Finite Group Databasebeta

← C2×C7×GL(2,ℤ/4) #13062105

Subgroups — C2×C7×GL(2,ℤ/4) #13062105

Order 1344.

542 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C23×C7×A4C211yesmaximal, characteristic, complemented
2C22×C7×S4C211yesmaximal, characteristic, complemented
3C7×GL(2,ℤ/4)C214yesmaximal, direct product factor
4C2×C7×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C2×C7×#120—31nomaximal, Hall
6C22×C7×A4—22no—
7C22×C7×A4C2211yescharacteristic
8C22×C7×A4C2212yescomplemented
9C2×C7×S4—22no—
10C2×C7×S4C2212yescomplemented
11C2×C7×#67—41nomaximal
12C7×SL(2,ℤ/4)C2212yescomplemented
13C25×C7D311yescharacteristic, complemented
14C22×C7×D4—31no—
15C22×C7×D4—31no—
16C22×C7×D4—31no—
17C7×#120—34no—
18C7×#120—34no—
19C2×C7×C22⋊C4—31no—
20C2×C7×C22⋊C4—31no—
21C2×C7×C22⋊C4—31no—
22C2×GL(2,ℤ/4)C711yesmaximal, characteristic, direct product factor, Hall
23C23×C3×C7—41no—
24C22×D3×C7—41no—
25C2×C7×A4—22no—
26C2×C7×A4D412yescomplemented
27C2×C7×A4C2311yescharacteristic
28C7×S4—22no—
29C7×#67—44no—
30C2×C7×Dic3—41no—
31C24×C7—22no—
32C24×C7—31no—
33C24×C7—31no—
34C24×C7—32no—
35C24×C7—31no—
36C24×C7—62no—
37C24×C7C2×D311yescharacteristic
38C24×C7C2×D312yescomplemented
39C2×C7×D4—32no—
40C2×C7×D4—34no—
41C2×C7×D4—34no—
42C2×C7×D4—32no—
43C2×C7×D4—62no—
44C2×C7×D4—64no—
45C2×C7×D4—62no—
46C2×C7×D4—61no—
47C2×C7×D4—62no—
48C2×C7×D4—61no—
49C22×C4×C7—31no—
50C22×C4×C7—31no—
51C22×C4×C7—31no—
52C7×C22⋊C4—32no—
53C7×C22⋊C4—34no—
54C7×C22⋊C4—34no—
55C7×C22⋊C4—32no—
56C23×A4C2×C711yescharacteristic, complemented
57C22×S4C2×C711yescharacteristic, complemented
58GL(2,ℤ/4)C2×C714yesdirect product factor
59C2×SL(2,ℤ/4)C2×C711yescharacteristic, complemented
60C22×C3×C7—41no—
61C22×C3×C7—42no—
62C22×C3×C7—82no—
63C2×D3×C7—42no—
64C2×D3×C7—82no—
65C7×A4C2×D411yescharacteristic, complemented
66C7×Dic3—42no—
67C2×#120—31noSylow
68C23×C7—22no—
69C23×C7—31no—
70C23×C7—31no—
71C23×C7—32no—
72C23×C7—32no—
73C23×C7—34no—
74C23×C7—31no—
75C23×C7—32no—
76C23×C7—32no—
77C23×C7—32no—
78C23×C7—62no—
79C23×C7—62no—
80C23×C7—62no—
81C23×C7—61no—
82C23×C7—64no—
83C23×C7—62no—
84C23×C7—64no—
85C23×C7—62no—
86C23×C7—62no—
87C23×C7—61no—
88C23×C7—61no—
89C23×C7#6712yescomplemented
90C23×C7S411yescharacteristic, complemented
91C23×C7C22×D311yescharacteristic
92C7×D4—64no—
93C7×D4—62no—
94C7×D4—68no—
95C7×D4—64no—
96C7×D4—62no—
97C7×D4—64no—
98C2×C4×C7—32no—
99C2×C4×C7—32no—
100C2×C4×C7—32no—
101C2×C4×C7—62no—
102C2×C4×C7—62no—
103C2×C4×C7—61no—
104C2×C4×C7—61no—
105C22×A4—22no—
106C22×A4C22×C711yescharacteristic
107C22×A4C22×C712yescomplemented
108C2×S4—22no—
109C2×S4C22×C712yescomplemented
110C2×#67—41no—
111SL(2,ℤ/4)C22×C712yescomplemented
112C2×C3×C7—42no—
113C2×C3×C7—41no—
114C2×C3×C7—82no—
115D3×C7—82no—
116C25D3×C711yescharacteristic, complemented
117C22×D4—31no—
118C22×D4—31no—
119C22×D4—31no—
120#120—34no—
121#120—34no—
122C2×C22⋊C4—31no—
123C2×C22⋊C4—31no—
124C2×C22⋊C4—31no—
125C22×C7—22no—
126C22×C7—32no—
127C22×C7—31no—
128C22×C7—32no—
129C22×C7—32no—
130C22×C7—31no—
131C22×C7—31no—
132C22×C7—34no—
133C22×C7—32no—
134C22×C7—62no—
135C22×C7—62no—
136C22×C7—64no—
137C22×C7—61no—
138C22×C7—62no—
139C22×C7—64no—
140C22×C7—62no—
141C22×C7—62no—
142C22×C7—62no—
143C22×C7—62no—
144C22×C7—122no—
145C22×C7—122no—
146C22×C7C2×#6711yescharacteristic, complemented
147C22×C7C2×S411yescharacteristic
148C22×C7C2×S412yescomplemented
149C4×C7—62no—
150C4×C7—62no—
151C4×C7—62no—
152C23×C3—41no—
153C22×D3—41no—
154C2×A4—22no—
155C2×A4C7×D412yescomplemented
156C2×A4C23×C711yescharacteristic
157S4—22no—
158#67—44no—
159C2×Dic3—41no—
160C3×C7—41noHall
161C24—22no—
162C24—31no—
163C24—31no—
164C24—32no—
165C24—31no—
166C24—62no—
167C24C2×D3×C711yescharacteristic
168C24C2×D3×C712yescomplemented
169C2×D4—32no—
170C2×D4—34no—
171C2×D4—34no—
172C2×D4—32no—
173C2×D4—62no—
174C2×D4—64no—
175C2×D4—62no—
176C2×D4—61no—
177C2×D4—62no—
178C2×D4—61no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22⋊C4—32no—
183C22⋊C4—34no—
184C22⋊C4—34no—
185C22⋊C4—32no—
186C2×C7—22no—
187C2×C7—31no—
188C2×C7—31no—
189C2×C7—32no—
190C2×C7—62no—
191C2×C7—122no—
192C2×C7GL(2,ℤ/4)12yesdirect product factor
193C2×C7C22×S411yescharacteristic
194C22×C3—41no—
195C22×C3—42no—
196C22×C3—82no—
197C2×D3—42no—
198C2×D3—82no—
199A4C2×C7×D411yescharacteristic, complemented
200Dic3—42no—
201C23—22no—
202C23—31no—
203C23—31no—
204C23—32no—
205C23—32no—
206C23—34no—
207C23—31no—
208C23—32no—
209C23—32no—
210C23—32no—
211C23—62no—
212C23—62no—
213C23—62no—
214C23—61no—
215C23—64no—
216C23—62no—
217C23—64no—
218C23—62no—
219C23—62no—
220C23—61no—
221C23—61no—
222C23C7×#6712yescomplemented
223C23C7×S411yescharacteristic, complemented
224C23C22×D3×C711yescharacteristic
225D4—64no—
226D4—62no—
227D4—68no—
228D4—64no—
229D4—62no—
230D4—64no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—61no—
237C2×C4—61no—
238C7C2×GL(2,ℤ/4)11yescharacteristic, direct product factor, Sylow
239C2×C3—42no—
240C2×C3—41no—
241C2×C3—82no—
242D3—82no—
243C22—22no—
244C22—32no—
245C22—31no—
246C22—32no—
247C22—32no—
248C22—31no—
249C22—31no—
250C22—34no—
251C22—32no—
252C22—62no—
253C22—62no—
254C22—64no—
255C22—61no—
256C22—62no—
257C22—64no—
258C22—62no—
259C22—62no—
260C22—62no—
261C22—62no—
262C22—122no—
263C22—122no—
264C22C2×C7×#6711yescharacteristic, complemented
265C22C2×C7×S411yescharacteristic
266C22C2×C7×S412yescomplemented
267C4—62no—
268C4—62no—
269C4—62no—
270C3—41noSylow
271C2—22no—
272C2—31no—
273C2—31no—
274C2—32no—
275C2—62no—
276C2—122no—
277C2C7×GL(2,ℤ/4)12yesdirect product factor
278C2C22×C7×S411yescharacteristic
subgroupG/H
centerC22×C7C2×S4
commutator, cosocleC2×A4C23×C7
FrattiniC2C22×C7×S4
FittingC25×C7D3
socleC24×C7C2×D3
hypercenterC23×C7S4
hypocenterA4C2×C7×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C7 ◁ C23×C7

Complements

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

NH ≅ G/Ntype
1. C23×C7×A4C2semidirect
2. C22×C7×S4C2semidirect
3. C7×GL(2,ℤ/4)C2direct (N × H)
4. C2×C7×SL(2,ℤ/4)C2semidirect
8. C22×C7×A4C22semidirect
10. C2×C7×S4C22semidirect
12. C7×SL(2,ℤ/4)C22semidirect
13. C25×C7D3semidirect
22. C2×GL(2,ℤ/4)C7direct (N × H)
26. C2×C7×A4D4semidirect
38. C24×C7C2×D3semidirect
56. C23×A4C2×C7semidirect
57. C22×S4C2×C7semidirect
58. GL(2,ℤ/4)C2×C7direct (N × H)
59. C2×SL(2,ℤ/4)C2×C7semidirect
65. C7×A4C2×D4semidirect
89. C23×C7#67semidirect
90. C23×C7S4semidirect
107. C22×A4C22×C7semidirect
109. C2×S4C22×C7semidirect
111. SL(2,ℤ/4)C22×C7semidirect
116. C25D3×C7semidirect
146. C22×C7C2×#67semidirect
148. C22×C7C2×S4semidirect
155. C2×A4C7×D4semidirect
168. C24C2×D3×C7semidirect
192. C2×C7GL(2,ℤ/4)direct (N × H)
199. A4C2×C7×D4semidirect
222. C23C7×#67semidirect
223. C23C7×S4semidirect
238. C7C2×GL(2,ℤ/4)direct (N × H)
264. C22C2×C7×#67semidirect
266. C22C2×C7×S4semidirect
277. C2C7×GL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C7×A42
C22×C7×S42
C7×GL(2,ℤ/4)2
C2×C7×SL(2,ℤ/4)2
C2×C7×#1203
C2×C7×#674
C2×GL(2,ℤ/4)7