Finite Group Databasebeta

← C2×C4×C7×S4 #13062086

Subgroups — C2×C4×C7×S4 #13062086

Order 1344.

466 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C22×C7×S4C211yesmaximal, characteristic
2C2×C4×C7×A4C211yesmaximal, characteristic, complemented
3C2×C7×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C4×C7×S4C214yesmaximal, direct product factor
5C2×C4×C7×D4—31nomaximal, Hall
6C22×C7×A4C2211yescharacteristic
7C2×C7×S4C414yesdirect product factor
8C2×C7×S4C2212yes—
9C2×C4×D3×C7—41nomaximal
10C4×C7×A4C2212yescomplemented
11C7×SL(2,ℤ/4)C2212yescomplemented
12C22×C7×D4—31no—
13C23×C4×C7—31no—
14C23×C4×C7D311yescharacteristic, complemented
15C4×C7×D4—34no—
16C4×C7×D4—34no—
17C2×C7×C4⋊C4—31no—
18C2×C7×C22⋊C4—31no—
19C2×C7×C22⋊C4—31no—
20C2×C42×C7—31no—
21C2×C4×S4C711yesmaximal, characteristic, direct product factor, Hall
22C22×D3×C7—41no—
23C2×C7×A4C2×C412yescomplemented
24C2×C7×A4C2311yescharacteristic
25C7×S4C2×C414yesdirect product factor
26C2×C3×C4×C7—41no—
27C2×C7×Dic3—41no—
28C4×D3×C7—44no—
29C24×C7—31no—
30C24×C7C2×D311yescharacteristic
31C2×C7×D4—32no—
32C2×C7×D4—34no—
33C2×C7×D4—34no—
34C2×C7×D4—32no—
35C22×C4×C7—31no—
36C22×C4×C7—31no—
37C22×C4×C7—31no—
38C22×C4×C7—32no—
39C22×C4×C7—31no—
40C22×C4×C7—31no—
41C22×C4×C7—34no—
42C22×C4×C7—61no—
43C22×C4×C7—62no—
44C22×C4×C7—61no—
45C22×C4×C7C2×D312yescomplemented
46C7×C4⋊C4—34no—
47C7×C22⋊C4—34no—
48C7×C22⋊C4—32no—
49C7×C22⋊C4—32no—
50C42×C7—34no—
51C22×S4C2×C711yescharacteristic
52C2×C4×A4C2×C711yescharacteristic, complemented
53C2×SL(2,ℤ/4)C2×C711yescharacteristic, complemented
54C4×S4C2×C714yesdirect product factor
55C22×C3×C7—41no—
56C2×D3×C7—42no—
57C2×D3×C7—44no—
58C7×A4C22×C411yescharacteristic, complemented
59C3×C4×C7—42no—
60C7×Dic3—42no—
61C2×C4×D4—31noSylow
62C23×C7—31no—
63C23×C7—31no—
64C23×C7—32no—
65C23×C7—32no—
66C23×C7—34no—
67C23×C7—61no—
68C23×C7—62no—
69C23×C7—61no—
70C23×C7C4×D312yescomplemented
71C23×C7C22×D311yescharacteristic
72C7×D4—38no—
73C7×D4—34no—
74C7×D4—34no—
75C2×C4×C7—32no—
76C2×C4×C7—34no—
77C2×C4×C7—31no—
78C2×C4×C7—32no—
79C2×C4×C7—32no—
80C2×C4×C7—32no—
81C2×C4×C7—32no—
82C2×C4×C7—32no—
83C2×C4×C7—61no—
84C2×C4×C7—64no—
85C2×C4×C7—62no—
86C2×C4×C7—61no—
87C2×C4×C7—61no—
88C2×C4×C7—64no—
89C2×C4×C7S411yescharacteristic, direct product factor
90C22×A4C22×C711yescharacteristic
91C2×S4C4×C714yesdirect product factor
92C2×S4C22×C712yes—
93C2×C4×D3—41no—
94C4×A4C22×C712yescomplemented
95SL(2,ℤ/4)C22×C712yescomplemented
96C2×C3×C7—42no—
97C2×C3×C7—41no—
98D3×C7—44no—
99C22×D4—31no—
100C23×C4—31no—
101C23×C4D3×C711yescharacteristic, complemented
102C4×D4—34no—
103C4×D4—34no—
104C2×C4⋊C4—31no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C42—31no—
108C22×C7—32no—
109C22×C7—31no—
110C22×C7—32no—
111C22×C7—32no—
112C22×C7—31no—
113C22×C7—31no—
114C22×C7—34no—
115C22×C7—61no—
116C22×C7—62no—
117C22×C7—64no—
118C22×C7—64no—
119C22×C7—62no—
120C22×C7C2×C4×D311yescharacteristic, complemented
121C22×C7C2×S411yescharacteristic
122C4×C7—34no—
123C4×C7—32no—
124C4×C7—62no—
125C4×C7—62no—
126C4×C7C2×S412yesdirect product factor
127C22×D3—41no—
128C2×A4C2×C4×C712yescomplemented
129C2×A4C23×C711yescharacteristic
130S4C2×C4×C714yesdirect product factor
131C2×C3×C4—41no—
132C2×Dic3—41no—
133C4×D3—44no—
134C3×C7—41noHall
135C24—31no—
136C24C2×D3×C711yescharacteristic
137C2×D4—32no—
138C2×D4—34no—
139C2×D4—34no—
140C2×D4—32no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—32no—
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—34no—
148C22×C4—61no—
149C22×C4—62no—
150C22×C4—61no—
151C22×C4C2×D3×C712yescomplemented
152C4⋊C4—34no—
153C22⋊C4—34no—
154C22⋊C4—32no—
155C22⋊C4—32no—
156C42—34no—
157C2×C7—31no—
158C2×C7—31no—
159C2×C7—32no—
160C2×C7—64no—
161C2×C7C4×S412yesdirect product factor
162C2×C7C22×S411yescharacteristic
163C22×C3—41no—
164C2×D3—42no—
165C2×D3—44no—
166A4C22×C4×C711yescharacteristic, complemented
167C3×C4—42no—
168Dic3—42no—
169C23—31no—
170C23—31no—
171C23—32no—
172C23—32no—
173C23—34no—
174C23—61no—
175C23—62no—
176C23—61no—
177C23C4×D3×C712yescomplemented
178C23C22×D3×C711yescharacteristic
179D4—38no—
180D4—34no—
181D4—34no—
182C2×C4—32no—
183C2×C4—34no—
184C2×C4—31no—
185C2×C4—32no—
186C2×C4—32no—
187C2×C4—32no—
188C2×C4—32no—
189C2×C4—32no—
190C2×C4—61no—
191C2×C4—64no—
192C2×C4—62no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—64no—
196C2×C4C7×S411yescharacteristic, direct product factor
197C7C2×C4×S411yescharacteristic, direct product factor, Sylow
198C2×C3—42no—
199C2×C3—41no—
200D3—44no—
201C22—32no—
202C22—31no—
203C22—32no—
204C22—32no—
205C22—31no—
206C22—31no—
207C22—34no—
208C22—61no—
209C22—62no—
210C22—64no—
211C22—64no—
212C22—62no—
213C22C2×C4×D3×C711yescharacteristic, complemented
214C22C2×C7×S411yescharacteristic
215C4—34no—
216C4—32no—
217C4—62no—
218C4—62no—
219C4C2×C7×S412yesdirect product factor
220C3—41noSylow
221C2—31no—
222C2—31no—
223C2—32no—
224C2—64no—
225C2C4×C7×S412yesdirect product factor
226C2C22×C7×S411yescharacteristic
subgroupG/H
center, hypercenterC2×C4×C7S4
commutator, hypocenterA4C22×C4×C7
FrattiniC2C22×C7×S4
FittingC23×C4×C7D3
socleC24×C7C2×D3
prime radicalC22×C7×S4C2
cosocleC2×A4C23×C7
LCSG ▷ A4
UCSC1 ◁ C2×C4×C7

Complements

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

NH ≅ G/Ntype
2. C2×C4×C7×A4C2semidirect
3. C2×C7×SL(2,ℤ/4)C2semidirect
4. C4×C7×S4C2direct (N × H)
7. C2×C7×S4C4direct (N × H)
10. C4×C7×A4C22semidirect
11. C7×SL(2,ℤ/4)C22semidirect
14. C23×C4×C7D3semidirect
21. C2×C4×S4C7direct (N × H)
23. C2×C7×A4C2×C4semidirect
25. C7×S4C2×C4direct (N × H)
45. C22×C4×C7C2×D3semidirect
52. C2×C4×A4C2×C7semidirect
53. C2×SL(2,ℤ/4)C2×C7semidirect
54. C4×S4C2×C7direct (N × H)
58. C7×A4C22×C4semidirect
70. C23×C7C4×D3semidirect
89. C2×C4×C7S4direct (N × H)
91. C2×S4C4×C7direct (N × H)
94. C4×A4C22×C7semidirect
95. SL(2,ℤ/4)C22×C7semidirect
101. C23×C4D3×C7semidirect
120. C22×C7C2×C4×D3semidirect
126. C4×C7C2×S4direct (N × H)
128. C2×A4C2×C4×C7semidirect
130. S4C2×C4×C7direct (N × H)
151. C22×C4C2×D3×C7semidirect
161. C2×C7C4×S4direct (N × H)
166. A4C22×C4×C7semidirect
177. C23C4×D3×C7semidirect
196. C2×C4C7×S4direct (N × H)
197. C7C2×C4×S4direct (N × H)
213. C22C2×C4×D3×C7semidirect
219. C4C2×C7×S4direct (N × H)
225. C2C4×C7×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×S42
C2×C4×C7×A42
C2×C7×SL(2,ℤ/4)2
C4×C7×S42
C2×C4×C7×D43
C2×C4×D3×C74
C2×C4×S47