Finite Group Databasebeta

← A4×#4004 #14512561

Subgroups — A4×#4004 #14512561

Order 1920.

340 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×A4×#170C212yesmaximal
2C4×A4×Dic5C211yesmaximal, characteristic
3C22×#4004C311yesmaximal, characteristic, complemented, Hall
4C2×C4×C5×A4C411yescharacteristic
5C2×A4×Dic5C411yescharacteristic
6C2×A4×Dic5C2211yescharacteristic
7A4×#170—22no—
8C3×#4004—41nomaximal
9A4×C4⋊C8—51nomaximal, Hall
10C23×#170C2×C312yes—
11C22×C4×Dic5C2×C311yescharacteristic
12C2×#4004—31no—
13C2×#4004—32no—
14C2×#4004—31no—
15C22×C5×A4C2×C411yescharacteristic
16C4×C5×A4—21no—
17C2×C3×#170—42no—
18A4×Dic5C812yescomplemented
19A4×Dic5D411yescharacteristic
20A4×Dic5Q811yescharacteristic
21C3×C4×Dic5—41no—
22C2×C8×A4—52no—
23C42×A4—51no—
24C23×C4×C5C3×C411yescharacteristic
25C23×Dic5C3×C411yescharacteristic
26C23×Dic5C22×C311yescharacteristic
27C22×#170—22no—
28C22×#170—32no—
29C22×#170—32no—
30C22×#170—62no—
31C2×C4×Dic5—31no—
32C2×C4×Dic5—31no—
33C2×C4×Dic5—31no—
34C2×C4×Dic5—31no—
35#4004—32no—
36#4004—32no—
37#4004—31no—
38#4004A411yescharacteristic, direct product factor
39C22×C4⋊C8—51noSylow
40C2×C5×A4C4⋊C411yescharacteristic
41C2×C5×A4C2×C811yescharacteristic
42C2×C5×A4M1611yescharacteristic
43C2×C3×C4×C5—41no—
44C2×C3×Dic5—41no—
45C2×C3×Dic5—41no—
46C3×#170—82no—
47C2×C4×A4—51no—
48C2×C4×A4—51no—
49C2×C4×A4AGL(1,𝔽5)11yescharacteristic
50C8×A4—102no—
51C3×C4⋊C8—201no—
52C24×C5C2×C3×C411yescharacteristic
53C22×C4×C5—21no—
54C22×C4×C5—31no—
55C22×C4×C5—31no—
56C22×C4×C5—61no—
57C22×Dic5—31no—
58C22×Dic5—31no—
59C22×Dic5—31no—
60C22×Dic5—32no—
61C22×Dic5—31no—
62C22×Dic5—31no—
63C22×Dic5—31no—
64C22×Dic5C3×C812yescomplemented
65C22×Dic5C3×D411yescharacteristic
66C22×Dic5C3×Q811yescharacteristic
67C2×#170—32no—
68C2×#170—62no—
69C2×#170—62no—
70C2×#170—62no—
71C2×#170—62no—
72C2×#170C2×A412yes—
73C4×Dic5—32no—
74C4×Dic5—31no—
75C4×Dic5—31no—
76C4×Dic5—31no—
77C4×Dic5C2×A411yescharacteristic
78C23×C8—52no—
79C22×C42—51no—
80C2×C4⋊C8—151no—
81C2×C4⋊C8—151no—
82C2×C4⋊C8—152no—
83C22×C3×C5—41no—
84C5×A4C4⋊C811yescharacteristic, complemented
85C3×C4×C5—81no—
86C3×Dic5—42no—
87C3×Dic5—41no—
88C3×Dic5—41no—
89C22×A4C2×AGL(1,𝔽5)11yescharacteristic
90C4×A4—21no—
91C4×A4—52no—
92C4×A4—51no—
93C4×A4—51no—
94C2×C3×C8—202no—
95C3×C42—201no—
96C23×C5—31no—
97C23×C5—31no—
98C23×C5—31no—
99C23×C5—31no—
100C23×C5C3×C4⋊C411yescharacteristic
101C23×C5C2×C3×C811yescharacteristic
102C23×C5C3×M1611yescharacteristic
103C2×C4×C5—31no—
104C2×C4×C5—61no—
105C2×C4×C5—61no—
106C2×C4×C5—61no—
107C2×C4×C5—61no—
108C2×C4×C5C4×A411yescharacteristic
109C2×Dic5—32no—
110C2×Dic5—32no—
111C2×Dic5—31no—
112C2×Dic5—31no—
113C2×Dic5—32no—
114C2×Dic5—32no—
115C2×Dic5—31no—
116C2×Dic5—31no—
117C2×Dic5—31no—
118C2×Dic5—31no—
119C2×Dic5—32no—
120C2×Dic5—31no—
121C2×Dic5—31no—
122C2×Dic5C4×A411yescharacteristic
123C2×Dic5C22×A411yescharacteristic
124#170—22no—
125#170—62no—
126C23×C4—51no—
127C23×C4—51no—
128C23×C4C3×AGL(1,𝔽5)11yescharacteristic
129C22×C8—102no—
130C22×C8—152no—
131C22×C8—152no—
132C22×C8—302no—
133C2×C42—151no—
134C2×C42—151no—
135C2×C42—151no—
136C2×C42—151no—
137C4⋊C8—51no—
138C4⋊C8—151no—
139C4⋊C8—152no—
140C4⋊C8—152no—
141C2×C3×C5—41no—
142C2×C3×C5—41no—
143C2×C3×C5—41no—
144C2×A4D5⋊C811yescharacteristic
145C2×A4C4⋊AGL(1,𝔽5)11yescharacteristic
146C2×A4#70411yescharacteristic
147C2×C3×C4—41no—
148C2×C3×C4—201no—
149C2×C3×C4—201no—
150C3×C8—402no—
151C22×C5—31no—
152C22×C5—31no—
153C22×C5—31no—
154C22×C5—32no—
155C22×C5—31no—
156C22×C5—31no—
157C22×C5—31no—
158C22×C5—31no—
159C22×C5—31no—
160C22×C5—31no—
161C22×C5C3×C4⋊C811yescharacteristic, complemented
162C22×C5C2×C4×A411yescharacteristic
163C4×C5—21no—
164C4×C5—61no—
165Dic5—31no—
166Dic5—31no—
167Dic5—32no—
168Dic5C8×A412yescomplemented
169Dic5D4×A411yescharacteristic
170Dic5Q8×A411yescharacteristic
171C24C2×C3×AGL(1,𝔽5)11yescharacteristic
172C22×C4—21no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—52no—
176C22×C4—51no—
177C22×C4—51no—
178C22×C4—61no—
179C22×C4—151no—
180C22×C4—151no—
181C22×C4—151no—
182C22×C4—151no—
183C22×C4—151no—
184C22×C4—151no—
185C22×C4—152no—
186C2×C8—52no—
187C2×C8—152no—
188C2×C8—302no—
189C2×C8—302no—
190C2×C8—302no—
191C2×C8—302no—
192C42—51no—
193C42—151no—
194C42—151no—
195C42—152no—
196C42—151no—
197C3×C5—41noHall
198C22×C3—41no—
199A4#400411yescharacteristic, direct product factor
200C3×C4—81no—
201C3×C4—201no—
202C3×C4—202no—
203C3×C4—201no—
204C2×C5—31no—
205C2×C5—31no—
206C2×C5—31no—
207C2×C5—31no—
208C2×C5A4×C4⋊C411yescharacteristic
209C2×C5C2×C8×A411yescharacteristic
210C2×C5A4×M1611yescharacteristic
211C23—31no—
212C23—31no—
213C23—31no—
214C23—31no—
215C23C3×D5⋊C811yescharacteristic
216C23C3×C4⋊AGL(1,𝔽5)11yescharacteristic
217C23C3×#70411yescharacteristic
218C2×C4—31no—
219C2×C4—51no—
220C2×C4—51no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—151no—
226C2×C4—151no—
227C2×C4—152no—
228C2×C4—151no—
229C2×C4—152no—
230C2×C4—151no—
231C2×C4—152no—
232C2×C4—152no—
233C2×C4—151no—
234C2×C4—151no—
235C2×C4—152no—
236C2×C4—151no—
237C2×C4—151no—
238C2×C4A4×AGL(1,𝔽5)11yescharacteristic
239C8—102no—
240C8—302no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—41no—
244C5A4×C4⋊C811yescharacteristic, complemented, Sylow
245C22—31no—
246C22—31no—
247C22—31no—
248C22—32no—
249C22—31no—
250C22—31no—
251C22—31no—
252C22—31no—
253C22—31no—
254C22—31no—
255C22C3×#400411yescharacteristic, complemented
256C22C2×A4×AGL(1,𝔽5)11yescharacteristic
257C4—21no—
258C4—52no—
259C4—51no—
260C4—51no—
261C4—61no—
262C4—152no—
263C4—151no—
264C4—151no—
265C3—41noSylow
266C2—31no—
267C2—31no—
268C2—31no—
269C2—31no—
270C2A4×D5⋊C811yescharacteristic
271C2A4×C4⋊AGL(1,𝔽5)11yescharacteristic
272C2A4×#70411yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×AGL(1,𝔽5)
commutatorC23×C5C2×C3×C8
FittingC23×C4×C5C3×C4
socleC24×C5C2×C3×C4
prime radicalC22×C5×A4C2×C4
cosocleC23×Dic5C22×C3
hypercenterC2×C4A4×AGL(1,𝔽5)
hypocenterC22×C5C3×C4⋊C8
LCSG ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C22×#4004C3semidirect
18. A4×Dic5C8semidirect
38. #4004A4direct (N × H)
64. C22×Dic5C3×C8semidirect
84. C5×A4C4⋊C8semidirect
161. C22×C5C3×C4⋊C8semidirect
168. Dic5C8×A4semidirect
199. A4#4004direct (N × H)
244. C5A4×C4⋊C8semidirect
255. C22C3×#4004semidirect

Maximal subgroups

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

subgroupindex
C2×A4×#1702
C4×A4×Dic52
C22×#40043
C3×#40044
A4×C4⋊C85