Subgroups — AGL(1,𝔽₅)×#64424 #72219853

Order 5880.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#64424	C₂	1	1	yes	maximal, characteristic
2	AGL(1,𝔽₅)×#3861	C₂	1	1	yes	maximal, characteristic, complemented
3	#15668765	C₂	1	1	yes	maximal, characteristic, complemented
4	AGL(1,𝔽₅)×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	D₅×#3861	C₂²	1	1	yes	characteristic
6	C₅⋊#64424	C₄	1	1	yes	characteristic, complemented
7	C₅×#64424	C₄	1	1	yes	characteristic, complemented
8	C₄×#64424	—	5	1	no	maximal, Hall
9	D₅×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic
10	C₇⋊(C₇⋊AGL(1,𝔽₅))	C₂×C₃	1	1	yes	characteristic, complemented
11	C₇²×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic, complemented
12	AGL(1,𝔽₅)×AGL(1,𝔽₇)	—	7	8	no	maximal
13	C₅×#3861	C₂×C₄	1	1	yes	characteristic, complemented, Hall
14	C₂×#64424	—	5	1	no	—
15	C₄×#3861	—	5	1	no	—
16	#10596874	—	5	1	no	—
17	C₅⋊(C₇⋊D₇)	C₃×C₄	1	1	yes	characteristic, complemented
18	C₇²×D₅	C₂²×C₃	1	1	yes	characteristic
19	C₅×C₇⋊D₇	C₃×C₄	1	1	yes	characteristic, complemented
20	D₅×AGL(1,𝔽₇)	—	7	8	no	—
21	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	—	7	8	no	—
22	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	7	8	no	—
23	C₄×C₇⋊D₇	—	5	1	no	Hall
24	C₂×#3861	—	5	1	no	—
25	#64424	—	5	1	no	—
26	#64424	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
27	D₇×AGL(1,𝔽₅)	—	7	8	no	—
28	C₅×C₇²	C₂×C₃×C₄	1	1	yes	characteristic, complemented, Hall
29	C₅⋊AGL(1,𝔽₇)	—	7	8	no	—
30	D₅×ΓL(1,𝔽₈)	—	7	8	no	—
31	C₅×AGL(1,𝔽₇)	—	7	8	no	—
32	C₂×C₇⋊D₇	—	5	1	no	—
33	C₄×C₇²	—	5	1	no	—
34	C₇⋊Dic₇	—	5	1	no	—
35	C₄×AGL(1,𝔽₇)	—	35	8	no	—
36	#3861	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
37	D₅×D₇	—	7	8	no	—
38	C₇⋊AGL(1,𝔽₅)	—	7	8	no	—
39	C₇×AGL(1,𝔽₅)	AGL(1,𝔽₇)	1	8	yes	complemented
40	C₂×C₃×AGL(1,𝔽₅)	—	49	1	no	Hall
41	C₅×ΓL(1,𝔽₈)	—	7	8	no	—
42	C₂×C₇²	—	5	1	no	—
43	C₇⋊D₇	—	5	1	no	—
44	C₇⋊D₇	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
45	C₂×AGL(1,𝔽₇)	—	35	8	no	—
46	C₄×ΓL(1,𝔽₈)	—	35	8	no	—
47	ΓL(1,𝔽₈)⋊C₄	—	35	8	no	—
48	D₃₅	—	7	8	no	—
49	C₅×D₇	—	7	8	no	—
50	C₇×D₅	C₂×AGL(1,𝔽₇)	1	8	yes	—
51	C₂×C₃×D₅	—	49	1	no	—
52	C₃×AGL(1,𝔽₅)	—	49	1	no	—
53	C₃×AGL(1,𝔽₅)	—	49	1	no	—
54	C₄×D₇	—	35	8	no	—
55	C₇²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
56	C₂×ΓL(1,𝔽₈)	—	35	8	no	—
57	AGL(1,𝔽₇)	—	7	8	no	—
58	AGL(1,𝔽₇)	—	35	8	no	—
59	C₂×AGL(1,𝔽₅)	—	49	1	no	Hall
60	C₅×C₇	C₄×AGL(1,𝔽₇)	1	8	yes	complemented
61	C₂×C₃×C₅	—	49	1	no	—
62	C₃×D₅	—	49	1	no	—
63	C₃×D₅	—	49	1	no	—
64	C₂×D₇	—	35	8	no	—
65	C₄×C₇	—	5	8	no	—
66	Dic₇	—	35	8	no	—
67	C₂×C₃×C₄	—	245	1	no	Hall
68	ΓL(1,𝔽₈)	—	7	8	no	—
69	C₂×D₅	—	49	1	no	—
70	AGL(1,𝔽₅)	—	49	1	no	—
71	AGL(1,𝔽₅)	#64424	1	1	yes	characteristic, direct product factor
72	C₃×C₅	—	49	1	no	Hall
73	C₂×C₇	—	5	8	no	—
74	D₇	—	7	8	no	—
75	D₇	—	35	8	no	—
76	C₂²×C₃	—	245	1	no	—
77	C₃×C₄	—	245	1	no	—
78	C₃×C₄	—	245	1	no	—
79	C₂×C₅	—	49	1	no	—
80	D₅	—	49	1	no	—
81	D₅	C₂×#64424	1	1	yes	characteristic
82	C₂×C₄	—	245	1	no	Sylow
83	C₇	AGL(1,𝔽₅)×AGL(1,𝔽₇)	1	8	yes	complemented
84	C₂×C₃	—	49	1	no	—
85	C₂×C₃	—	245	1	no	—
86	C₂×C₃	—	245	1	no	—
87	C₅	C₄×#64424	1	1	yes	characteristic, complemented, Sylow
88	C₂²	—	245	1	no	—
89	C₄	—	5	1	no	—
90	C₄	—	245	1	no	—
91	C₃	—	49	1	no	Sylow
92	C₂	—	5	1	no	—
93	C₂	—	49	1	no	—
94	C₂	—	245	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²	C₂×C₃×C₄
prime radical	D₅×#64424	C₂
cosocle	C₇²×D₅	C₂²×C₃
LCS	G ▷ C₅×C₇²
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. AGL(1,𝔽₅)×#3861	C₂	semidirect
3. #15668765	C₂	semidirect
4. AGL(1,𝔽₅)×C₇⋊D₇	C₃	semidirect
6. C₅⋊#64424	C₄	semidirect
7. C₅×#64424	C₄	semidirect
10. C₇⋊(C₇⋊AGL(1,𝔽₅))	C₂×C₃	semidirect
11. C₇²×AGL(1,𝔽₅)	C₂×C₃	semidirect
13. C₅×#3861	C₂×C₄	semidirect
17. C₅⋊(C₇⋊D₇)	C₃×C₄	semidirect
19. C₅×C₇⋊D₇	C₃×C₄	semidirect
26. #64424	AGL(1,𝔽₅)	direct (N × H)
28. C₅×C₇²	C₂×C₃×C₄	semidirect
36. #3861	C₂×AGL(1,𝔽₅)	semidirect
39. C₇×AGL(1,𝔽₅)	AGL(1,𝔽₇)	semidirect
44. C₇⋊D₇	C₃×AGL(1,𝔽₅)	semidirect
55. C₇²	C₂×C₃×AGL(1,𝔽₅)	semidirect
60. C₅×C₇	C₄×AGL(1,𝔽₇)	semidirect
71. AGL(1,𝔽₅)	#64424	direct (N × H)
83. C₇	AGL(1,𝔽₅)×AGL(1,𝔽₇)	semidirect
87. C₅	C₄×#64424	semidirect

Maximal subgroups

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

subgroup	index
D₅×#64424	2
AGL(1,𝔽₅)×#3861	2
#15668765	2
AGL(1,𝔽₅)×C₇⋊D₇	3
C₄×#64424	5
AGL(1,𝔽₅)×AGL(1,𝔽₇)	7

Subgroups — AGL(1,𝔽5)×#64424 #72219853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽₅)×#64424 #72219853