Finite Group Databasebeta

← C4×A4×#157 #13180665

Subgroups — C4×A4×#157 #13180665

Order 1728.

422 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 54 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
3C22×C4×#157C311yesmaximal, characteristic, complemented
4C2×A4×C3⋊D3C2211yescharacteristic
5A4×#157C414yesdirect product factor
6C32×C4×A4C411yescharacteristic, complemented
7C3×C4×#157—41nomaximal
8A4×C3⋊Dic3C411yescharacteristic, complemented
9C23×#157C2×C312yes—
10C22×C4×C3⋊D3C2×C311yescharacteristic
11C2×C4×#157—31no—
12C2×C4×#157—32no—
13C2×C4×#157—31no—
14C4×D3×A4—62no—
15C2×C32×A4C2×C411yescharacteristic
16C2×C3×#157—42no—
17A4×C3⋊D3C2×C412yes—
18C3×C4×C3⋊D3—41no—
19C42×A4—91nomaximal
20C23×C3⋊D3C22×C311yescharacteristic
21C22×#157—32no—
22C22×#157—32no—
23C22×#157—34no—
24C22×#157C3×C414yescomplemented
25C2×D3×A4—62no—
26C22×C32×C4C3×C411yescharacteristic, complemented
27C22×C3⋊Dic3C3×C411yescharacteristic, complemented
28C2×C4×C3⋊D3—31no—
29C2×C4×C3⋊D3—31no—
30C2×C4×C3⋊D3—31no—
31C2×C4×C3⋊D3—31no—
32C3×C4×A4—22no—
33C3×C4×A4—42no—
34C4×#157—32no—
35C4×#157—32no—
36C4×#157—31no—
37C4×#157A411yescharacteristic, direct product factor
38Dic3×A4—62no—
39C2×C3×C3⋊D3—41no—
40C32×A4C4211yescharacteristic, complemented
41C3×#157—44no—
42C33×C4—41no—
43C3×C3⋊Dic3—41no—
44C22×C4×D3—62no—
45C2×C4×A4—92no—
46C2×C4×A4—91no—
47C23×C32C2×C3×C411yescharacteristic
48C22×C3⋊D3—31no—
49C22×C3⋊D3—32no—
50C22×C3⋊D3—31no—
51C22×C3⋊D3C2×C3×C412yes—
52C2×C3×A4—22no—
53C2×C3×A4—42no—
54C2×#157—34no—
55C2×#157—32no—
56C2×#157—34no—
57C2×#157—34no—
58C2×#157—34no—
59C2×#157C2×A412yes—
60D3×A4—64no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C3⋊Dic3—31no—
64C2×C3⋊Dic3—31no—
65C4×C3⋊D3—32no—
66C4×C3⋊D3—31no—
67C4×C3⋊D3—31no—
68C4×C3⋊D3—31no—
69C4×C3⋊D3C2×A411yescharacteristic
70C3×C4×D3—242no—
71C22×C42—91noSylow
72C2×C33—41no—
73C3×C3⋊D3—42no—
74C23×D3—62no—
75C22×A4—91no—
76C22×C3×C4—22no—
77C22×Dic3—62no—
78C2×C4×D3—182no—
79C2×C4×D3—182no—
80C2×C4×D3—182no—
81C2×C4×D3—182no—
82C4×A4—42no—
83C4×A4—94no—
84C4×A4—91no—
85C4×A4#15711yescharacteristic, direct product factor
86C3×C42—361no—
87C22×C32—31no—
88C22×C32—31no—
89C22×C32C3×C4211yescharacteristic, complemented
90C2×C3⋊D3—32no—
91C2×C3⋊D3—31no—
92C2×C3⋊D3—32no—
93C2×C3⋊D3—32no—
94C2×C3⋊D3—32no—
95C2×C3⋊D3C22×A411yescharacteristic
96C2×C3×D3—242no—
97C3×A4—22no—
98C3×A4—42no—
99#157—34no—
100#157C4×A414yesdirect product factor
101C32×C4—31no—
102C32×C4—82no—
103C32×C4—162no—
104C32×C4C4×A411yescharacteristic, complemented
105C3⋊Dic3—31no—
106C3⋊Dic3C4×A411yescharacteristic, complemented
107C3×Dic3—242no—
108C23×C4—92no—
109C23×C4—91no—
110C2×C42—271no—
111C2×C42—271no—
112C2×C42—272no—
113C33—41noSylow
114C23×C3—22no—
115C22×D3—64no—
116C22×D3—182no—
117C22×D3—182no—
118C22×D3—184no—
119C2×A4—42no—
120C2×A4—92no—
121C2×A4C2×#15711yescharacteristic
122C2×C3×C4—62no—
123C2×C3×C4—62no—
124C2×C3×C4—361no—
125C2×C3×C4—362no—
126C2×Dic3—182no—
127C2×Dic3—182no—
128C4×D3—62no—
129C4×D3—182no—
130C4×D3—182no—
131C4×D3—182no—
132C4×D3—184no—
133C2×C32—31no—
134C2×C32—31no—
135C2×C32—82no—
136C2×C32—162no—
137C2×C32C2×C4×A411yescharacteristic
138C3⋊D3—32no—
139C3⋊D3C2×C4×A412yes—
140C3×D3—244no—
141C24—91no—
142C22×C4—94no—
143C22×C4—91no—
144C22×C4—271no—
145C22×C4—271no—
146C22×C4—271no—
147C22×C4—271no—
148C22×C4—272no—
149C22×C4—272no—
150C22×C4—274no—
151C22×C4C3×#15711yescharacteristic, complemented
152C42—91no—
153C42—272no—
154C42—271no—
155C42—272no—
156C22×C3—22no—
157C22×C3—62no—
158C22×C3—62no—
159C22×C3—361no—
160C2×D3—62no—
161C2×D3—184no—
162C2×D3—184no—
163C2×D3—182no—
164C2×D3—184no—
165C2×D3—184no—
166A4—42no—
167A4C4×#15711yescharacteristic, direct product factor
168C3×C4—22no—
169C3×C4—41no—
170C3×C4—62no—
171C3×C4—162no—
172C3×C4—361no—
173C3×C4—364no—
174Dic3—62no—
175Dic3—182no—
176C32—82no—
177C32—162no—
178C32C42×A411yescharacteristic, complemented
179C23—92no—
180C23—271no—
181C23—272no—
182C23—271no—
183C23C2×C3×#15711yescharacteristic
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—92no—
187C2×C4—91no—
188C2×C4—271no—
189C2×C4—271no—
190C2×C4—272no—
191C2×C4—271no—
192C2×C4—274no—
193C2×C4—274no—
194C2×C4—272no—
195C2×C4—274no—
196C2×C4—271no—
197C2×C4—271no—
198C2×C4—274no—
199C2×C3—22no—
200C2×C3—41no—
201C2×C3—62no—
202C2×C3—62no—
203C2×C3—162no—
204C2×C3—362no—
205D3—64no—
206D3—184no—
207C22—31no—
208C22—31no—
209C22—91no—
210C22—272no—
211C22—272no—
212C22—272no—
213C22—272no—
214C22—271no—
215C22C3×C4×#15711yescharacteristic, complemented
216C4—31no—
217C4—94no—
218C4—91no—
219C4—274no—
220C4—271no—
221C4A4×#15711yescharacteristic, direct product factor
222C3—22no—
223C3—41no—
224C3—162no—
225C2—31no—
226C2—31no—
227C2—92no—
228C2—272no—
229C2C2×A4×#15711yescharacteristic
subgroupG/H
center, hypercenterC4A4×#157
commutator, hypocenterC22×C32C3×C42
FrattiniC2C2×A4×#157
FittingC22×C32×C4C3×C4
socleC23×C32C2×C3×C4
prime radicalC2×A4×C3⋊D3C22
cosocleC23×C3⋊D3C22×C3
LCSG ▷ C22×C32
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
3. C22×C4×#157C3semidirect
5. A4×#157C4direct (N × H)
6. C32×C4×A4C4semidirect
8. A4×C3⋊Dic3C4semidirect
24. C22×#157C3×C4semidirect
26. C22×C32×C4C3×C4semidirect
27. C22×C3⋊Dic3C3×C4semidirect
37. C4×#157A4direct (N × H)
40. C32×A4C42semidirect
85. C4×A4#157direct (N × H)
89. C22×C32C3×C42semidirect
100. #157C4×A4direct (N × H)
104. C32×C4C4×A4semidirect
106. C3⋊Dic3C4×A4semidirect
151. C22×C4C3×#157semidirect
167. A4C4×#157direct (N × H)
178. C32C42×A4semidirect
215. C22C3×C4×#157semidirect
221. C4A4×#157direct (N × H)

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
C22×C4×#1573
C3×C4×#1574
C42×A49