Subgroups — C₁₃×AGL(1,𝔽₅)×#3862 #50303936

Order 38220.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×C₁₃×#3862	C₂	1	1	yes	maximal, characteristic
2	C₇²×C₁₃×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₃×#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	C₄×C₁₃×#3862	—	5	1	no	maximal, Hall
5	C₇²×D₅×C₁₃	C₂×C₃	1	1	yes	characteristic
6	C₁₃×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	7	2	no	maximal
7	C₂×C₁₃×#3862	—	5	1	no	—
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	AGL(1,𝔽₅)×#3862	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	D₅×C₁₃×ΓL(1,𝔽₈)	—	7	2	no	—
11	C₄×C₇²×C₁₃	—	5	1	no	Hall
12	C₁₃×#3862	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
13	C₇×C₁₃×AGL(1,𝔽₅)	—	3	2	no	—
14	C₇×C₁₃×AGL(1,𝔽₅)	ΓL(1,𝔽₈)	1	2	yes	complemented
15	D₅×#3862	C₂×C₁₃	1	1	yes	characteristic
16	C₅×C₁₃×ΓL(1,𝔽₈)	—	7	2	no	—
17	C₂×C₇²×C₁₃	—	5	1	no	—
18	C₄×C₁₃×ΓL(1,𝔽₈)	—	35	2	no	—
19	C₇²×AGL(1,𝔽₅)	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
20	C₇×D₅×C₁₃	—	3	2	no	—
21	C₇×D₅×C₁₃	C₂×ΓL(1,𝔽₈)	1	2	yes	—
22	C₃×C₁₃×AGL(1,𝔽₅)	—	49	1	no	Hall
23	C₅×#3862	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
24	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
25	C₄×#3862	—	5	1	no	Hall
26	C₂×C₁₃×ΓL(1,𝔽₈)	—	35	2	no	—
27	C₇²×D₅	C₂×C₃×C₁₃	1	1	yes	characteristic
28	C₅×C₇×C₁₃	—	3	2	no	—
29	C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	2	yes	complemented
30	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	7	2	no	—
31	C₃×D₅×C₁₃	—	49	1	no	—
32	C₄×C₇×C₁₃	—	5	2	no	—
33	C₄×C₇×C₁₃	—	15	2	no	—
34	C₂×#3862	—	5	1	no	—
35	C₁₃×ΓL(1,𝔽₈)	—	7	2	no	—
36	C₁₃×AGL(1,𝔽₅)	#3862	1	1	yes	characteristic, direct product factor, Hall
37	C₅×C₇²	C₃×C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
38	D₅×ΓL(1,𝔽₈)	—	7	2	no	—
39	C₄×C₇²	—	5	1	no	Hall
40	C₃×C₅×C₁₃	—	49	1	no	Hall
41	C₂×C₇×C₁₃	—	5	2	no	—
42	C₂×C₇×C₁₃	—	15	2	no	—
43	C₃×C₄×C₁₃	—	245	1	no	Hall
44	#3862	C₁₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
45	C₇×AGL(1,𝔽₅)	—	3	2	no	—
46	C₇×AGL(1,𝔽₅)	C₁₃×ΓL(1,𝔽₈)	1	2	yes	complemented
47	D₅×C₁₃	C₂×#3862	1	1	yes	characteristic
48	C₅×ΓL(1,𝔽₈)	—	7	2	no	—
49	C₂×C₇²	—	5	1	no	—
50	C₇×C₁₃	—	3	2	no	—
51	C₇×C₁₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	2	yes	complemented
52	C₄×ΓL(1,𝔽₈)	—	35	2	no	—
53	C₂×C₃×C₁₃	—	245	1	no	—
54	C₇×D₅	—	3	2	no	—
55	C₇×D₅	C₂×C₁₃×ΓL(1,𝔽₈)	1	2	yes	—
56	C₅×C₁₃	C₄×#3862	1	1	yes	characteristic, complemented, Hall
57	C₃×AGL(1,𝔽₅)	—	49	1	no	Hall
58	C₄×C₁₃	—	5	1	no	Hall
59	C₇²	C₃×C₁₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
60	C₂×ΓL(1,𝔽₈)	—	35	2	no	—
61	C₃×C₁₃	—	49	1	no	Hall
62	C₅×C₇	—	3	2	no	—
63	C₅×C₇	C₄×C₁₃×ΓL(1,𝔽₈)	1	2	yes	complemented
64	C₃×D₅	—	49	1	no	—
65	C₄×C₇	—	5	2	no	—
66	C₄×C₇	—	15	2	no	—
67	C₂×C₁₃	—	5	1	no	—
68	ΓL(1,𝔽₈)	—	7	2	no	—
69	AGL(1,𝔽₅)	C₁₃×#3862	1	1	yes	characteristic, direct product factor, Hall
70	C₃×C₅	—	49	1	no	Hall
71	C₂×C₇	—	5	2	no	—
72	C₂×C₇	—	15	2	no	—
73	C₁₃	AGL(1,𝔽₅)×#3862	1	1	yes	characteristic, direct product factor, Sylow
74	C₃×C₄	—	245	1	no	Hall
75	D₅	C₂×C₁₃×#3862	1	1	yes	characteristic
76	C₇	—	3	2	no	—
77	C₇	C₁₃×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	2	yes	complemented
78	C₂×C₃	—	245	1	no	—
79	C₅	C₄×C₁₃×#3862	1	1	yes	characteristic, complemented, Sylow
80	C₄	—	5	1	no	Sylow
81	C₃	—	49	1	no	Sylow
82	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃	AGL(1,𝔽₅)×#3862
commutator, hypocenter	C₅×C₇²	C₃×C₄×C₁₃
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₁₃	C₃×C₄
prime radical	D₅×C₁₃×#3862	C₂
cosocle	C₇²×D₅	C₂×C₃×C₁₃
LCS	G ▷ C₅×C₇²
UCS	C₁ ◁ C₁₃

Complements

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

N	H ≅ G/N	type
2. C₇²×C₁₃×AGL(1,𝔽₅)	C₃	semidirect
3. C₅×C₁₃×#3862	C₄	semidirect
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
9. AGL(1,𝔽₅)×#3862	C₁₃	direct (N × H)
12. C₁₃×#3862	AGL(1,𝔽₅)	direct (N × H)
14. C₇×C₁₃×AGL(1,𝔽₅)	ΓL(1,𝔽₈)	semidirect
19. C₇²×AGL(1,𝔽₅)	C₃×C₁₃	semidirect
23. C₅×#3862	C₄×C₁₃	semidirect
24. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
29. C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	semidirect
36. C₁₃×AGL(1,𝔽₅)	#3862	direct (N × H)
37. C₅×C₇²	C₃×C₄×C₁₃	semidirect
44. #3862	C₁₃×AGL(1,𝔽₅)	direct (N × H)
46. C₇×AGL(1,𝔽₅)	C₁₃×ΓL(1,𝔽₈)	semidirect
51. C₇×C₁₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect
56. C₅×C₁₃	C₄×#3862	semidirect
59. C₇²	C₃×C₁₃×AGL(1,𝔽₅)	semidirect
63. C₅×C₇	C₄×C₁₃×ΓL(1,𝔽₈)	semidirect
69. AGL(1,𝔽₅)	C₁₃×#3862	direct (N × H)
73. C₁₃	AGL(1,𝔽₅)×#3862	direct (N × H)
77. C₇	C₁₃×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect
79. C₅	C₄×C₁₃×#3862	semidirect

Maximal subgroups

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

subgroup	index
D₅×C₁₃×#3862	2
C₇²×C₁₃×AGL(1,𝔽₅)	3
C₄×C₁₃×#3862	5
C₁₃×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	7
AGL(1,𝔽₅)×#3862	13

Subgroups — C13×AGL(1,𝔽5)×#3862 #50303936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×AGL(1,𝔽₅)×#3862 #50303936