Finite Group Databasebeta

← S4×C7⋊AGL(1,𝔽5) #72218822

Subgroups — S4×C7⋊AGL(1,𝔽5) #72218822

Order 3360.

224 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 31 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
1C7×D5×S4C211yesmaximal, characteristic
2A4×C7⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C7⋊(A4⋊AGL(1,𝔽5))C211yesmaximal, characteristic, complemented
4D4×C7⋊AGL(1,𝔽5)—31nomaximal, Hall
5C7×D5×A4C2211yescharacteristic
6C7×C5⋊S4C411yescharacteristic, complemented
7C5×C7×S4C411yescharacteristic, complemented
8D3×C7⋊AGL(1,𝔽5)—41nomaximal
9S4×Dic7—51nomaximal, Hall
10C22×C7⋊AGL(1,𝔽5)—31no—
11C22×C7⋊AGL(1,𝔽5)D311yescharacteristic, complemented
12C7×D4×D5—31no—
13C22⋊(C7⋊AGL(1,𝔽5))—31no—
14C22⋊(C7⋊AGL(1,𝔽5))—31no—
15C4⋊(C7⋊AGL(1,𝔽5))—31no—
16C4×C7⋊AGL(1,𝔽5)—31no—
17AGL(1,𝔽5)×S4—71nomaximal, Hall
18C5×C7×A4C2×C411yescharacteristic, complemented
19D3×C7×D5—41no—
20C3⋊(C7⋊AGL(1,𝔽5))—41no—
21C3×C7⋊AGL(1,𝔽5)—41no—
22C2×C7×S4—51no—
23A4×Dic7—51no—
24C7⋊SL(2,ℤ/4)—51no—
25C22×C7×D5—31no—
26C22×C7×D5C2×D311yescharacteristic
27C2×C7⋊AGL(1,𝔽5)—31no—
28C2×C7⋊AGL(1,𝔽5)—31no—
29C2×C7⋊AGL(1,𝔽5)—31no—
30C2×C7⋊AGL(1,𝔽5)—31no—
31C2×C7⋊AGL(1,𝔽5)—61no—
32C2×C7⋊AGL(1,𝔽5)—61no—
33C5×C7×D4—31no—
34C7×C22⋊D5—31no—
35C7×C22⋊D5—31no—
36C7×D20—31no—
37C4×C7×D5—31no—
38D5×S4D711yescharacteristic
39A4×AGL(1,𝔽5)—71no—
40A4⋊AGL(1,𝔽5)—71no—
41D4×Dic7—151noHall
42C7×D15—41no—
43C5×D3×C7—41no—
44C3×C7×D5—41no—
45C2×C7×A4—51no—
46C7×S4—51no—
47C7×S4AGL(1,𝔽5)11yescharacteristic, complemented
48D3×Dic7—201no—
49D4×AGL(1,𝔽5)—211noHall
50C22×C5×C7—31no—
51C22×C5×C7C4×D311yescharacteristic, complemented
52C2×C7×D5—31no—
53C2×C7×D5—31no—
54C2×C7×D5—31no—
55C2×C7×D5—61no—
56C2×C7×D5—61no—
57C7⋊AGL(1,𝔽5)—31no—
58C7⋊AGL(1,𝔽5)—61no—
59C7⋊AGL(1,𝔽5)—61no—
60C7⋊AGL(1,𝔽5)S411yescharacteristic, direct product factor
61C4×C5×C7—31no—
62C7×Dic5—31no—
63D5×A4C2×D711yescharacteristic
64C5⋊S4Dic711yescharacteristic, complemented
65C5×S4Dic711yescharacteristic, complemented
66D3×AGL(1,𝔽5)—281no—
67C2×C7×D4—151no—
68C22×Dic7—51no—
69C22×Dic7—151no—
70C22⋊Dic7—151no—
71C22⋊Dic7—151no—
72C4⋊Dic7—151no—
73C4×Dic7—151no—
74C3×C5×C7—41noHall
75C4×S4—351noHall
76C2×D3×C7—201no—
77C7×A4C2×AGL(1,𝔽5)11yescharacteristic, complemented
78Dic21—201no—
79C3×Dic7—201no—
80C22×AGL(1,𝔽5)—71no—
81C22×AGL(1,𝔽5)—211no—
82D4×D5—31no—
83C22⋊AGL(1,𝔽5)—211no—
84C22⋊AGL(1,𝔽5)—211no—
85C4⋊AGL(1,𝔽5)—211no—
86C4×AGL(1,𝔽5)—211no—
87C2×C5×C7—31no—
88C2×C5×C7—61no—
89C7×D5—31no—
90C7×D5—61no—
91C7×D5C2×S411yescharacteristic
92C5×A4C2×Dic711yescharacteristic, complemented
93D3×D5—41no—
94ΓL(1,𝔽16)—281no—
95C3×AGL(1,𝔽5)—281no—
96C23×C7—51no—
97C23×C7—151no—
98C7×D4—31no—
99C7×D4—151no—
100C7×D4—151no—
101C7×D4—151no—
102C2×C4×C7—151no—
103C2×Dic7—151no—
104C2×Dic7—151no—
105C2×Dic7—151no—
106C2×Dic7—151no—
107C2×Dic7—301no—
108C2×Dic7—301no—
109C2×S4—51no—
110C4×A4—351no—
111SL(2,ℤ/4)—351no—
112C2×C3×C7—201no—
113D3×C7—41no—
114D3×C7—201no—
115C22×D5—31no—
116C22×D5D3×D711yescharacteristic
117C2×AGL(1,𝔽5)—211no—
118C2×AGL(1,𝔽5)—211no—
119C2×AGL(1,𝔽5)—211no—
120C2×AGL(1,𝔽5)—211no—
121C2×AGL(1,𝔽5)—421no—
122C2×AGL(1,𝔽5)—421no—
123C5×D4—31no—
124C22⋊D5—31no—
125C22⋊D5—31no—
126D20—31no—
127C4×D5—31no—
128C5×C7C4×S411yescharacteristic, complemented, Hall
129C4×D4—1051noSylow
130D15—41no—
131C3×D5—41no—
132C5×D3—41no—
133C22×C7—31no—
134C22×C7—151no—
135C22×C7—151no—
136C22×C7—151no—
137C22×C7—301no—
138C22×C7—301no—
139C22×C7D3×AGL(1,𝔽5)11yescharacteristic, complemented
140C4×C7—31no—
141C4×C7—151no—
142Dic7—51no—
143Dic7—151no—
144Dic7—301no—
145Dic7—301no—
146C2×A4—51no—
147S4—51no—
148S4C7⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
149C4×D3—1401no—
150C3×C7—41noHall
151C22×C5—31no—
152C22×C5D3×Dic711yescharacteristic, complemented
153C2×D5—31no—
154C2×D5—31no—
155C2×D5—31no—
156C2×D5—61no—
157C2×D5—61no—
158AGL(1,𝔽5)—71no—
159AGL(1,𝔽5)—211no—
160AGL(1,𝔽5)—421no—
161AGL(1,𝔽5)—421no—
162C4×C5—31no—
163Dic5—31no—
164C2×D4—151no—
165C22×C4—351no—
166C22×C4—1051no—
167C4⋊C4—1051no—
168C22⋊C4—1051no—
169C22⋊C4—1051no—
170C42—1051no—
171C3×C5—41noHall
172C2×C7—31no—
173C2×C7—51no—
174C2×C7—61no—
175C2×C7—151no—
176C2×C7—301no—
177C2×D3—201no—
178A4C2×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
179C3×C4—1401no—
180Dic3—1401no—
181C2×C5—31no—
182C2×C5—61no—
183D5—31no—
184D5—61no—
185D5D7×S411yescharacteristic
186C23—51no—
187C23—151no—
188D4—31no—
189D4—151no—
190D4—151no—
191D4—151no—
192C2×C4—151no—
193C2×C4—1051no—
194C2×C4—1051no—
195C2×C4—1051no—
196C2×C4—1051no—
197C2×C4—2101no—
198C2×C4—2101no—
199C7AGL(1,𝔽5)×S411yescharacteristic, complemented, Sylow
200C2×C3—201no—
201D3—41no—
202D3—201no—
203C5S4×Dic711yescharacteristic, complemented, Sylow
204C22—31no—
205C22—151no—
206C22—151no—
207C22—151no—
208C22—301no—
209C22—301no—
210C22D3×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
211C4—31no—
212C4—151no—
213C4—351no—
214C4—1051no—
215C4—2101no—
216C4—2101no—
217C3—41noSylow
218C2—31no—
219C2—51no—
220C2—61no—
221C2—151no—
222C2—301no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, hypocenterC5×C7×A4C2×C4
Fitting, socleC22×C5×C7C4×D3
prime radicalC7×D5×S4C2
cosocleC7×D5×A4C22
LCSG ▷ C5×C7×A4
UCSC1

Complements

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

NH ≅ G/Ntype
2. A4×C7⋊AGL(1,𝔽5)C2semidirect
3. C7⋊(A4⋊AGL(1,𝔽5))C2semidirect
6. C7×C5⋊S4C4semidirect
7. C5×C7×S4C4semidirect
11. C22×C7⋊AGL(1,𝔽5)D3semidirect
18. C5×C7×A4C2×C4semidirect
47. C7×S4AGL(1,𝔽5)semidirect
51. C22×C5×C7C4×D3semidirect
60. C7⋊AGL(1,𝔽5)S4direct (N × H)
64. C5⋊S4Dic7semidirect
65. C5×S4Dic7semidirect
77. C7×A4C2×AGL(1,𝔽5)semidirect
92. C5×A4C2×Dic7semidirect
128. C5×C7C4×S4semidirect
139. C22×C7D3×AGL(1,𝔽5)semidirect
148. S4C7⋊AGL(1,𝔽5)direct (N × H)
152. C22×C5D3×Dic7semidirect
178. A4C2×C7⋊AGL(1,𝔽5)semidirect
199. C7AGL(1,𝔽5)×S4semidirect
203. C5S4×Dic7semidirect
210. C22D3×C7⋊AGL(1,𝔽5)semidirect

Maximal subgroups

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

subgroupindex
C7×D5×S42
A4×C7⋊AGL(1,𝔽5)2
C7⋊(A4⋊AGL(1,𝔽5))2
D4×C7⋊AGL(1,𝔽5)3
D3×C7⋊AGL(1,𝔽5)4
S4×Dic75
AGL(1,𝔽5)×S47