Finite Group Databasebeta

← AGL(1,𝔽5)×#4177 #72268653

Subgroups — AGL(1,𝔽5)×#4177 #72268653

Order 3240.

172 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 36 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×#4177C211yesmaximal, characteristic
2#13108185C211yesmaximal, characteristic, complemented
3AGL(1,𝔽5)×#731C211yesmaximal, characteristic, complemented
4AGL(1,𝔽5)×C3⋊D9C311yesmaximal, characteristic, complemented
5AGL(1,𝔽5)×(C3⋊D3)⋊C3—31nomaximal
6C5⋊#4177C411yescharacteristic, complemented
7C5×#4177C411yescharacteristic, complemented
8D5×#731C2211yescharacteristic
9C4×#4177—51nomaximal, Hall
10D5×C3⋊D9C2×C311yescharacteristic
11D5×(C3⋊D3)⋊C3—31no—
12C3⋊(C9⋊AGL(1,𝔽5))C2×C311yescharacteristic, complemented
13(C3⋊ΓL(1,𝔽16))⋊C3—31no—
14AGL(1,𝔽5)×3−1+2—21no—
15AGL(1,𝔽5)×H(3,𝔽3)D311yescharacteristic
16C3×C9×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
17C5×#731C2×C411yescharacteristic, complemented, Hall
18C3⋊D3×AGL(1,𝔽5)—31no—
19C3×D3×AGL(1,𝔽5)—91no—
20D9×AGL(1,𝔽5)—91no—
21C2×#4177—51no—
22C4×#731—51no—
23#66324—51no—
24C3⋊D45C3×C411yescharacteristic, complemented
25C5×C3⋊D9C3×C411yescharacteristic, complemented
26C5⋊((C3⋊D3)⋊C3)—31no—
27C5×(C3⋊D3)⋊C3—31no—
28D5×3−1+2—21no—
29D5×H(3,𝔽3)C2×D311yescharacteristic
30C3×C9×D5C22×C311yescharacteristic
31C4×C3⋊D9—51no—
32C4×(C3⋊D3)⋊C3—151no—
33D5×C3⋊D3—31no—
34C3×D3×D5—91no—
35C3⋊ΓL(1,𝔽16)—31no—
36C3×ΓL(1,𝔽16)—91no—
37C32×AGL(1,𝔽5)—31no—
38C32×AGL(1,𝔽5)C3×D311yescharacteristic
39D5×D9—91no—
40C9⋊AGL(1,𝔽5)—91no—
41C9×AGL(1,𝔽5)—31no—
42C9×AGL(1,𝔽5)—61no—
43C2×#731—51no—
44#4177—51no—
45#4177AGL(1,𝔽5)11yescharacteristic, direct product factor
46C5×3−1+2—21no—
47C5×H(3,𝔽3)C4×D311yescharacteristic
48C3×C5×C9C2×C3×C411yescharacteristic, complemented
49C2×C3×AGL(1,𝔽5)—271no—
50D3×AGL(1,𝔽5)—91no—
51D3×AGL(1,𝔽5)—271no—
52C2×C3⋊D9—51no—
53C2×(C3⋊D3)⋊C3—151no—
54C4×3−1+2—101no—
55C4×H(3,𝔽3)—51no—
56C3×C4×C9—51no—
57C3⋊Dic9—51no—
58(C3⋊Dic3)⋊C3—151no—
59C3⋊D15—31no—
60C5×C3⋊D3—31no—
61C3×D15—91no—
62C3×C5×D3—91no—
63C32×D5—31no—
64C32×D5C2×C3×D311yescharacteristic
65D45—91no—
66C9×D5—31no—
67C9×D5—61no—
68C5×D9—91no—
69#731C2×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
70C4×C3⋊D3—151no—
71C3×C4×D3—451no—
72C4×D9—451no—
73C2×C3×D5—271no—
74D3×D5—91no—
75D3×D5—271no—
76ΓL(1,𝔽16)—91no—
77ΓL(1,𝔽16)—271no—
78C3×AGL(1,𝔽5)—31no—
79C3×AGL(1,𝔽5)—181no—
80C3×AGL(1,𝔽5)—271no—
81C3×AGL(1,𝔽5)(C3⋊D3)⋊C311yescharacteristic
82C2×3−1+2—101no—
83C2×H(3,𝔽3)—51no—
84C2×C3×C9—51no—
85C3⋊D9—51no—
86C3⋊D9C3×AGL(1,𝔽5)11yescharacteristic, complemented
87(C3⋊D3)⋊C3—31no—
88(C3⋊D3)⋊C3—151no—
89C32×C5—31no—
90C32×C5C3×C4×D311yescharacteristic
91C5×C9—31no—
92C5×C9—61no—
93C2×AGL(1,𝔽5)—271noHall
94C2×C3⋊D3—151no—
95C2×C3×D3—451no—
96C32×C4—51no—
97C32×C4—301no—
98C3⋊Dic3—151no—
99C3×Dic3—451no—
100C2×D9—451no—
101C4×C9—151no—
102C4×C9—301no—
103Dic9—451no—
104C2×C3×C5—271no—
105D15—91no—
106D15—271no—
107C3×D5—31no—
108C3×D5—181no—
109C3×D5—271no—
110C3×D5C2×(C3⋊D3)⋊C311yescharacteristic
111C5×D3—91no—
112C5×D3—271no—
1133−1+2—21no—
114H(3,𝔽3)D3×AGL(1,𝔽5)11yescharacteristic
115C3×C9C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
116C2×C3×C4—1351no—
117C4×D3—451no—
118C4×D3—1351no—
119C2×D5—271no—
120AGL(1,𝔽5)—271no—
121AGL(1,𝔽5)#417711yescharacteristic, direct product factor
122C2×C32—51no—
123C2×C32—301no—
124C3⋊D3—31no—
125C3⋊D3—151no—
126C3×D3—91no—
127C3×D3—451no—
128C2×C9—151no—
129C2×C9—301no—
130D9—91no—
131D9—451no—
132C3×C5—31no—
133C3×C5—181no—
134C3×C5C4×(C3⋊D3)⋊C311yescharacteristic
135C22×C3—1351no—
136C2×D3—451no—
137C2×D3—1351no—
138C3×C4—51no—
139C3×C4—151no—
140C3×C4—901no—
141C3×C4—1351no—
142Dic3—451no—
143Dic3—1351no—
144C2×C5—271no—
145D5—271no—
146D5C2×#417711yescharacteristic
147C32—61no—
148C32C3×D3×AGL(1,𝔽5)11yescharacteristic
149C9—31no—
150C9—61no—
151C2×C4—1351noSylow
152C2×C3—51no—
153C2×C3—151no—
154C2×C3—271no—
155C2×C3—901no—
156C2×C3—1351no—
157D3—91no—
158D3—271no—
159D3—451no—
160D3—1351no—
161C5C4×#417711yescharacteristic, complemented, Sylow
162C22—1351no—
163C4—51no—
164C4—1351no—
165C3—31no—
166C3—181no—
167C3AGL(1,𝔽5)×(C3⋊D3)⋊C311yescharacteristic
168C2—51no—
169C2—271no—
170C2—1351no—
subgroupG/H
center, hypercenterC1G
commutator, hypocenterC3×C5×C9C2×C3×C4
FrattiniC32C3×D3×AGL(1,𝔽5)
FittingC5×#731C2×C4
socleC3×C5C4×(C3⋊D3)⋊C3
prime radicalD5×#4177C2
cosocleC3×C9×D5C22×C3
LCSG ▷ C3×C5×C9
UCSC1

Complements

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

NH ≅ G/Ntype
2. #13108185C2semidirect
3. AGL(1,𝔽5)×#731C2semidirect
4. AGL(1,𝔽5)×C3⋊D9C3semidirect
6. C5⋊#4177C4semidirect
7. C5×#4177C4semidirect
12. C3⋊(C9⋊AGL(1,𝔽5))C2×C3semidirect
16. C3×C9×AGL(1,𝔽5)C2×C3semidirect
17. C5×#731C2×C4semidirect
24. C3⋊D45C3×C4semidirect
25. C5×C3⋊D9C3×C4semidirect
45. #4177AGL(1,𝔽5)direct (N × H)
48. C3×C5×C9C2×C3×C4semidirect
69. #731C2×AGL(1,𝔽5)semidirect
86. C3⋊D9C3×AGL(1,𝔽5)semidirect
115. C3×C9C2×C3×AGL(1,𝔽5)semidirect
121. AGL(1,𝔽5)#4177direct (N × H)
161. C5C4×#4177semidirect

Maximal subgroups

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

subgroupindex
D5×#41772
#131081852
AGL(1,𝔽5)×#7312
AGL(1,𝔽5)×C3⋊D93
AGL(1,𝔽5)×(C3⋊D3)⋊C33
C4×#41775