Subgroups — C₃×C₅×C₇×AGL(1,𝔽₁₃) #30655310

Order 16380.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×D₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₇×#261	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅×C₇×AGL(1,𝔽₁₃)	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅×C₇×C₁₃⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₇×AGL(1,𝔽₁₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₅×C₇×D₁₃⋊C₃	C₂×C₃	1	3	yes	—
7	C₃×C₅×C₇×D₁₃	C₂×C₃	1	1	yes	characteristic
8	C₃×C₅×AGL(1,𝔽₁₃)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇×#261	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃×C₇×D₁₃⋊C₃	C₂×C₅	1	1	yes	characteristic
11	C₃×C₅×C₇×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented
12	C₅×C₇×C₁₃⋊C₃	C₃×C₄	1	3	yes	complemented
13	C₃²×C₄×C₅×C₇	—	13	1	no	maximal, Hall
14	C₃×C₅×D₁₃⋊C₃	C₂×C₇	1	1	yes	characteristic
15	C₃×C₇×#261	C₃×C₅	1	1	yes	characteristic, complemented
16	C₇×AGL(1,𝔽₁₃)	C₃×C₅	1	3	yes	direct product factor
17	C₅×C₇×D₁₃	C₂×C₃²	1	1	yes	characteristic
18	C₃×C₇×C₁₃⋊C₃	C₄×C₅	1	1	yes	characteristic, complemented, Hall
19	C₃×C₅×#261	C₃×C₇	1	1	yes	characteristic, complemented
20	C₅×AGL(1,𝔽₁₃)	C₃×C₇	1	3	yes	direct product factor
21	C₂×C₃²×C₅×C₇	—	13	1	no	—
22	C₃×C₅×C₁₃⋊C₃	C₄×C₇	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇×D₁₃	C₂×C₃×C₅	1	1	yes	characteristic
24	C₇×D₁₃⋊C₃	C₂×C₃×C₅	1	3	yes	—
25	C₃×AGL(1,𝔽₁₃)	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
26	C₅×C₇×C₁₃	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
27	C₃×C₄×C₅×C₇	—	13	3	no	—
28	C₃×C₄×C₅×C₇	—	39	1	no	—
29	C₃×C₅×D₁₃	C₂×C₃×C₇	1	1	yes	characteristic
30	C₅×D₁₃⋊C₃	C₂×C₃×C₇	1	3	yes	—
31	C₇×#261	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
32	C₃²×C₅×C₇	—	13	1	no	Hall
33	C₃×C₇×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic, complemented
34	C₇×C₁₃⋊C₃	C₃×C₄×C₅	1	3	yes	complemented
35	C₅×#261	C₃²×C₇	1	1	yes	characteristic, complemented, Hall
36	C₃²×C₄×C₇	—	13	1	no	Hall
37	C₃×D₁₃⋊C₃	C₂×C₅×C₇	1	1	yes	characteristic
38	C₂×C₃×C₅×C₇	—	13	3	no	—
39	C₂×C₃×C₅×C₇	—	39	1	no	—
40	C₃×C₅×C₁₃	C₃×C₄×C₇	1	1	yes	characteristic, complemented
41	C₅×C₁₃⋊C₃	C₃×C₄×C₇	1	3	yes	complemented
42	C₇×D₁₃	C₂×C₃²×C₅	1	1	yes	characteristic
43	C₃²×C₄×C₅	—	13	1	no	Hall
44	C₃×#261	C₃×C₅×C₇	1	1	yes	characteristic, complemented
45	AGL(1,𝔽₁₃)	C₃×C₅×C₇	1	3	yes	direct product factor
46	C₄×C₅×C₇	—	39	1	no	Hall
47	C₅×D₁₃	C₂×C₃²×C₇	1	1	yes	characteristic
48	C₂×C₃²×C₇	—	13	1	no	—
49	C₃×C₁₃⋊C₃	C₄×C₅×C₇	1	1	yes	characteristic, complemented, Hall
50	C₃×C₅×C₇	—	13	3	no	—
51	C₃×C₅×C₇	AGL(1,𝔽₁₃)	1	1	yes	characteristic, direct product factor
52	C₇×C₁₃	C₃²×C₄×C₅	1	1	yes	characteristic, complemented, Hall
53	C₂×C₃²×C₅	—	13	1	no	—
54	C₃×C₄×C₇	—	13	3	no	—
55	C₃×C₄×C₇	—	39	1	no	—
56	C₃×D₁₃	C₂×C₃×C₅×C₇	1	1	yes	characteristic
57	D₁₃⋊C₃	C₂×C₃×C₅×C₇	1	3	yes	—
58	C₂×C₅×C₇	—	39	1	no	—
59	C₅×C₁₃	C₃²×C₄×C₇	1	1	yes	characteristic, complemented, Hall
60	C₃²×C₇	—	13	1	no	Hall
61	C₃×C₄×C₅	—	13	3	no	—
62	C₃×C₄×C₅	—	39	1	no	—
63	#261	C₃²×C₅×C₇	1	1	yes	characteristic, complemented, Hall
64	C₃²×C₅	—	13	1	no	Hall
65	C₂×C₃×C₇	—	13	3	no	—
66	C₂×C₃×C₇	—	39	1	no	—
67	C₃×C₁₃	C₃×C₄×C₅×C₇	1	1	yes	characteristic, complemented
68	C₁₃⋊C₃	C₃×C₄×C₅×C₇	1	3	yes	complemented
69	C₃²×C₄	—	13	1	no	Hall
70	C₅×C₇	C₃×AGL(1,𝔽₁₃)	1	1	yes	characteristic, direct product factor, Hall
71	C₂×C₃×C₅	—	13	3	no	—
72	C₂×C₃×C₅	—	39	1	no	—
73	C₄×C₇	—	39	1	no	Hall
74	D₁₃	C₂×C₃²×C₅×C₇	1	1	yes	characteristic
75	C₃×C₇	—	13	3	no	—
76	C₃×C₇	C₅×AGL(1,𝔽₁₃)	1	1	yes	characteristic, direct product factor
77	C₄×C₅	—	39	1	no	Hall
78	C₂×C₃²	—	13	1	no	—
79	C₃×C₅	—	13	3	no	—
80	C₃×C₅	C₇×AGL(1,𝔽₁₃)	1	1	yes	characteristic, direct product factor
81	C₂×C₇	—	39	1	no	—
82	C₁₃	C₃²×C₄×C₅×C₇	1	1	yes	characteristic, complemented, Sylow
83	C₃×C₄	—	13	3	no	—
84	C₃×C₄	—	39	1	no	—
85	C₂×C₅	—	39	1	no	—
86	C₃²	—	13	1	no	Sylow
87	C₇	C₃×C₅×AGL(1,𝔽₁₃)	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×C₃	—	13	3	no	—
89	C₂×C₃	—	39	1	no	—
90	C₅	C₃×C₇×AGL(1,𝔽₁₃)	1	1	yes	characteristic, direct product factor, Sylow
91	C₄	—	39	1	no	Sylow
92	C₃	—	13	3	no	—
93	C₃	C₅×C₇×AGL(1,𝔽₁₃)	1	1	yes	characteristic, direct product factor
94	C₂	—	39	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅×C₇	AGL(1,𝔽₁₃)
commutator, hypocenter	C₁₃	C₃²×C₄×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₁₃	C₃×C₄
prime radical	C₃×C₅×C₇×D₁₃⋊C₃	C₂
cosocle	D₁₃	C₂×C₃²×C₅×C₇
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃×C₅×C₇

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₇×#261	C₃	semidirect
3. C₅×C₇×AGL(1,𝔽₁₃)	C₃	direct (N × H)
4. C₃×C₅×C₇×C₁₃⋊C₃	C₄	semidirect
5. C₃×C₇×AGL(1,𝔽₁₃)	C₅	direct (N × H)
8. C₃×C₅×AGL(1,𝔽₁₃)	C₇	direct (N × H)
9. C₅×C₇×#261	C₃²	semidirect
11. C₃×C₅×C₇×C₁₃	C₃×C₄	semidirect
12. C₅×C₇×C₁₃⋊C₃	C₃×C₄	semidirect
15. C₃×C₇×#261	C₃×C₅	semidirect
16. C₇×AGL(1,𝔽₁₃)	C₃×C₅	direct (N × H)
18. C₃×C₇×C₁₃⋊C₃	C₄×C₅	semidirect
19. C₃×C₅×#261	C₃×C₇	semidirect
20. C₅×AGL(1,𝔽₁₃)	C₃×C₇	direct (N × H)
22. C₃×C₅×C₁₃⋊C₃	C₄×C₇	semidirect
25. C₃×AGL(1,𝔽₁₃)	C₅×C₇	direct (N × H)
26. C₅×C₇×C₁₃	C₃²×C₄	semidirect
31. C₇×#261	C₃²×C₅	semidirect
33. C₃×C₇×C₁₃	C₃×C₄×C₅	semidirect
34. C₇×C₁₃⋊C₃	C₃×C₄×C₅	semidirect
35. C₅×#261	C₃²×C₇	semidirect
40. C₃×C₅×C₁₃	C₃×C₄×C₇	semidirect
41. C₅×C₁₃⋊C₃	C₃×C₄×C₇	semidirect
44. C₃×#261	C₃×C₅×C₇	semidirect
45. AGL(1,𝔽₁₃)	C₃×C₅×C₇	direct (N × H)
49. C₃×C₁₃⋊C₃	C₄×C₅×C₇	semidirect
51. C₃×C₅×C₇	AGL(1,𝔽₁₃)	direct (N × H)
52. C₇×C₁₃	C₃²×C₄×C₅	semidirect
59. C₅×C₁₃	C₃²×C₄×C₇	semidirect
63. #261	C₃²×C₅×C₇	semidirect
67. C₃×C₁₃	C₃×C₄×C₅×C₇	semidirect
68. C₁₃⋊C₃	C₃×C₄×C₅×C₇	semidirect
70. C₅×C₇	C₃×AGL(1,𝔽₁₃)	direct (N × H)
76. C₃×C₇	C₅×AGL(1,𝔽₁₃)	direct (N × H)
80. C₃×C₅	C₇×AGL(1,𝔽₁₃)	direct (N × H)
82. C₁₃	C₃²×C₄×C₅×C₇	semidirect
87. C₇	C₃×C₅×AGL(1,𝔽₁₃)	direct (N × H)
90. C₅	C₃×C₇×AGL(1,𝔽₁₃)	direct (N × H)
93. C₃	C₅×C₇×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.

subgroup	index
C₃×C₅×C₇×D₁₃⋊C₃	2
C₃×C₅×C₇×#261	3
C₅×C₇×AGL(1,𝔽₁₃)	3
C₃×C₇×AGL(1,𝔽₁₃)	5
C₃×C₅×AGL(1,𝔽₁₃)	7
C₃²×C₄×C₅×C₇	13

Subgroups — C3×C5×C7×AGL(1,𝔽13) #30655310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₇×AGL(1,𝔽₁₃) #30655310