Finite Group Databasebeta

← AGL(1,𝔽5)×#131⋊C3 #14512548

Subgroups — AGL(1,𝔽5)×#131⋊C3 #14512548

Order 1920.

314 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 62 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
1AGL(1,𝔽5)×#41⋊C3C211yesmaximal, characteristic
2D5⋊(#131⋊C3)C211yesmaximal, characteristic
3D5×#131⋊C3C211yesmaximal, characteristic
4AGL(1,𝔽5)×#131C311yesmaximal, characteristic, complemented, Hall
5D5×#41⋊C3C2211yescharacteristic
6AGL(1,𝔽5)×SL(2,𝔽3)C412yes—
7#4133⋊C3C412yescomplemented
8C5×#131⋊C3C411yescharacteristic, complemented
9C3×C8×AGL(1,𝔽5)—41nomaximal
10#4094⋊C3C411yescharacteristic, complemented
11C4×#131⋊C3—51nomaximal, Hall
12#41×AGL(1,𝔽5)C2×C311yescharacteristic
13D5⋊#131C2×C311yescharacteristic
14D5×#131C2×C311yescharacteristic
15#65728—31no—
16M16×AGL(1,𝔽5)—31no—
17#65718—31no—
18C2×C8×AGL(1,𝔽5)—31no—
19C5×#41⋊C3C2×C411yescharacteristic
20C3×C4×AGL(1,𝔽5)—41no—
21C3×D5⋊C8—41no—
22D5×SL(2,𝔽3)C2×C411yescharacteristic
23C5⋊(#41⋊C3)C2×C411yescharacteristic
24C3×C8×D5—41no—
25C2×#131⋊C3—51no—
26C2×#131⋊C3—51no—
27C4×#41⋊C3—51no—
28D5×#41C22×C311yescharacteristic
29Q8×AGL(1,𝔽5)C3×C412yes—
30#4133C3×C412yescomplemented
31D4×AGL(1,𝔽5)—32no—
32#4131—32no—
33#4130—31no—
34C2×C4×AGL(1,𝔽5)—31no—
35D5⋊M16—31no—
36C2×D5⋊C8—31no—
37C5×#131C3×C411yescharacteristic, complemented
38#4094C3×C411yescharacteristic, complemented
39#4053—31no—
40D5×M16—31no—
41#4047—31no—
42C2×C8×D5—31no—
43#3992—32no—
44C8×AGL(1,𝔽5)—31no—
45C8×AGL(1,𝔽5)—31no—
46C8×AGL(1,𝔽5)—31no—
47C8×AGL(1,𝔽5)A411yescharacteristic
48C4×#131—51noSylow
49C2×C3×AGL(1,𝔽5)—42no—
50C3×C4×D5—41no—
51C5×SL(2,𝔽3)C4211yescharacteristic
52C3×#170—42no—
53C3×C5×C8—41no—
54C3×#168—41no—
55C2×#41⋊C3—51no—
56#131⋊C3—52no—
57#131⋊C3—51no—
58#131⋊C3AGL(1,𝔽5)11yescharacteristic, direct product factor
59C4×SL(2,𝔽3)—52no—
60C3×C4×C8—201no—
61C22×AGL(1,𝔽5)—32no—
62C5×#41C2×C3×C411yescharacteristic
63#713C2×C3×C411yescharacteristic
64Q8×D5C2×C3×C411yescharacteristic
65#711—31no—
66D4×D5—31no—
67#709—31no—
68C2×C4×D5—31no—
69C22⋊AGL(1,𝔽5)—32no—
70#704—32no—
71C2×#170—32no—
72C4⋊AGL(1,𝔽5)—32no—
73C4×AGL(1,𝔽5)—32no—
74C4×AGL(1,𝔽5)—31no—
75C4×AGL(1,𝔽5)C2×A411yescharacteristic
76#700—32no—
77D5⋊C8—31no—
78D5⋊C8—32no—
79D5⋊C8C2×A411yescharacteristic
80C5×M16—31no—
81C2×C5×C8—31no—
82#681—31no—
83C2×#168—31no—
84#676—32no—
85C8×D5—31no—
86C8×D5—31no—
87C8×D5—31no—
88C8×D5C2×A411yescharacteristic
89C2×#131—51no—
90C2×#131—51no—
91C4×#41—51no—
92#405—151no—
93#405—151no—
94C4×M16—151no—
95C2×C4×C8—151no—
96C2×C3×D5—41no—
97C3×AGL(1,𝔽5)—44no—
98C3×C4×C5—41no—
99C3×Dic5—41no—
100#41⋊C3—51no—
101#41⋊C3C2×AGL(1,𝔽5)11yescharacteristic
102C2×SL(2,𝔽3)—51no—
103C2×C3×C8—201no—
104C2×C3×C8—201no—
105C3×C42—201no—
106C22×D5—31no—
107C2×AGL(1,𝔽5)—32no—
108C2×AGL(1,𝔽5)—64no—
109C2×AGL(1,𝔽5)C4×A412yes—
110C5×Q8C3×C4211yescharacteristic
111C5×D4—31no—
112C2×C4×C5—31no—
113C22⋊D5—32no—
114C2×Dic5—31no—
115D20—31no—
116C4×D5—31no—
117C4×D5—31no—
118C4×D5—31no—
119C4×D5C22×A411yescharacteristic
120Dic10—31no—
121#170—32no—
122#170C4×A412yes—
123C5×C8—31no—
124C5×C8C4×A411yescharacteristic
125#168—31no—
126#168C4×A411yescharacteristic
127C2×#41—51no—
128#131—52no—
129#131—51no—
130#131—151no—
131#131—151no—
132#131—152no—
133#131C3×AGL(1,𝔽5)11yescharacteristic, complemented
134C2×M16—151no—
135C2×M16—151no—
136C22×C8—151no—
137C22×C8—151no—
138C4×Q8—52no—
139C4×D4—152no—
140#117—151no—
141C2×C42—151no—
142#97—152no—
143C4×C8—51no—
144C4×C8—151no—
145C4×C8—151no—
146C4×C8—151no—
147C2×C3×C5—41no—
148C3×D5—42no—
149C2×C3×C4—201no—
150C2×C3×C4—202no—
151SL(2,𝔽3)C4×AGL(1,𝔽5)11yescharacteristic
152C3×C8—41no—
153C3×C8—201no—
154C3×C8—202no—
155C22×C5—31no—
156C2×D5—31no—
157C2×D5—62no—
158C2×D5C2×C4×A411yescharacteristic
159AGL(1,𝔽5)—62no—
160AGL(1,𝔽5)#131⋊C314yesdirect product factor
161C4×C5—31no—
162C4×C5C2×C4×A411yescharacteristic
163Dic5—31no—
164Dic5C2×C4×A411yescharacteristic
165#41—51no—
166#41—151no—
167#41—151no—
168#41C2×C3×AGL(1,𝔽5)11yescharacteristic
169C2×Q8—51no—
170C2×D4—151no—
171C22×C4—151no—
172C22×C4—152no—
173M16—31no—
174M16—152no—
175M16—152no—
176M16—151no—
177M16—152no—
178C2×C8—31no—
179C2×C8—51no—
180C2×C8—51no—
181C2×C8—151no—
182C2×C8—151no—
183C2×C8—151no—
184C2×C8—152no—
185C2×C8—152no—
186C2×C8—151no—
187C2×C8—151no—
188C4⋊C4—152no—
189C22⋊C4—152no—
190C42—51no—
191C42—151no—
192C42—152no—
193C3×C5—41noHall
194C22×C3—201no—
195C3×C4—41no—
196C3×C4—201no—
197C3×C4—204no—
198C2×C5—61no—
199C2×C5C42×A411yescharacteristic
200D5—61no—
201D5C2×#131⋊C312yes—
202C23—151no—
203Q8—151no—
204Q8C3×C4×AGL(1,𝔽5)11yescharacteristic
205D4—31no—
206D4—151no—
207D4—152no—
208C2×C4—31no—
209C2×C4—52no—
210C2×C4—51no—
211C2×C4—151no—
212C2×C4—151no—
213C2×C4—151no—
214C2×C4—151no—
215C2×C4—152no—
216C2×C4—304no—
217C8—31no—
218C8—52no—
219C8—51no—
220C8—152no—
221C8—151no—
222C8A4×AGL(1,𝔽5)11yescharacteristic
223C2×C3—41no—
224C2×C3—202no—
225C5C4×#131⋊C311yescharacteristic, complemented, Sylow
226C22—31no—
227C22—51no—
228C22—151no—
229C22—302no—
230C4—31no—
231C4—54no—
232C4—51no—
233C4—151no—
234C4—302no—
235C4C2×A4×AGL(1,𝔽5)11yescharacteristic
236C3—41noSylow
237C2—52no—
238C2—61no—
239C2—301no—
240C2C4×A4×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, hypercenterC8A4×AGL(1,𝔽5)
commutator, hypocenterC5×Q8C3×C42
FrattiniC4C2×A4×AGL(1,𝔽5)
FittingC5×#131C3×C4
socleC2×C5C42×A4
prime radicalD5×#41⋊C3C22
cosocleD5×#41C22×C3
LCSG ▷ C5×Q8
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
4. AGL(1,𝔽5)×#131C3semidirect
7. #4133⋊C3C4semidirect
8. C5×#131⋊C3C4semidirect
10. #4094⋊C3C4semidirect
30. #4133C3×C4semidirect
37. C5×#131C3×C4semidirect
38. #4094C3×C4semidirect
58. #131⋊C3AGL(1,𝔽5)direct (N × H)
133. #131C3×AGL(1,𝔽5)semidirect
160. AGL(1,𝔽5)#131⋊C3direct (N × H)
225. C5C4×#131⋊C3semidirect

Maximal subgroups

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

subgroupindex
AGL(1,𝔽5)×#41⋊C32
D5⋊(#131⋊C3)2
D5×#131⋊C32
AGL(1,𝔽5)×#1313
C3×C8×AGL(1,𝔽5)4
C4×#131⋊C35