⋊(C3⋊#157)11yescharacteristic, complemented
241C4—31no—
242C4—91no—
243C4—271no—
244C4—1082no—
245C4—1082no—
246C4A4⋊#15711yescharacteristic, complemented
247C3—22no—
248C3—41no—
249C3—162no—
250C2—31no—
251C2—31no—
252C2—91no—
253C2—91no—
254C2—271no—
255C2—271no—
256C2C2×A4⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4⋊#157
commutatorC2×C32×A4C2×C4
FittingC22×C32×C4Dic3
socleC23×C32C2×Dic3
prime radical, cosocleC2×A4×C3⋊D3C22
hypercenterC4A4⋊#157
hypocenterC32×A4C4⋊C4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
6. C32×C4×A4C4semidirect
8. A4×C3⋊Dic3C4semidirect
25. C22×C32×C4Dic3semidirect
26. C22×C3⋊Dic3Dic3semidirect
41. C32×A4C4⋊C4semidirect
89. C4×A4#157semidirect
94. C22×C32C4⋊Dic3semidirect
112. C32×C4SL(2,ℤ/4)semidirect
114. C3⋊Dic3SL(2,ℤ/4)semidirect
163. C22×C4C3⋊#157semidirect
184. A4C4⋊#157semidirect
195. C32C4⋊SL(2,ℤ/4)semidirect
240. C22C4⋊(C3⋊#157)semidirect
246. C4A4⋊#157semidirect

Maximal subgroups

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

subgroupindex
C2×A4⋊#1572
C4×A4×C3⋊D32
#105938123
C4⋊(C3⋊#157)4
C4⋊SL(2,ℤ/4)9