Finite Group Databasebeta

← AGL(1,𝔽5)×D7⋊A4 #72218647

Subgroups — AGL(1,𝔽5)×D7⋊A4 #72218647

Order 3360.

264 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 38 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
1D5×D7⋊A4C211yesmaximal, characteristic
2(C22⋊ΓL(1,𝔽8))⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3AGL(1,𝔽5)×C22⋊ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4C22×D7×AGL(1,𝔽5)C311yesmaximal, characteristic, complemented, Hall
5C5⋊(D7⋊A4)C411yescharacteristic, complemented
6D5×C22⋊ΓL(1,𝔽8)C2211yescharacteristic
7C5×D7⋊A4C411yescharacteristic, complemented
8AGL(1,𝔽5)×AGL(1,𝔽7)—41nomaximal
9C4×D7⋊A4—51nomaximal, Hall
10C22×D5×D7C2×C311yescharacteristic
11C22×C7⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
12C22×C7×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
13C2×D7×AGL(1,𝔽5)—31no—
14C2×D7×AGL(1,𝔽5)—31no—
15C2×D7×AGL(1,𝔽5)—31no—
16C2×D7×AGL(1,𝔽5)—31no—
17C2×A4×AGL(1,𝔽5)—71nomaximal, Hall
18C5×C22⋊ΓL(1,𝔽8)C2×C411yescharacteristic, complemented
19D5×AGL(1,𝔽7)—41no—
20ΓL(1,𝔽8)⋊AGL(1,𝔽5)—41no—
21AGL(1,𝔽5)×ΓL(1,𝔽8)—41no—
22C2×D7⋊A4—51no—
23C4×C22⋊ΓL(1,𝔽8)—51no—
24Dic7⋊A4—51no—
25C22×D35C3×C411yescharacteristic, complemented
26C22×C7×D5C22×C311yescharacteristic
27C22×C5×D7C3×C411yescharacteristic, complemented
28C2×D5×D7—31no—
29C2×D5×D7—31no—
30C2×D5×D7—31no—
31C2×D5×D7—31no—
32C2×C7⋊AGL(1,𝔽5)—31no—
33C2×C7⋊AGL(1,𝔽5)—31no—
34C2×C7×AGL(1,𝔽5)—31no—
35C2×C7×AGL(1,𝔽5)—31no—
36D7×AGL(1,𝔽5)—32no—
37D7×AGL(1,𝔽5)—31no—
38D7×AGL(1,𝔽5)—31no—
39D7×AGL(1,𝔽5)—31no—
40D7×AGL(1,𝔽5)A411yescharacteristic, complemented
41C2×D5×A4—71no—
42A4×AGL(1,𝔽5)—71no—
43A4×AGL(1,𝔽5)—71no—
44C22×C4×D7—51noHall
45C5⋊AGL(1,𝔽7)—41no—
46D5×ΓL(1,𝔽8)—41no—
47C5×AGL(1,𝔽7)—41no—
48C2×C22⋊ΓL(1,𝔽8)—51no—
49D7⋊A4—51no—
50D7⋊A4AGL(1,𝔽5)11yescharacteristic, direct product factor
51C4×AGL(1,𝔽7)—201no—
52C23×AGL(1,𝔽5)—71noHall
53C22×C5×C7C2×C3×C411yescharacteristic, complemented
54C2×D35—31no—
55C2×D35—31no—
56C2×C7×D5—31no—
57C2×C7×D5—31no—
58C2×C5×D7—31no—
59C2×C5×D7—31no—
60D5×D7—32no—
61D5×D7—31no—
62D5×D7—31no—
63D5×D7—31no—
64D5×D7C2×A411yescharacteristic
65C7⋊AGL(1,𝔽5)—31no—
66C7⋊AGL(1,𝔽5)C2×A411yescharacteristic, complemented
67C7×AGL(1,𝔽5)—31no—
68C7×AGL(1,𝔽5)C2×A411yescharacteristic, complemented
69C2×C5×A4—71no—
70C2×C3×AGL(1,𝔽5)—281no—
71D5×A4—71no—
72D5×A4—71no—
73C23×D7—51no—
74C22×C4×C7—51no—
75C22×Dic7—51no—
76C2×C4×D7—151no—
77C2×C4×D7—151no—
78C2×C4×D7—151no—
79C2×C4×D7—151no—
80C5×ΓL(1,𝔽8)—41noHall
81C2×C4×A4—351noHall
82C22⋊ΓL(1,𝔽8)C2×AGL(1,𝔽5)11yescharacteristic, complemented
83C2×AGL(1,𝔽7)—201no—
84C4×ΓL(1,𝔽8)—201no—
85ΓL(1,𝔽8)⋊C4—201no—
86C23×D5—71no—
87C22×AGL(1,𝔽5)—71no—
88C22×AGL(1,𝔽5)—211no—
89C22×AGL(1,𝔽5)—211no—
90C22×AGL(1,𝔽5)—211no—
91C22×AGL(1,𝔽5)—211no—
92C22×AGL(1,𝔽5)AGL(1,𝔽7)11yescharacteristic, complemented
93C2×C5×C7—31no—
94D35—31no—
95D35C4×A411yescharacteristic, complemented
96C5×D7—31no—
97C5×D7C4×A411yescharacteristic, complemented
98C7×D5—31no—
99C7×D5C22×A411yescharacteristic
100C2×C3×D5—281no—
101C5×A4—71no—
102C3×AGL(1,𝔽5)—281no—
103C3×AGL(1,𝔽5)—281no—
104C23×C7—51no—
105C22×D7—51no—
106C22×D7—151no—
107C22×D7—151no—
108C22×D7—151no—
109C22×D7—151no—
110C22×D7C3×AGL(1,𝔽5)11yescharacteristic, complemented
111C2×C4×C7—151no—
112C2×C4×C7—151no—
113C2×Dic7—151no—
114C2×Dic7—151no—
115C4×D7—51no—
116C4×D7—151no—
117C4×D7—151no—
118C4×D7—151no—
119C4×D7—152no—
120C22×A4—351no—
121C4×A4—351no—
122C4×A4—351no—
123C2×ΓL(1,𝔽8)—201no—
124AGL(1,𝔽7)—41no—
125AGL(1,𝔽7)—201no—
126C23×C5—71no—
127C22×D5—71no—
128C22×D5—211no—
129C22×D5—211no—
130C22×D5—211no—
131C22×D5—211no—
132C22×D5C2×AGL(1,𝔽7)11yescharacteristic
133C2×AGL(1,𝔽5)—31no—
134C2×AGL(1,𝔽5)—31no—
135C2×AGL(1,𝔽5)—71no—
136C2×AGL(1,𝔽5)—211no—
137C2×AGL(1,𝔽5)—211no—
138C2×AGL(1,𝔽5)—212no—
139C2×AGL(1,𝔽5)—211no—
140C2×AGL(1,𝔽5)—211no—
141C2×AGL(1,𝔽5)—211no—
142C5×C7C2×C4×A411yescharacteristic, complemented, Hall
143C23×C4—351noSylow
144C2×C3×C5—281no—
145C3×D5—281no—
146C3×D5—281no—
147C22×C7—151no—
148C22×C7—151no—
149C22×C7C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
150C2×D7—31no—
151C2×D7—31no—
152C2×D7—51no—
153C2×D7—151no—
154C2×D7—151no—
155C2×D7—152no—
156C2×D7—151no—
157C2×D7—151no—
158C2×D7—151no—
159C4×C7—51no—
160C4×C7—151no—
161Dic7—51no—
162Dic7—151no—
163C2×A4—71no—
164C2×A4—351no—
165C2×A4—351no—
166C2×C3×C4—1401no—
167ΓL(1,𝔽8)—41noHall
168C22×C5—211no—
169C22×C5—211no—
170C22×C5C4×AGL(1,𝔽7)11yescharacteristic, complemented
171C2×D5—31no—
172C2×D5—31no—
173C2×D5—71no—
174C2×D5—211no—
175C2×D5—211no—
176C2×D5—212no—
177C2×D5—211no—
178C2×D5—211no—
179C2×D5—211no—
180AGL(1,𝔽5)—31no—
181AGL(1,𝔽5)—71no—
182AGL(1,𝔽5)—211no—
183AGL(1,𝔽5)D7⋊A411yescharacteristic, direct product factor
184C24—351no—
185C22×C4—51no—
186C22×C4—351no—
187C22×C4—1051no—
188C22×C4—1051no—
189C22×C4—1051no—
190C22×C4—1051no—
191C3×C5—281noHall
192C2×C7—31no—
193C2×C7—51no—
194C2×C7—151no—
195D7—31no—
196D7—51no—
197D7—151no—
198D7A4×AGL(1,𝔽5)11yescharacteristic, complemented
199C22×C3—1401no—
200A4—71no—
201C3×C4—1401no—
202C3×C4—1401no—
203C2×C5—31no—
204C2×C5—71no—
205C2×C5—211no—
206D5—31no—
207D5—71no—
208D5—211no—
209D5C2×D7⋊A411yescharacteristic
210C23—51no—
211C23—71no—
212C23—351no—
213C23—1051no—
214C23—1051no—
215C23—1051no—
216C23—1051no—
217C2×C4—151no—
218C2×C4—151no—
219C2×C4—351no—
220C2×C4—1051no—
221C2×C4—1051no—
222C2×C4—1051no—
223C2×C4—1052no—
224C2×C4—1051no—
225C2×C4—1051no—
226C7C2×A4×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
227C2×C3—281no—
228C2×C3—1401no—
229C2×C3—1401no—
230C5C4×D7⋊A411yescharacteristic, complemented, Sylow
231C22—151no—
232C22—151no—
233C22—211no—
234C22—211no—
235C22—351no—
236C22—1051no—
237C22—1051no—
238C22—1051no—
239C22—1051no—
240C22—1051no—
241C22—1052no—
242C22AGL(1,𝔽5)×AGL(1,𝔽7)11yescharacteristic, complemented
243C4—51no—
244C4—151no—
245C4—351no—
246C4—1051no—
247C3—281noSylow
248C2—31no—
249C2—51no—
250C2—71no—
251C2—151no—
252C2—211no—
253C2—351no—
254C2—1051no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC22×C5×C7C2×C3×C4
prime radicalD5×D7⋊A4C2
cosocleC22×C7×D5C22×C3
LCSG ▷ C22×C5×C7
UCSC1

Complements

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

NH ≅ G/Ntype
2. (C22⋊ΓL(1,𝔽8))⋊AGL(1,𝔽5)C2semidirect
3. AGL(1,𝔽5)×C22⋊ΓL(1,𝔽8)C2semidirect
4. C22×D7×AGL(1,𝔽5)C3semidirect
5. C5⋊(D7⋊A4)C4semidirect
7. C5×D7⋊A4C4semidirect
11. C22×C7⋊AGL(1,𝔽5)C2×C3semidirect
12. C22×C7×AGL(1,𝔽5)C2×C3semidirect
18. C5×C22⋊ΓL(1,𝔽8)C2×C4semidirect
25. C22×D35C3×C4semidirect
27. C22×C5×D7C3×C4semidirect
40. D7×AGL(1,𝔽5)A4semidirect
50. D7⋊A4AGL(1,𝔽5)direct (N × H)
53. C22×C5×C7C2×C3×C4semidirect
66. C7⋊AGL(1,𝔽5)C2×A4semidirect
68. C7×AGL(1,𝔽5)C2×A4semidirect
82. C22⋊ΓL(1,𝔽8)C2×AGL(1,𝔽5)semidirect
92. C22×AGL(1,𝔽5)AGL(1,𝔽7)semidirect
95. D35C4×A4semidirect
97. C5×D7C4×A4semidirect
110. C22×D7C3×AGL(1,𝔽5)semidirect
142. C5×C7C2×C4×A4semidirect
149. C22×C7C2×C3×AGL(1,𝔽5)semidirect
170. C22×C5C4×AGL(1,𝔽7)semidirect
183. AGL(1,𝔽5)D7⋊A4direct (N × H)
198. D7A4×AGL(1,𝔽5)semidirect
226. C7C2×A4×AGL(1,𝔽5)semidirect
230. C5C4×D7⋊A4semidirect
242. C22AGL(1,𝔽5)×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
D5×D7⋊A42
(C22⋊ΓL(1,𝔽8))⋊AGL(1,𝔽5)2
AGL(1,𝔽5)×C22⋊ΓL(1,𝔽8)2
C22×D7×AGL(1,𝔽5)3
AGL(1,𝔽5)×AGL(1,𝔽7)4
C4×D7⋊A45
C2×A4×AGL(1,𝔽5)7