Finite Group Databasebeta

← C7×(C2×S4)⋊C4 #13057133

Subgroups — C7×(C2×S4)⋊C4 #13057133

Order 1344.

286 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 46 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
4C7×#386—31nomaximal, Hall
5C22×C7×A4C2211yescharacteristic
6C2×C7×S4—21no—
7C2×C7×S4—21no—
8C2×C7×S4C412yescomplemented
9C4×C7×A4—21no—
10C7×SL(2,ℤ/4)—21no—
11C7×(C2×D3)⋊C4—41nomaximal
12C22×C7×D4—31no—
13C23×C4×C7D311yescharacteristic, complemented
14C2×C7×C22⋊C4—31no—
15C2×C7×C22⋊C4—31no—
16C2×C7×C22⋊C4—31no—
17C7×#95—31no—
18C7×#95—31no—
19(C2×S4)⋊C4C711yesmaximal, characteristic, direct product factor, Hall
20C22×D3×C7—41no—
21C2×C7×A4C2×C411yescharacteristic
22C2×C7×A4D411yescharacteristic
23C2×C7×A4D411yescharacteristic
24C7×S4—22no—
25C2×C3×C4×C7—41no—
26C2×C7×Dic3—41no—
27C24×C7—31no—
28C24×C7C2×D311yescharacteristic
29C2×C7×D4—32no—
30C2×C7×D4—32no—
31C2×C7×D4—61no—
32C2×C7×D4—61no—
33C2×C7×D4—61no—
34C2×C7×D4—61no—
35C22×C4×C7—21no—
36C22×C4×C7—31no—
37C22×C4×C7—31no—
38C22×C4×C7—31no—
39C22×C4×C7—31no—
40C22×C4×C7—31no—
41C22×C4×C7—61no—
42C7×C22⋊C4—62no—
43C7×C22⋊C4—61no—
44C7×C22⋊C4—61no—
45C7×C22⋊C4—61no—
46C7×C22⋊C4—61no—
47C22×S4C2×C711yescharacteristic
48C2×C4×A4C2×C711yescharacteristic, complemented
49C2×SL(2,ℤ/4)C2×C711yescharacteristic, complemented
50C22×C3×C7—41no—
51C2×D3×C7—42no—
52C2×D3×C7—81no—
53C2×D3×C7—81no—
54C7×A4C22⋊C411yescharacteristic, complemented
55C3×C4×C7—81no—
56C7×Dic3—81no—
57#386—31noSylow
58C23×C7—31no—
59C23×C7—31no—
60C23×C7—31no—
61C23×C7—31no—
62C23×C7—32no—
63C23×C7—61no—
64C23×C7—61no—
65C23×C7—61no—
66C23×C7—61no—
67C23×C7—61no—
68C23×C7—61no—
69C23×C7C4×D311yescharacteristic
70C23×C7D1211yescharacteristic
71C23×C7#6711yescharacteristic
72C7×D4—62no—
73C7×D4—62no—
74C7×D4—62no—
75C7×D4—62no—
76C2×C4×C7—32no—
77C2×C4×C7—32no—
78C2×C4×C7—31no—
79C2×C4×C7—61no—
80C2×C4×C7—61no—
81C2×C4×C7—61no—
82C2×C4×C7—61no—
83C2×C4×C7—61no—
84C2×C4×C7—61no—
85C2×C4×C7—61no—
86C2×C4×C7—61no—
87C2×C4×C7—61no—
88C2×C4×C7—61no—
89C2×C4×C7—61no—
90C2×C4×C7S411yescharacteristic, complemented
91C22×A4C22×C711yescharacteristic
92C2×S4—21no—
93C2×S4—21no—
94C2×S4C4×C712yescomplemented
95C4×A4—21no—
96SL(2,ℤ/4)—21no—
97(C2×D3)⋊C4—41no—
98C2×C3×C7—41no—
99C2×C3×C7—41no—
100C2×C3×C7—41no—
101D3×C7—82no—
102C22×D4—31no—
103C23×C4D3×C711yescharacteristic, complemented
104C2×C22⋊C4—31no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107#95—31no—
108#95—31no—
109C22×C7—31no—
110C22×C7—31no—
111C22×C7—31no—
112C22×C7—31no—
113C22×C7—31no—
114C22×C7—31no—
115C22×C7—31no—
116C22×C7—31no—
117C22×C7—31no—
118C22×C7—61no—
119C22×C7—62no—
120C22×C7—62no—
121C22×C7—62no—
122C22×C7—121no—
123C22×C7—121no—
124C22×C7—121no—
125C22×C7—121no—
126C22×C7(C2×D3)⋊C411yescharacteristic, complemented
127C22×C7C2×S411yescharacteristic
128C4×C7—21no—
129C4×C7—62no—
130C4×C7—61no—
131C4×C7—121no—
132C4×C7—121no—
133C22×D3—41no—
134C2×A4C2×C4×C711yescharacteristic
135C2×A4C7×D411yescharacteristic
136C2×A4C7×D411yescharacteristic
137S4—22no—
138C2×C3×C4—41no—
139C2×Dic3—41no—
140C3×C7—41noHall
141C24—31no—
142C24C2×D3×C711yescharacteristic
143C2×D4—32no—
144C2×D4—32no—
145C2×D4—61no—
146C2×D4—61no—
147C2×D4—61no—
148C2×D4—61no—
149C22×C4—21no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—31no—
155C22×C4—61no—
156C22⋊C4—62no—
157C22⋊C4—61no—
158C22⋊C4—61no—
159C22⋊C4—61no—
160C22⋊C4—61no—
161C2×C7—31no—
162C2×C7—31no—
163C2×C7—31no—
164C2×C7—31no—
165C2×C7—122no—
166C2×C7C4×S411yescharacteristic
167C2×C7C4⋊S411yescharacteristic
168C2×C7GL(2,ℤ/4)11yescharacteristic
169C22×C3—41no—
170C2×D3—42no—
171C2×D3—81no—
172C2×D3—81no—
173A4C7×C22⋊C411yescharacteristic, complemented
174C3×C4—81no—
175Dic3—81no—
176C23—31no—
177C23—31no—
178C23—31no—
179C23—31no—
180C23—32no—
181C23—61no—
182C23—61no—
183C23—61no—
184C23—61no—
185C23—61no—
186C23—61no—
187C23C4×D3×C711yescharacteristic
188C23C7×D1211yescharacteristic
189C23C7×#6711yescharacteristic
190D4—62no—
191D4—62no—
192D4—62no—
193D4—62no—
194C2×C4—32no—
195C2×C4—32no—
196C2×C4—31no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4C7×S411yescharacteristic, complemented
209C7(C2×S4)⋊C411yescharacteristic, direct product factor, Sylow
210C2×C3—41no—
211C2×C3—41no—
212C2×C3—41no—
213D3—82no—
214C22—31no—
215C22—31no—
216C22—31no—
217C22—31no—
218C22—31no—
219C22—31no—
220C22—31no—
221C22—31no—
222C22—31no—
223C22—61no—
224C22—62no—
225C22—62no—
226C22—62no—
227C22—121no—
228C22—121no—
229C22—121no—
230C22—121no—
231C22C7×(C2×D3)⋊C411yescharacteristic, complemented
232C22C2×C7×S411yescharacteristic
233C4—21no—
234C4—62no—
235C4—61no—
236C4—121no—
237C4—121no—
238C3—41noSylow
239C2—31no—
240C2—31no—
241C2—31no—
242C2—31no—
243C2—122no—
244C2C4×C7×S411yescharacteristic
245C2C7×C4⋊S411yescharacteristic
246C2C7×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C7C2×S4
commutatorC2×A4C2×C4×C7
FrattiniC22C2×C7×S4
FittingC23×C4×C7D3
socleC24×C7C2×D3
prime radicalC22×C7×S4C2
cosocleC22×A4C22×C7
hypercenterC2×C4×C7S4
hypocenterA4C7×C22⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C7 ◁ 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
8. C2×C7×S4C4semidirect
13. C23×C4×C7D3semidirect
19. (C2×S4)⋊C4C7direct (N × H)
48. C2×C4×A4C2×C7semidirect
49. C2×SL(2,ℤ/4)C2×C7semidirect
54. C7×A4C22⋊C4semidirect
90. C2×C4×C7S4semidirect
94. C2×S4C4×C7semidirect
103. C23×C4D3×C7semidirect
126. C22×C7(C2×D3)⋊C4semidirect
173. A4C7×C22⋊C4semidirect
208. C2×C4C7×S4semidirect
209. C7(C2×S4)⋊C4direct (N × H)
231. C22C7×(C2×D3)⋊C4semidirect

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
C7×#3863
C7×(C2×D3)⋊C44
(C2×S4)⋊C47