Subgroups — C₂×AGL(1,𝔽₅)×#3861 #72243263

Order 5880.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×#3861	C₂	1	1	yes	maximal, characteristic
2	AGL(1,𝔽₅)×#3861	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇²×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×C₅×#3861	C₄	1	1	yes	characteristic, complemented
5	D₅×#3861	C₄	1	1	yes	characteristic, complemented
6	D₅×#3861	C₂²	1	1	yes	characteristic
7	C₂×C₄×#3861	—	5	1	no	maximal, Hall
8	C₂×C₇²×D₅	C₂×C₃	1	1	yes	characteristic
9	C₇²×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	complemented
10	C₂×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	7	8	no	maximal
11	C₅×#3861	C₂×C₄	1	1	yes	characteristic, complemented, Hall
12	C₂²×#3861	—	5	1	no	—
13	C₄×#3861	—	5	2	no	—
14	C₂×C₅×C₇²	C₃×C₄	1	1	yes	characteristic, complemented
15	C₇²×D₅	C₃×C₄	1	1	yes	characteristic, complemented
16	C₇²×D₅	C₂²×C₃	1	1	yes	characteristic
17	C₂×D₅×ΓL(1,𝔽₈)	—	7	8	no	—
18	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	7	16	no	—
19	C₂×C₄×C₇²	—	5	1	no	Hall
20	C₂×#3861	—	5	1	no	—
21	C₂×#3861	—	5	1	no	—
22	C₂×#3861	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
23	C₂×C₇×AGL(1,𝔽₅)	ΓL(1,𝔽₈)	1	8	yes	complemented
24	C₅×C₇²	C₂×C₃×C₄	1	1	yes	characteristic, complemented, Hall
25	C₂×C₅×ΓL(1,𝔽₈)	—	7	8	no	—
26	D₅×ΓL(1,𝔽₈)	—	7	8	no	—
27	D₅×ΓL(1,𝔽₈)	—	7	8	no	—
28	C₂²×C₇²	—	5	1	no	—
29	C₄×C₇²	—	5	2	no	—
30	C₂×C₄×ΓL(1,𝔽₈)	—	35	8	no	—
31	#3861	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₇×D₅	C₂×ΓL(1,𝔽₈)	1	8	yes	—
33	C₇×AGL(1,𝔽₅)	C₂×ΓL(1,𝔽₈)	1	16	yes	complemented
34	C₂×C₃×AGL(1,𝔽₅)	—	49	1	no	Hall
35	C₅×ΓL(1,𝔽₈)	—	7	8	no	—
36	C₂×C₇²	—	5	1	no	—
37	C₂×C₇²	—	5	1	no	—
38	C₂×C₇²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₂²×ΓL(1,𝔽₈)	—	35	8	no	—
40	C₄×ΓL(1,𝔽₈)	—	35	16	no	—
41	C₂×C₅×C₇	C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
42	C₇×D₅	C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
43	C₇×D₅	C₂²×ΓL(1,𝔽₈)	1	8	yes	—
44	C₂×C₃×D₅	—	49	1	no	—
45	C₃×AGL(1,𝔽₅)	—	49	2	no	—
46	C₂×C₄×C₇	—	5	8	no	—
47	C₇²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
48	C₂×ΓL(1,𝔽₈)	—	7	8	no	—
49	C₂×ΓL(1,𝔽₈)	—	35	8	no	—
50	C₂×ΓL(1,𝔽₈)	—	35	8	no	—
51	C₂×AGL(1,𝔽₅)	#3861	1	1	yes	characteristic, direct product factor, Hall
52	C₅×C₇	C₂×C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
53	C₂×C₃×C₅	—	49	1	no	—
54	C₃×D₅	—	49	1	no	—
55	C₃×D₅	—	49	1	no	—
56	C₂²×C₇	—	5	8	no	—
57	C₄×C₇	—	5	16	no	—
58	C₂×C₃×C₄	—	245	1	no	Hall
59	ΓL(1,𝔽₈)	—	7	8	no	—
60	C₂×D₅	C₂×#3861	1	1	yes	characteristic
61	AGL(1,𝔽₅)	C₂×#3861	1	2	yes	direct product factor
62	C₃×C₅	—	49	1	no	Hall
63	C₂×C₇	—	5	8	no	—
64	C₂×C₇	—	5	8	no	—
65	C₂×C₇	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	8	yes	complemented
66	C₂²×C₃	—	245	1	no	—
67	C₃×C₄	—	245	2	no	—
68	C₂×C₅	C₄×#3861	1	1	yes	characteristic, complemented
69	D₅	C₄×#3861	1	1	yes	characteristic, complemented
70	D₅	C₂²×#3861	1	1	yes	characteristic
71	C₂×C₄	—	5	1	no	Sylow
72	C₇	C₂×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	8	yes	complemented
73	C₂×C₃	—	49	1	no	—
74	C₂×C₃	—	245	1	no	—
75	C₂×C₃	—	245	1	no	—
76	C₅	C₂×C₄×#3861	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	5	1	no	—
78	C₄	—	5	2	no	—
79	C₃	—	49	1	no	Sylow
80	C₂	—	5	1	no	—
81	C₂	—	5	1	no	—
82	C₂	AGL(1,𝔽₅)×#3861	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	AGL(1,𝔽₅)×#3861
commutator, hypocenter	C₅×C₇²	C₂×C₃×C₄
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇²	C₃×C₄
prime radical	C₂×D₅×#3861	C₂
cosocle	C₇²×D₅	C₂²×C₃
LCS	G ▷ C₅×C₇²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. AGL(1,𝔽₅)×#3861	C₂	direct (N × H)
3. C₂×C₇²×AGL(1,𝔽₅)	C₃	semidirect
4. C₂×C₅×#3861	C₄	semidirect
5. D₅×#3861	C₄	semidirect
9. C₇²×AGL(1,𝔽₅)	C₂×C₃	semidirect
11. C₅×#3861	C₂×C₄	semidirect
14. C₂×C₅×C₇²	C₃×C₄	semidirect
15. C₇²×D₅	C₃×C₄	semidirect
22. C₂×#3861	AGL(1,𝔽₅)	direct (N × H)
23. C₂×C₇×AGL(1,𝔽₅)	ΓL(1,𝔽₈)	semidirect
24. C₅×C₇²	C₂×C₃×C₄	semidirect
31. #3861	C₂×AGL(1,𝔽₅)	direct (N × H)
33. C₇×AGL(1,𝔽₅)	C₂×ΓL(1,𝔽₈)	semidirect
38. C₂×C₇²	C₃×AGL(1,𝔽₅)	semidirect
41. C₂×C₅×C₇	C₄×ΓL(1,𝔽₈)	semidirect
42. C₇×D₅	C₄×ΓL(1,𝔽₈)	semidirect
47. C₇²	C₂×C₃×AGL(1,𝔽₅)	semidirect
51. C₂×AGL(1,𝔽₅)	#3861	direct (N × H)
52. C₅×C₇	C₂×C₄×ΓL(1,𝔽₈)	semidirect
61. AGL(1,𝔽₅)	C₂×#3861	direct (N × H)
65. C₂×C₇	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect
68. C₂×C₅	C₄×#3861	semidirect
69. D₅	C₄×#3861	semidirect
72. C₇	C₂×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect
76. C₅	C₂×C₄×#3861	semidirect
82. C₂	AGL(1,𝔽₅)×#3861	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₅×#3861	2
AGL(1,𝔽₅)×#3861	2
C₂×C₇²×AGL(1,𝔽₅)	3
C₂×C₄×#3861	5
C₂×AGL(1,𝔽₅)×ΓL(1,𝔽₈)	7

Subgroups — C2×AGL(1,𝔽5)×#3861 #72243263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×AGL(1,𝔽₅)×#3861 #72243263