Finite Group Databasebeta

← C2×AGL(1,𝔽5)×C22⋊A4 #14514708

Subgroups — C2×AGL(1,𝔽5)×C22⋊A4 #14514708

Order 1920.

3704 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 3704 conjugacy classes of subgroups — too many to draw. Only its 80 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×D5×C22⋊A4C211yesmaximal, characteristic
2AGL(1,𝔽5)×C22⋊A4C212yesmaximal, direct product factor
3C25×AGL(1,𝔽5)C311yesmaximal, characteristic, complemented, Hall
4C2×C5×C22⋊A4C411yescharacteristic, complemented
5D5×C22⋊A4C411yescharacteristic, complemented
6D5×C22⋊A4C2211yescharacteristic
7C2×A4×AGL(1,𝔽5)—45nomaximal
8C2×C4×C22⋊A4—51nomaximal, Hall
9C25×D5C2×C311yescharacteristic
10C24×AGL(1,𝔽5)—35no—
11C24×AGL(1,𝔽5)—35no—
12C24×AGL(1,𝔽5)—310no—
13C24×AGL(1,𝔽5)C2×C312yescomplemented
14C5×C22⋊A4C2×C411yescharacteristic, complemented
15C2×D5×A4—45no—
16A4×AGL(1,𝔽5)—410no—
17C22×C22⋊A4—51no—
18C4×C22⋊A4—52no—
19C25×C5C3×C411yescharacteristic, complemented
20C24×D5—35no—
21C24×D5—35no—
22C24×D5—35no—
23C24×D5—35no—
24C24×D5C3×C411yescharacteristic, complemented
25C24×D5C22×C311yescharacteristic
26C23×AGL(1,𝔽5)—360no—
27C23×AGL(1,𝔽5)—360no—
28C23×AGL(1,𝔽5)—330no—
29C23×AGL(1,𝔽5)—310no—
30C23×AGL(1,𝔽5)—310no—
31C23×AGL(1,𝔽5)—35no—
32C23×AGL(1,𝔽5)—310no—
33C23×AGL(1,𝔽5)—310no—
34C23×AGL(1,𝔽5)—310no—
35C23×AGL(1,𝔽5)A415yescomplemented
36C25×C4—51noSylow
37C2×C5×A4—45no—
38C2×C3×AGL(1,𝔽5)—161no—
39D5×A4—45no—
40D5×A4—45no—
41C2×C22⋊A4—51no—
42C2×C22⋊A4—51no—
43C2×C22⋊A4AGL(1,𝔽5)11yescharacteristic, direct product factor
44C2×C4×A4—205no—
45C24×C5—35no—
46C24×C5—35no—
47C24×C5C2×C3×C411yescharacteristic, complemented
48C23×D5—330no—
49C23×D5—310no—
50C23×D5—360no—
51C23×D5—330no—
52C23×D5—330no—
53C23×D5—310no—
54C23×D5—35no—
55C23×D5—35no—
56C23×D5—35no—
57C23×D5—35no—
58C23×D5—35no—
59C23×D5—35no—
60C23×D5—35no—
61C23×D5C2×A415yes—
62C22×AGL(1,𝔽5)—3120no—
63C22×AGL(1,𝔽5)—330no—
64C22×AGL(1,𝔽5)—360no—
65C22×AGL(1,𝔽5)—360no—
66C22×AGL(1,𝔽5)—330no—
67C22×AGL(1,𝔽5)—360no—
68C22×AGL(1,𝔽5)—320no—
69C22×AGL(1,𝔽5)—310no—
70C22×AGL(1,𝔽5)—35no—
71C22×AGL(1,𝔽5)—35no—
72C22×AGL(1,𝔽5)—310no—
73C22×AGL(1,𝔽5)C2×A4110yescomplemented
74C26—51no—
75C24×C4—52no—
76C24×C4—155no—
77C24×C4—1510no—
78C24×C4—155no—
79C2×C3×D5—161no—
80C5×A4—45no—
81C3×AGL(1,𝔽5)—162no—
82C22⋊A4C2×AGL(1,𝔽5)11yescharacteristic, direct product factor
83C22×A4—205no—
84C4×A4—2010no—
85C23×C5—330no—
86C23×C5—310no—
87C23×C5—35no—
88C23×C5—35no—
89C23×C5C4×A415yescomplemented
90C22×D5—330no—
91C22×D5—310no—
92C22×D5—330no—
93C22×D5—3120no—
94C22×D5—330no—
95C22×D5—330no—
96C22×D5—330no—
97C22×D5—330no—
98C22×D5—330no—
99C22×D5—330no—
100C22×D5—310no—
101C22×D5—35no—
102C22×D5—35no—
103C22×D5—35no—
104C22×D5—35no—
105C22×D5—35no—
106C22×D5—35no—
107C22×D5C4×A415yescomplemented
108C22×D5C22×A415yes—
109C2×AGL(1,𝔽5)—360no—
110C2×AGL(1,𝔽5)—310no—
111C2×AGL(1,𝔽5)—360no—
112C2×AGL(1,𝔽5)—35no—
113C2×AGL(1,𝔽5)—310no—
114C2×AGL(1,𝔽5)—310no—
115C2×AGL(1,𝔽5)—310no—
116C2×AGL(1,𝔽5)C22⋊A411yescharacteristic, direct product factor
117C25—51no—
118C25—51no—
119C25—155no—
120C25—155no—
121C25—155no—
122C25—155no—
123C25C3×AGL(1,𝔽5)11yescharacteristic, complemented
124C23×C4—55no—
125C23×C4—1510no—
126C23×C4—1510no—
127C23×C4—155no—
128C23×C4—1510no—
129C23×C4—1510no—
130C23×C4—1560no—
131C23×C4—1560no—
132C23×C4—1530no—
133C23×C4—1510no—
134C2×C3×C5—161no—
135C3×D5—161no—
136C3×D5—161no—
137C2×A4—45no—
138C2×A4—205no—
139C2×A4—205no—
140C2×C3×C4—801no—
141C22×C5—35no—
142C22×C5—330no—
143C22×C5—310no—
144C22×C5—35no—
145C22×C5C2×C4×A415yescomplemented
146C2×D5—360no—
147C2×D5—310no—
148C2×D5—330no—
149C2×D5—330no—
150C2×D5—35no—
151C2×D5—35no—
152C2×D5—35no—
153C2×D5—35no—
154C2×D5—35no—
155C2×D5—35no—
156C2×D5—35no—
157C2×D5C2×C22⋊A411yescharacteristic
158AGL(1,𝔽5)—310no—
159AGL(1,𝔽5)C2×C22⋊A412yesdirect product factor
160C24—35no—
161C24—35no—
162C24—55no—
163C24—155no—
164C24—155no—
165C24—155no—
166C24—155no—
167C24—155no—
168C24—155no—
169C24—1510no—
170C24—1530no—
171C24—1530no—
172C24—1530no—
173C24—1560no—
174C24—1510no—
175C24—155no—
176C24C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
177C22×C4—510no—
178C22×C4—155no—
179C22×C4—1510no—
180C22×C4—1520no—
181C22×C4—1560no—
182C22×C4—1530no—
183C22×C4—1530no—
184C22×C4—15120no—
185C22×C4—1560no—
186C22×C4—1560no—
187C22×C4—1510no—
188C22×C4—155no—
189C3×C5—161noHall
190C22×C3—801no—
191A4—45no—
192C3×C4—802no—
193C2×C5—35no—
194C2×C5—35no—
195C2×C5C4×C22⋊A411yescharacteristic, complemented
196D5—35no—
197D5—35no—
198D5C4×C22⋊A411yescharacteristic, complemented
199D5C22×C22⋊A411yescharacteristic
200C23—330no—
201C23—310no—
202C23—35no—
203C23—35no—
204C23—55no—
205C23—55no—
206C23—155no—
207C23—155no—
208C23—155no—
209C23—155no—
210C23—1510no—
211C23—1530no—
212C23—1530no—
213C23—1510no—
214C23—1530no—
215C23—1530no—
216C23—1530no—
217C23—1530no—
218C23—1530no—
219C23—15120no—
220C23—1530no—
221C23—155no—
222C23—155no—
223C23A4×AGL(1,𝔽5)15yescomplemented
224C2×C4—51no—
225C2×C4—1510no—
226C2×C4—1510no—
227C2×C4—155no—
228C2×C4—1560no—
229C2×C4—1560no—
230C2×C4—1510no—
231C2×C4—1510no—
232C2×C3—161no—
233C2×C3—801no—
234C2×C3—801no—
235C5C2×C4×C22⋊A411yescharacteristic, complemented, Sylow
236C22—35no—
237C22—310no—
238C22—330no—
239C22—35no—
240C22—51no—
241C22—155no—
242C22—155no—
243C22—155no—
244C22—155no—
245C22—155no—
246C22—155no—
247C22—1530no—
248C22—1530no—
249C22—1510no—
250C22—1560no—
251C22—155no—
252C22C2×A4×AGL(1,𝔽5)15yescomplemented
253C4—52no—
254C4—1510no—
255C3—161noSylow
256C2—35no—
257C2—35no—
258C2—51no—
259C2—51no—
260C2—155no—
261C2—155no—
262C2AGL(1,𝔽5)×C22⋊A411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2AGL(1,𝔽5)×C22⋊A4
commutator, hypocenterC24×C5C2×C3×C4
FrattiniC1G
Fitting, socleC25×C5C3×C4
prime radicalC2×D5×C22⋊A4C2
cosocleC24×D5C22×C3
LCSG ▷ C24×C5
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. AGL(1,𝔽5)×C22⋊A4C2direct (N × H)
3. C25×AGL(1,𝔽5)C3semidirect
4. C2×C5×C22⋊A4C4semidirect
5. D5×C22⋊A4C4semidirect
13. C24×AGL(1,𝔽5)C2×C3semidirect
14. C5×C22⋊A4C2×C4semidirect
19. C25×C5C3×C4semidirect
24. C24×D5C3×C4semidirect
35. C23×AGL(1,𝔽5)A4semidirect
43. C2×C22⋊A4AGL(1,𝔽5)direct (N × H)
47. C24×C5C2×C3×C4semidirect
73. C22×AGL(1,𝔽5)C2×A4semidirect
82. C22⋊A4C2×AGL(1,𝔽5)direct (N × H)
89. C23×C5C4×A4semidirect
107. C22×D5C4×A4semidirect
116. C2×AGL(1,𝔽5)C22⋊A4direct (N × H)
123. C25C3×AGL(1,𝔽5)semidirect
145. C22×C5C2×C4×A4semidirect
159. AGL(1,𝔽5)C2×C22⋊A4direct (N × H)
176. C24C2×C3×AGL(1,𝔽5)semidirect
195. C2×C5C4×C22⋊A4semidirect
198. D5C4×C22⋊A4semidirect
223. C23A4×AGL(1,𝔽5)semidirect
235. C5C2×C4×C22⋊A4semidirect
252. C22C2×A4×AGL(1,𝔽5)semidirect
262. C2AGL(1,𝔽5)×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×C22⋊A42
AGL(1,𝔽5)×C22⋊A42
C25×AGL(1,𝔽5)3
C2×A4×AGL(1,𝔽5)4
C2×C4×C22⋊A45