Subgroups — C₂×C₅²×#64425 #28413402

Order 14700.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#3862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×#64425	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅²×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅²×#3862	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×#64425	C₅	1	6	yes	maximal, direct product factor
6	C₅²×C₇⋊D₇	C₂×C₃	1	2	yes	complemented
7	C₂×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₅²×AGL(1,𝔽₇)	—	7	2	no	maximal
9	C₂×C₅×#3862	C₂×C₅	1	6	yes	complemented
10	C₅×#64425	C₂×C₅	1	12	yes	direct product factor
11	C₅²×C₇²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₅²×ΓL(1,𝔽₈)	—	7	2	no	—
13	C₅²×AGL(1,𝔽₇)	—	7	4	no	—
14	C₂×C₅×C₇⋊D₇	C₃×C₅	1	6	yes	complemented
15	C₅×#3862	C₂²×C₅	1	6	yes	complemented
16	C₂×C₅²×D₇	—	7	2	no	—
17	C₂×C₅²×D₇	—	21	2	no	—
18	C₂×#64425	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₅²×ΓL(1,𝔽₈)	—	7	2	no	—
20	C₂×C₅×C₇²	C₂×C₃×C₅	1	6	yes	complemented
21	C₅×C₇⋊D₇	C₂×C₃×C₅	1	12	yes	complemented
22	C₂×C₅×AGL(1,𝔽₇)	—	7	12	no	—
23	C₂×C₅²×C₇	—	3	2	no	—
24	C₂×C₅²×C₇	AGL(1,𝔽₇)	1	2	yes	complemented
25	C₅²×D₇	—	7	4	no	—
26	C₅²×D₇	—	21	4	no	—
27	C₂²×C₃×C₅²	—	49	1	no	Hall
28	C₂×#3862	C₂×C₅²	1	1	yes	characteristic, complemented
29	#64425	C₂×C₅²	1	2	yes	direct product factor
30	C₅×C₇²	C₂²×C₃×C₅	1	6	yes	complemented
31	C₂×C₅×ΓL(1,𝔽₈)	—	7	12	no	—
32	C₅×AGL(1,𝔽₇)	—	7	24	no	—
33	C₂×C₇⋊D₇	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
34	C₅²×C₇	—	3	2	no	—
35	C₅²×C₇	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
36	C₂×C₃×C₅²	—	49	2	no	—
37	C₂×C₃×C₅²	—	49	1	no	—
38	#3862	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
39	C₂×C₅×D₇	—	7	12	no	—
40	C₂×C₅×D₇	—	21	12	no	—
41	C₅×ΓL(1,𝔽₈)	—	7	12	no	—
42	C₂²×C₅²	—	49	1	no	Hall
43	C₂×C₇²	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
44	C₇⋊D₇	C₂×C₃×C₅²	1	2	yes	complemented
45	C₂×AGL(1,𝔽₇)	—	7	2	no	—
46	C₃×C₅²	—	49	1	no	Hall
47	C₂×C₅×C₇	—	3	12	no	—
48	C₂×C₅×C₇	C₅×AGL(1,𝔽₇)	1	12	yes	complemented
49	C₅×D₇	—	7	24	no	—
50	C₅×D₇	—	21	24	no	—
51	C₂²×C₃×C₅	—	49	6	no	—
52	C₂×C₅²	—	49	2	no	—
53	C₂×C₅²	#64425	1	1	yes	characteristic, direct product factor
54	C₇²	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented, Sylow
55	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
56	AGL(1,𝔽₇)	—	7	4	no	—
57	C₅×C₇	—	3	12	no	—
58	C₅×C₇	C₂×C₅×AGL(1,𝔽₇)	1	12	yes	complemented
59	C₂×C₃×C₅	—	49	12	no	—
60	C₂×C₃×C₅	—	49	6	no	—
61	C₂×D₇	—	7	2	no	—
62	C₂×D₇	—	21	2	no	—
63	C₅²	C₂×#64425	1	1	yes	characteristic, direct product factor, Sylow
64	ΓL(1,𝔽₈)	—	7	2	no	—
65	C₂²×C₅	—	49	6	no	—
66	C₃×C₅	—	49	6	no	—
67	C₂×C₇	—	3	2	no	—
68	C₂×C₇	C₅²×AGL(1,𝔽₇)	1	2	yes	complemented
69	D₇	—	7	4	no	—
70	D₇	—	21	4	no	—
71	C₂²×C₃	—	49	1	no	Hall
72	C₂×C₅	—	49	12	no	—
73	C₂×C₅	C₅×#64425	1	6	yes	direct product factor
74	C₇	—	3	2	no	—
75	C₇	C₂×C₅²×AGL(1,𝔽₇)	1	2	yes	complemented
76	C₂×C₃	—	49	2	no	—
77	C₂×C₃	—	49	1	no	—
78	C₅	C₂×C₅×#64425	1	6	yes	direct product factor
79	C₂²	—	49	1	no	Sylow
80	C₃	—	49	1	no	Sylow
81	C₂	—	49	2	no	—
82	C₂	C₅²×#64425	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²	#64425
commutator, cosocle, hypocenter	C₇²	C₂²×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₅²

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#3862	C₂	semidirect
2. C₅²×#64425	C₂	direct (N × H)
3. C₂×C₅²×C₇⋊D₇	C₃	semidirect
4. C₅²×#3862	C₂²	semidirect
5. C₂×C₅×#64425	C₅	direct (N × H)
6. C₅²×C₇⋊D₇	C₂×C₃	semidirect
7. C₂×C₅²×C₇²	C₂×C₃	semidirect
9. C₂×C₅×#3862	C₂×C₅	semidirect
10. C₅×#64425	C₂×C₅	direct (N × H)
11. C₅²×C₇²	C₂²×C₃	semidirect
14. C₂×C₅×C₇⋊D₇	C₃×C₅	semidirect
15. C₅×#3862	C₂²×C₅	semidirect
18. C₂×#64425	C₅²	direct (N × H)
20. C₂×C₅×C₇²	C₂×C₃×C₅	semidirect
21. C₅×C₇⋊D₇	C₂×C₃×C₅	semidirect
24. C₂×C₅²×C₇	AGL(1,𝔽₇)	semidirect
28. C₂×#3862	C₂×C₅²	semidirect
29. #64425	C₂×C₅²	direct (N × H)
30. C₅×C₇²	C₂²×C₃×C₅	semidirect
33. C₂×C₇⋊D₇	C₃×C₅²	semidirect
35. C₅²×C₇	C₂×AGL(1,𝔽₇)	semidirect
38. #3862	C₂²×C₅²	semidirect
43. C₂×C₇²	C₂×C₃×C₅²	semidirect
44. C₇⋊D₇	C₂×C₃×C₅²	semidirect
48. C₂×C₅×C₇	C₅×AGL(1,𝔽₇)	semidirect
53. C₂×C₅²	#64425	direct (N × H)
54. C₇²	C₂²×C₃×C₅²	semidirect
58. C₅×C₇	C₂×C₅×AGL(1,𝔽₇)	semidirect
63. C₅²	C₂×#64425	direct (N × H)
68. C₂×C₇	C₅²×AGL(1,𝔽₇)	semidirect
73. C₂×C₅	C₅×#64425	direct (N × H)
75. C₇	C₂×C₅²×AGL(1,𝔽₇)	semidirect
78. C₅	C₂×C₅×#64425	direct (N × H)
82. C₂	C₅²×#64425	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×C₅²×#3862	2
C₅²×#64425	2
C₂×C₅²×C₇⋊D₇	3
C₂×C₅×#64425	5
C₂×C₅²×AGL(1,𝔽₇)	7

Subgroups — C2×C52×#64425 #28413402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×#64425 #28413402