Finite Group Databasebeta

← A4×#3993 #14512542

Subgroups — A4×#3993 #14512542

Order 1920.

320 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 54 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
1A4×C4⋊AGL(1,𝔽5)C212yesmaximal
2C8×D5×A4C211yesmaximal, characteristic
3C22×#3993C311yesmaximal, characteristic, complemented, Hall
4C2×A4×AGL(1,𝔽5)—22no—
5C4×D5×A4C2211yescharacteristic
6C5×C8×A4C411yescharacteristic, complemented
7C3×#3993—41nomaximal
8A4×#168C411yescharacteristic, complemented
9A4×#106—51nomaximal, Hall
10C22×C4⋊AGL(1,𝔽5)C2×C312yes—
11C22×C8×D5C2×C311yescharacteristic
12C2×#3993—31no—
13C2×#3993—32no—
14C2×#3993—31no—
15C2×D5×A4D411yescharacteristic
16A4×AGL(1,𝔽5)—42no—
17C4×C5×A4C2×C411yescharacteristic
18C3×C4⋊AGL(1,𝔽5)—42no—
19A4×Dic5Q811yescharacteristic
20C3×C8×D5—41no—
21C2×C8×A4—51no—
22A4×C4⋊C4—52no—
23C23×AGL(1,𝔽5)—22no—
24C22×C4×D5C22×C311yescharacteristic
25C2×C4⋊AGL(1,𝔽5)—32no—
26C2×C4⋊AGL(1,𝔽5)—32no—
27C2×C4⋊AGL(1,𝔽5)—62no—
28C22×C5×C8C3×C411yescharacteristic, complemented
29C22×#168C3×C411yescharacteristic, complemented
30C2×C8×D5—31no—
31C2×C8×D5—31no—
32C2×C8×D5—31no—
33C2×C8×D5—31no—
34#3993—32no—
35#3993—32no—
36#3993—31no—
37#3993A411yescharacteristic, direct product factor
38C22×#106—51noSylow
39C2×C5×A4C4⋊C411yescharacteristic
40C2×C3×AGL(1,𝔽5)—82no—
41D5×A4SD1612yes—
42C3×C4×D5—41no—
43C3×C5×C8—41no—
44C3×#168—41no—
45C2×C4×A4—51no—
46C2×C4×A4—102no—
47C8×A4—51no—
48C8×A4AGL(1,𝔽5)11yescharacteristic, complemented
49C3×#106—201no—
50C23×D5C3×D411yescharacteristic
51C22×AGL(1,𝔽5)—42no—
52C22×AGL(1,𝔽5)—62no—
53C22×AGL(1,𝔽5)—62no—
54C22×AGL(1,𝔽5)—122no—
55C22×C4×C5C2×C3×C411yescharacteristic
56C22×Dic5C3×Q811yescharacteristic
57C2×C4×D5—31no—
58C2×C4×D5—31no—
59C2×C4×D5—31no—
60C2×C4×D5—31no—
61C4⋊AGL(1,𝔽5)—32no—
62C4⋊AGL(1,𝔽5)—62no—
63C4⋊AGL(1,𝔽5)—62no—
64C4⋊AGL(1,𝔽5)C2×A412yes—
65C2×C5×C8—31no—
66C2×C5×C8—31no—
67C2×#168—31no—
68C2×#168—31no—
69C8×D5—32no—
70C8×D5—31no—
71C8×D5—31no—
72C8×D5—31no—
73C8×D5C2×A411yescharacteristic
74C23×C8—51no—
75C22×C4⋊C4—52no—
76C2×#106—151no—
77C2×#106—152no—
78C2×#106—151no—
79C2×C3×D5—41no—
80C5×A4#10611yescharacteristic, complemented
81C3×AGL(1,𝔽5)—162no—
82C3×C4×C5—41no—
83C3×Dic5—41no—
84C22×A4—51no—
85C4×A4—51no—
86C4×A4—202no—
87C4×A4C2×AGL(1,𝔽5)11yescharacteristic
88C2×C3×C8—201no—
89C3×C4⋊C4—202no—
90C23×C5C3×C4⋊C411yescharacteristic
91C22×D5—31no—
92C22×D5—31no—
93C22×D5—32no—
94C22×D5C3×SD1612yes—
95C2×AGL(1,𝔽5)—22no—
96C2×AGL(1,𝔽5)—62no—
97C2×AGL(1,𝔽5)—122no—
98C2×AGL(1,𝔽5)—122no—
99C2×AGL(1,𝔽5)—122no—
100C2×AGL(1,𝔽5)—122no—
101C2×C4×C5—31no—
102C2×C4×C5—31no—
103C2×Dic5—31no—
104C2×Dic5—31no—
105C4×D5—32no—
106C4×D5—31no—
107C4×D5—31no—
108C4×D5—31no—
109C4×D5C22×A411yescharacteristic
110C5×C8—31no—
111C5×C8C4×A411yescharacteristic, complemented
112#168—31no—
113#168C4×A411yescharacteristic, complemented
114C23×C4—51no—
115C23×C4—102no—
116C22×C8—51no—
117C22×C8—151no—
118C22×C8—151no—
119C22×C8—151no—
120C22×C8—151no—
121C22×C8C3×AGL(1,𝔽5)11yescharacteristic, complemented
122C2×C4⋊C4—152no—
123C2×C4⋊C4—152no—
124C2×C4⋊C4—302no—
125#106—51no—
126#106—152no—
127#106—151no—
128#106—152no—
129C2×C3×C5—41no—
130C3×D5—42no—
131C2×A4—52no—
132C2×A4C4⋊AGL(1,𝔽5)11yescharacteristic
133C2×C3×C4—201no—
134C2×C3×C4—402no—
135C3×C8—41no—
136C3×C8—201no—
137C22×C5—31no—
138C22×C5—31no—
139C22×C5C3×#10611yescharacteristic, complemented
140C2×D5—32no—
141C2×D5—31no—
142C2×D5—32no—
143C2×D5—32no—
144C2×D5—32no—
145C2×D5D4×A411yescharacteristic
146AGL(1,𝔽5)—42no—
147AGL(1,𝔽5)—122no—
148C4×C5—31no—
149C4×C5C2×C4×A411yescharacteristic
150Dic5—31no—
151Dic5Q8×A411yescharacteristic
152C24—51no—
153C22×C4—51no—
154C22×C4—151no—
155C22×C4—151no—
156C22×C4—151no—
157C22×C4—151no—
158C22×C4—202no—
159C22×C4—302no—
160C22×C4—302no—
161C22×C4—602no—
162C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
163C2×C8—31no—
164C2×C8—31no—
165C2×C8—51no—
166C2×C8—151no—
167C2×C8—151no—
168C2×C8—152no—
169C2×C8—151no—
170C2×C8—151no—
171C2×C8—151no—
172C4⋊C4—52no—
173C4⋊C4—152no—
174C4⋊C4—302no—
175C4⋊C4—302no—
176C3×C5—41noHall
177C22×C3—201no—
178A4#399311yescharacteristic, direct product factor
179C3×C4—41no—
180C3×C4—201no—
181C3×C4—802no—
182C2×C5—31no—
183C2×C5—31no—
184C2×C5A4×C4⋊C411yescharacteristic
185D5—32no—
186D5A4×SD1612yes—
187C23—52no—
188C23—151no—
189C23—151no—
190C23—152no—
191C23C3×C4⋊AGL(1,𝔽5)11yescharacteristic
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—51no—
195C2×C4—102no—
196C2×C4—151no—
197C2×C4—151no—
198C2×C4—152no—
199C2×C4—151no—
200C2×C4—151no—
201C2×C4—151no—
202C2×C4—302no—
203C2×C4—602no—
204C2×C4—602no—
205C2×C4—602no—
206C2×C4—602no—
207C8—31no—
208C8—51no—
209C8—151no—
210C8A4×AGL(1,𝔽5)11yescharacteristic, complemented
211C2×C3—41no—
212C2×C3—202no—
213C5A4×#10611yescharacteristic, complemented, Sylow
214C22—31no—
215C22—31no—
216C22—51no—
217C22—152no—
218C22—152no—
219C22—152no—
220C22—152no—
221C22—151no—
222C22C3×#399311yescharacteristic, complemented
223C4—31no—
224C4—51no—
225C4—151no—
226C4—202no—
227C4—602no—
228C4C2×A4×AGL(1,𝔽5)11yescharacteristic
229C3—41noSylow
230C2—31no—
231C2—31no—
232C2—52no—
233C2—152no—
234C2A4×C4⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2A4×C4⋊AGL(1,𝔽5)
commutatorC22×C4×C5C2×C3×C4
FrattiniC4C2×A4×AGL(1,𝔽5)
FittingC22×C5×C8C3×C4
socleC23×C5C3×C4⋊C4
prime radicalC2×D5×A4D4
cosocleC22×C4×D5C22×C3
hypercenterC8A4×AGL(1,𝔽5)
hypocenterC22×C5C3×#106
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
3. C22×#3993C3semidirect
6. C5×C8×A4C4semidirect
8. A4×#168C4semidirect
28. C22×C5×C8C3×C4semidirect
29. C22×#168C3×C4semidirect
37. #3993A4direct (N × H)
48. C8×A4AGL(1,𝔽5)semidirect
80. C5×A4#106semidirect
111. C5×C8C4×A4semidirect
113. #168C4×A4semidirect
121. C22×C8C3×AGL(1,𝔽5)semidirect
139. C22×C5C3×#106semidirect
178. A4#3993direct (N × H)
210. C8A4×AGL(1,𝔽5)semidirect
213. C5A4×#106semidirect
222. C22C3×#3993semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊AGL(1,𝔽5)2
C8×D5×A42
C22×#39933
C3×#39934
A4×#1065