Finite Group Databasebeta

← C3×A4×AGL(1,𝔽13) #72242026

Subgroups — C3×A4×AGL(1,𝔽13) #72242026

Order 5616.

420 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C3×A4×D13⋊C3C211yesmaximal, characteristic
2C3×#261⋊A4C312yesmaximal, complemented
3C3×A4×#261C311yesmaximal, characteristic, complemented
4C22×C3×AGL(1,𝔽13)C311yesmaximal, characteristic, complemented
5A4×AGL(1,𝔽13)C319yesmaximal, direct product factor
6C3×A4×C13⋊C3C411yescharacteristic, complemented
7C32×AGL(1,𝔽13)—41nomaximal
8C3×D13⋊A4C2×C312yes—
9C3×A4×D13C2×C311yescharacteristic
10C22×C3×D13⋊C3C2×C311yescharacteristic
11C2×C3×AGL(1,𝔽13)—31no—
12C2×C3×AGL(1,𝔽13)—31no—
13A4×D13⋊C3C2×C319yes—
14C32×D13⋊C3—41no—
15C22×C3×#261C3211yescharacteristic, complemented
16#261⋊A4C3216yescomplemented
17A4×#261C3213yescomplemented
18C22×AGL(1,𝔽13)C3213yescomplemented
19C3×C22⋊(C13⋊C3)C3×C412yescomplemented
20C3×A4×C13C3×C411yescharacteristic, complemented
21C22×C3×C13⋊C3C3×C411yescharacteristic, complemented
22C32×#261—41no—
23C2×C3×D13⋊C3—31no—
24C2×C3×D13⋊C3—31no—
25A4×C13⋊C3C3×C419yescomplemented
26C3×AGL(1,𝔽13)—31no—
27C3×AGL(1,𝔽13)—42no—
28C3×AGL(1,𝔽13)—49no—
29C3×AGL(1,𝔽13)A411yescharacteristic, direct product factor
30C32×C4×A4—131nomaximal, Hall
31C32×C13⋊C3—41noHall
32C22×C3×D13C2×C3211yescharacteristic
33C2×C3×#261—31no—
34C2×C3×#261—31no—
35D13⋊A4C2×C3216yes—
36A4×D13C2×C3213yes—
37C22×D13⋊C3C2×C3213yes—
38C2×AGL(1,𝔽13)—33no—
39C2×AGL(1,𝔽13)—33no—
40C32×D13—41no—
41C2×C3×C13⋊C3—31no—
42C3×D13⋊C3—31no—
43C3×D13⋊C3—42no—
44C3×D13⋊C3—49no—
45C3×D13⋊C3C2×A411yescharacteristic
46C2×C32×A4—131no—
47C22×#261C3311yescharacteristic, complemented, Hall
48C22×C3×C13C32×C411yescharacteristic, complemented
49C2×C3×D13—31no—
50C2×C3×D13—31no—
51C22⋊(C13⋊C3)C32×C416yescomplemented
52A4×C13C32×C413yescomplemented
53C22×C13⋊C3C32×C413yescomplemented
54C3×#261—31no—
55C3×#261—43no—
56C3×#261C3×A411yescharacteristic, complemented
57C2×D13⋊C3—33no—
58C2×D13⋊C3—33no—
59AGL(1,𝔽13)—33no—
60AGL(1,𝔽13)—46no—
61AGL(1,𝔽13)C3×A413yesdirect product factor
62C22×C32×C4—131no—
63C3×C4×A4—132no—
64C3×C4×A4—139no—
65C3×C4×A4—391no—
66C32×C13—41no—
67C3×C13⋊C3—42no—
68C3×C13⋊C3—49no—
69C3×C13⋊C3C4×A411yescharacteristic, complemented
70C32×A4—131no—
71C33×C4—521no—
72C22×D13C2×C3311yescharacteristic
73C2×#261—31no—
74C2×#261—31no—
75C2×C3×C13—31no—
76C3×D13—31no—
77C3×D13—43no—
78C3×D13C2×C3×A411yescharacteristic
79C2×C13⋊C3—33no—
80D13⋊C3—33no—
81D13⋊C3—46no—
82D13⋊C3C2×C3×A413yes—
83C23×C32—131no—
84C2×C3×A4—132no—
85C2×C3×A4—139no—
86C2×C3×A4—391no—
87C2×C32×C4—391no—
88C2×C32×C4—391no—
89C2×C33—521no—
90C22×C13C33×C411yescharacteristic, complemented
91C2×D13—31no—
92C2×D13—31no—
93#261—31no—
94#261C32×A411yescharacteristic, complemented
95C22×C3×C4—133no—
96C22×C3×C4—391no—
97C4×A4—136no—
98C4×A4—393no—
99C3×C13—43no—
100C3×C13C3×C4×A411yescharacteristic, complemented
101C13⋊C3—46no—
102C13⋊C3C3×C4×A413yescomplemented
103C22×C32—131no—
104C22×C32—391no—
105C22×C32—391no—
106C3×A4—132no—
107C3×A4—139no—
108C3×A4AGL(1,𝔽13)11yescharacteristic, direct product factor
109C32×C4—131no—
110C32×C4—391no—
111C32×C4—529no—
112C32×C4—522no—
113C32×C4—1561no—
114C33—521noSylow
115C2×C13—31no—
116D13—31no—
117D13C2×C32×A411yescharacteristic
118C23×C3—133no—
119C23×C3—391no—
120C2×A4—136no—
121C2×A4—393no—
122C2×C3×C4—393no—
123C2×C3×C4—393no—
124C2×C3×C4—1171no—
125C2×C3×C4—1171no—
126C2×C32—131no—
127C2×C32—391no—
128C2×C32—391no—
129C2×C32—521no—
130C2×C32—529no—
131C2×C32—522no—
132C22×C4—391noSylow
133C13C32×C4×A411yescharacteristic, complemented, Sylow
134C22×C3—133no—
135C22×C3—393no—
136C22×C3—393no—
137C22×C3—1171no—
138C22×C3—1171no—
139C22×C3C3×AGL(1,𝔽13)11yescharacteristic, complemented
140A4—136no—
141A4C3×AGL(1,𝔽13)13yesdirect product factor
142C3×C4—133no—
143C3×C4—391no—
144C3×C4—393no—
145C3×C4—526no—
146C3×C4—1171no—
147C3×C4—1563no—
148C32—41no—
149C32—131no—
150C32—529no—
151C32—522no—
152C23—391no—
153C2×C4—1171no—
154C2×C4—1171no—
155C2×C3—31no—
156C2×C3—133no—
157C2×C3—131no—
158C2×C3—393no—
159C2×C3—393no—
160C2×C3—523no—
161C2×C3—526no—
162C2×C3—1171no—
163C22—1171no—
164C22—1171no—
165C22C32×AGL(1,𝔽13)11yescharacteristic, complemented
166C4—391no—
167C4—1171no—
168C3—43no—
169C3—133no—
170C3—526no—
171C3A4×AGL(1,𝔽13)11yescharacteristic, direct product factor
172C2—31no—
173C2—131no—
174C2—1171no—
subgroupG/H
center, hypercenterC3A4×AGL(1,𝔽13)
commutator, hypocenterC22×C13C33×C4
FrattiniC1G
Fitting, socleC22×C3×C13C32×C4
prime radicalC3×A4×D13⋊C3C2
cosocleC22×D13C2×C33
LCSG ▷ C22×C13
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
2. C3×#261⋊A4C3semidirect
3. C3×A4×#261C3semidirect
4. C22×C3×AGL(1,𝔽13)C3semidirect
5. A4×AGL(1,𝔽13)C3direct (N × H)
6. C3×A4×C13⋊C3C4semidirect
15. C22×C3×#261C32semidirect
16. #261⋊A4C32semidirect
17. A4×#261C32semidirect
18. C22×AGL(1,𝔽13)C32semidirect
19. C3×C22⋊(C13⋊C3)C3×C4semidirect
20. C3×A4×C13C3×C4semidirect
21. C22×C3×C13⋊C3C3×C4semidirect
25. A4×C13⋊C3C3×C4semidirect
29. C3×AGL(1,𝔽13)A4direct (N × H)
47. C22×#261C33semidirect
48. C22×C3×C13C32×C4semidirect
51. C22⋊(C13⋊C3)C32×C4semidirect
52. A4×C13C32×C4semidirect
53. C22×C13⋊C3C32×C4semidirect
56. C3×#261C3×A4semidirect
61. AGL(1,𝔽13)C3×A4direct (N × H)
69. C3×C13⋊C3C4×A4semidirect
90. C22×C13C33×C4semidirect
94. #261C32×A4semidirect
100. C3×C13C3×C4×A4semidirect
102. C13⋊C3C3×C4×A4semidirect
108. C3×A4AGL(1,𝔽13)direct (N × H)
133. C13C32×C4×A4semidirect
139. C22×C3C3×AGL(1,𝔽13)semidirect
141. A4C3×AGL(1,𝔽13)direct (N × H)
165. C22C32×AGL(1,𝔽13)semidirect
171. C3A4×AGL(1,𝔽13)direct (N × H)

Maximal subgroups

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

subgroupindex
C3×A4×D13⋊C32
C3×#261⋊A43
C3×A4×#2613
C22×C3×AGL(1,𝔽13)3
A4×AGL(1,𝔽13)3
C32×AGL(1,𝔽13)4
C32×C4×A413