Finite Group Databasebeta

← C2×C4×D7⋊A4 #13062183

Subgroups — C2×C4×D7⋊A4 #13062183

Order 1344.

634 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 97 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×D7⋊A4C211yesmaximal, characteristic
2C2×C4×C22⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C2×Dic7⋊A4C211yesmaximal, characteristic, complemented
4C4×D7⋊A4C214yesmaximal, direct product factor
5C23×C4×D7C311yesmaximal, characteristic, complemented, Hall
6C22×C22⋊ΓL(1,𝔽8)C2211yescharacteristic
7C2×D7⋊A4C414yesdirect product factor
8C2×D7⋊A4C2212yes—
9C4×C22⋊ΓL(1,𝔽8)C2212yescomplemented
10Dic7⋊A4C2212yescomplemented
11C2×C4×AGL(1,𝔽7)—41nomaximal
12C24×D7C2×C311yescharacteristic
13C23×C4×C7C2×C311yescharacteristic, complemented
14C23×Dic7C2×C311yescharacteristic, complemented
15C22×C4×D7—31no—
16C22×C4×D7—31no—
17C22×C4×D7—31no—
18C22×C4×D7—31no—
19C22×C4×D7—34no—
20C22×C4×D7C2×C314yescomplemented
21C22×C4×A4—71nomaximal, Hall
22C2×C22⋊ΓL(1,𝔽8)C2×C412yescomplemented
23C2×C22⋊ΓL(1,𝔽8)C2311yescharacteristic
24D7⋊A4C2×C414yesdirect product factor
25C22×AGL(1,𝔽7)—41no—
26C2×C4×ΓL(1,𝔽8)—41no—
27C2×ΓL(1,𝔽8)⋊C4—41no—
28C4×AGL(1,𝔽7)—44no—
29C24×C7C22×C311yescharacteristic
30C23×D7—31no—
31C23×D7—31no—
32C23×D7—32no—
33C23×D7—34no—
34C23×D7C3×C414yescomplemented
35C23×D7C22×C312yes—
36C22×C4×C7—31no—
37C22×C4×C7—31no—
38C22×C4×C7—32no—
39C22×C4×C7C22×C312yescomplemented
40C22×Dic7—31no—
41C22×Dic7—31no—
42C22×Dic7—32no—
43C22×Dic7C22×C312yescomplemented
44C2×C4×D7—32no—
45C2×C4×D7—34no—
46C2×C4×D7—34no—
47C2×C4×D7—34no—
48C2×C4×D7—31no—
49C2×C4×D7—31no—
50C2×C4×D7—31no—
51C2×C4×D7—34no—
52C2×C4×D7—34no—
53C2×C4×D7—34no—
54C2×C4×D7—34no—
55C2×C4×D7—34no—
56C2×C4×D7A411yescharacteristic, complemented
57C23×A4—71no—
58C2×C4×A4—74no—
59C2×C4×A4—71no—
60C2×C4×A4—71no—
61C22⋊ΓL(1,𝔽8)C22×C411yescharacteristic, complemented
62C22×ΓL(1,𝔽8)—41no—
63C2×AGL(1,𝔽7)—44no—
64C2×AGL(1,𝔽7)—42no—
65C4×ΓL(1,𝔽8)—42no—
66ΓL(1,𝔽8)⋊C4—42no—
67C24×C4—71noSylow
68C23×C7—31no—
69C23×C7—32no—
70C23×C7—31no—
71C23×C7C2×C3×C412yescomplemented
72C23×C7C23×C311yescharacteristic
73C22×D7—32no—
74C22×D7—34no—
75C22×D7—38no—
76C22×D7—31no—
77C22×D7—34no—
78C22×D7—34no—
79C22×D7—32no—
80C22×D7—32no—
81C22×D7—32no—
82C22×D7—34no—
83C22×D7—38no—
84C22×D7—34no—
85C22×D7C2×C3×C414yescomplemented
86C22×D7C2×A411yescharacteristic
87C2×C4×C7—32no—
88C2×C4×C7—31no—
89C2×C4×C7—32no—
90C2×C4×C7—32no—
91C2×C4×C7—32no—
92C2×C4×C7C2×A411yescharacteristic, complemented
93C2×Dic7—32no—
94C2×Dic7—31no—
95C2×Dic7—32no—
96C2×Dic7—32no—
97C2×Dic7—32no—
98C2×Dic7C2×A411yescharacteristic, complemented
99C4×D7—38no—
100C4×D7—34no—
101C4×D7—34no—
102C4×D7—34no—
103C4×D7C2×A414yescomplemented
104C22×A4—74no—
105C22×A4—72no—
106C22×A4—71no—
107C22×C3×C4—281no—
108C4×A4—72no—
109C4×A4—72no—
110C2×ΓL(1,𝔽8)—42no—
111C2×ΓL(1,𝔽8)—41no—
112AGL(1,𝔽7)—44no—
113C25—71no—
114C23×C4—71no—
115C23×C4—74no—
116C23×C4—211no—
117C23×C4—211no—
118C23×C4—214no—
119C23×C4—211no—
120C23×C4—211no—
121C23×C4AGL(1,𝔽7)11yescharacteristic, complemented
122C22×C7—32no—
123C22×C7—31no—
124C22×C7—32no—
125C22×C7—32no—
126C22×C7—32no—
127C22×C7—31no—
128C22×C7—31no—
129C22×C7C22×C3×C411yescharacteristic, complemented
130C22×C7C22×A411yescharacteristic
131C2×D7—38no—
132C2×D7—34no—
133C2×D7—34no—
134C2×D7—32no—
135C2×D7—34no—
136C2×D7—38no—
137C2×D7—34no—
138C2×D7—34no—
139C2×D7C4×A414yescomplemented
140C2×D7C22×A412yes—
141C4×C7—32no—
142C4×C7C22×A412yescomplemented
143Dic7—32no—
144Dic7C22×A412yescomplemented
145C23×C3—281no—
146C2×A4—74no—
147C2×A4—72no—
148C2×A4—71no—
149C2×C3×C4—284no—
150C2×C3×C4—281no—
151C2×C3×C4—281no—
152ΓL(1,𝔽8)—41noHall
153C24—74no—
154C24—72no—
155C24—211no—
156C24—212no—
157C24—211no—
158C24—214no—
159C24C2×AGL(1,𝔽7)11yescharacteristic
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—32no—
163C22×C4—71no—
164C22×C4—72no—
165C22×C4—214no—
166C22×C4—211no—
167C22×C4—214no—
168C22×C4—211no—
169C22×C4—214no—
170C22×C4—214no—
171C22×C4—212no—
172C22×C4—214no—
173C22×C4—214no—
174C22×C4—214no—
175C22×C4—214no—
176C22×C4—211no—
177C22×C4—211no—
178C22×C4—212no—
179C22×C4—211no—
180C22×C4C2×AGL(1,𝔽7)12yescomplemented
181C2×C7—32no—
182C2×C7—31no—
183C2×C7—31no—
184C2×C7C2×C4×A412yescomplemented
185C2×C7C23×A411yescharacteristic
186D7—34no—
187D7C2×C4×A414yescomplemented
188C22×C3—282no—
189C22×C3—284no—
190C22×C3—281no—
191A4—71no—
192C3×C4—282no—
193C3×C4—282no—
194C23—31no—
195C23—32no—
196C23—31no—
197C23—71no—
198C23—74no—
199C23—212no—
200C23—212no—
201C23—214no—
202C23—214no—
203C23—212no—
204C23—218no—
205C23—214no—
206C23—214no—
207C23—212no—
208C23—218no—
209C23—214no—
210C23—211no—
211C23C4×AGL(1,𝔽7)12yescomplemented
212C23C22×AGL(1,𝔽7)11yescharacteristic
213C2×C4—32no—
214C2×C4—31no—
215C2×C4—32no—
216C2×C4—32no—
217C2×C4—32no—
218C2×C4—71no—
219C2×C4—74no—
220C2×C4—214no—
221C2×C4—214no—
222C2×C4—218no—
223C2×C4—212no—
224C2×C4—212no—
225C2×C4—212no—
226C2×C4—211no—
227C2×C4—214no—
228C2×C4—212no—
229C2×C4D7⋊A411yescharacteristic, direct product factor
230C7C22×C4×A411yescharacteristic, complemented, Sylow
231C2×C3—284no—
232C2×C3—282no—
233C2×C3—281no—
234C22—32no—
235C22—31no—
236C22—32no—
237C22—32no—
238C22—32no—
239C22—31no—
240C22—31no—
241C22—74no—
242C22—72no—
243C22—214no—
244C22—214no—
245C22—218no—
246C22—214no—
247C22—214no—
248C22—218no—
249C22—214no—
250C22—212no—
251C22C2×C4×AGL(1,𝔽7)11yescharacteristic, complemented
252C22C2×D7⋊A411yescharacteristic
253C4—32no—
254C4—72no—
255C4—212no—
256C4C2×D7⋊A412yesdirect product factor
257C3—281noSylow
258C2—31no—
259C2—32no—
260C2—31no—
261C2—74no—
262C2—214no—
263C2C4×D7⋊A412yesdirect product factor
264C2C22×D7⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D7⋊A4
commutator, hypocenterC22×C7C22×C3×C4
FrattiniC2C22×D7⋊A4
FittingC23×C4×C7C2×C3
socleC24×C7C22×C3
prime radicalC22×D7⋊A4C2
cosocleC23×C7C23×C3
LCSG ▷ C22×C7
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊ΓL(1,𝔽8)C2semidirect
3. C2×Dic7⋊A4C2semidirect
4. C4×D7⋊A4C2direct (N × H)
5. C23×C4×D7C3semidirect
7. C2×D7⋊A4C4direct (N × H)
9. C4×C22⋊ΓL(1,𝔽8)C22semidirect
10. Dic7⋊A4C22semidirect
13. C23×C4×C7C2×C3semidirect
14. C23×Dic7C2×C3semidirect
20. C22×C4×D7C2×C3semidirect
22. C2×C22⋊ΓL(1,𝔽8)C2×C4semidirect
24. D7⋊A4C2×C4direct (N × H)
34. C23×D7C3×C4semidirect
39. C22×C4×C7C22×C3semidirect
43. C22×Dic7C22×C3semidirect
56. C2×C4×D7A4semidirect
61. C22⋊ΓL(1,𝔽8)C22×C4semidirect
71. C23×C7C2×C3×C4semidirect
85. C22×D7C2×C3×C4semidirect
92. C2×C4×C7C2×A4semidirect
98. C2×Dic7C2×A4semidirect
103. C4×D7C2×A4semidirect
121. C23×C4AGL(1,𝔽7)semidirect
129. C22×C7C22×C3×C4semidirect
139. C2×D7C4×A4semidirect
142. C4×C7C22×A4semidirect
144. Dic7C22×A4semidirect
180. C22×C4C2×AGL(1,𝔽7)semidirect
184. C2×C7C2×C4×A4semidirect
187. D7C2×C4×A4semidirect
211. C23C4×AGL(1,𝔽7)semidirect
229. C2×C4D7⋊A4direct (N × H)
230. C7C22×C4×A4semidirect
251. C22C2×C4×AGL(1,𝔽7)semidirect
256. C4C2×D7⋊A4direct (N × H)
263. C2C4×D7⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D7⋊A42
C2×C4×C22⋊ΓL(1,𝔽8)2
C2×Dic7⋊A42
C4×D7⋊A42
C23×C4×D73
C2×C4×AGL(1,𝔽7)4
C22×C4×A47