Subgroups — C₂×#16838470 #21439727

Order 7644.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#63270	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838470	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₁₃×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×#63270	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
6	C₁₃×C₇⋊D₇	C₂×C₃	1	2	yes	complemented
7	C₂×#10587720	—	7	1	no	maximal
8	C₂×D₇×C₁₃⋊C₃	—	7	1	no	maximal
9	C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×ΓL(1,𝔽₈)⋊D₇	—	13	1	no	maximal, Hall
11	C₂×#63270	D₇	1	1	yes	characteristic, complemented
12	C₂×C₇×C₁₃⋊C₃	—	7	1	no	—
13	#10587720	—	7	2	no	—
14	D₇×C₁₃⋊C₃	—	7	2	no	—
15	C₂×C₁₃×D₇	—	7	1	no	—
16	C₂×C₁₃×D₇	—	7	1	no	—
17	C₂×C₁₃×D₇	—	21	2	no	—
18	C₂×C₇×ΓL(1,𝔽₈)	—	13	1	no	—
19	ΓL(1,𝔽₈)⋊D₇	—	13	2	no	—
20	#63270	C₂×D₇	1	1	yes	characteristic, complemented
21	C₇×C₁₃⋊C₃	—	7	1	no	—
22	C₂×C₇⋊D₇	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
23	C₂×C₇×C₁₃	—	3	2	no	—
24	C₂×C₇×C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
25	C₂×C₇×C₁₃	C₃×D₇	1	1	yes	characteristic, complemented
26	C₁₃×D₇	—	7	2	no	—
27	C₁₃×D₇	—	7	2	no	—
28	C₁₃×D₇	—	21	4	no	—
29	C₂²×C₁₃⋊C₃	—	49	1	no	Hall
30	C₇×ΓL(1,𝔽₈)	—	13	1	no	Hall
31	C₂×C₇²	C₂×C₁₃⋊C₃	1	1	yes	characteristic, complemented
32	C₇⋊D₇	C₂×C₁₃⋊C₃	1	2	yes	complemented
33	C₇×C₁₃	—	3	2	no	—
34	C₇×C₁₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
35	C₇×C₁₃	C₂×C₃×D₇	1	1	yes	characteristic, complemented
36	C₂×C₃×D₇	—	91	1	no	—
37	C₂×AGL(1,𝔽₇)	—	91	1	no	—
38	C₂×C₁₃⋊C₃	—	7	1	no	—
39	C₂×C₁₃⋊C₃	—	49	2	no	—
40	C₂²×C₁₃	—	49	1	no	Hall
41	C₇²	C₂²×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃×C₇	—	91	1	no	—
43	C₃×D₇	—	91	2	no	—
44	C₂×ΓL(1,𝔽₈)	—	13	1	no	—
45	AGL(1,𝔽₇)	—	91	2	no	—
46	C₁₃⋊C₃	—	7	1	no	Hall
47	C₂×D₇	—	7	1	no	—
48	C₂×D₇	—	7	1	no	—
49	C₂×D₇	—	21	2	no	—
50	C₂×C₁₃	—	49	2	no	—
51	C₂×C₁₃	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented
52	C₃×C₇	—	91	1	no	—
53	ΓL(1,𝔽₈)	—	13	1	no	—
54	C₂×C₇	—	3	2	no	—
55	C₂×C₇	D₇×C₁₃⋊C₃	1	1	yes	characteristic, complemented
56	C₂×C₇	#10587720	1	1	yes	characteristic, complemented
57	D₇	—	7	2	no	—
58	D₇	—	7	2	no	—
59	D₇	—	21	4	no	—
60	C₁₃	C₂×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₃	—	637	1	no	Hall
62	C₇	—	3	2	no	—
63	C₇	C₂×D₇×C₁₃⋊C₃	1	1	yes	characteristic, complemented
64	C₇	C₂×#10587720	1	1	yes	characteristic, complemented
65	C₂×C₃	—	91	1	no	—
66	C₂×C₃	—	637	2	no	—
67	C₂²	—	49	1	no	Sylow
68	C₃	—	91	1	no	Sylow
69	C₂	—	49	2	no	—
70	C₂	#16838470	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#16838470
commutator, hypocenter	C₇²×C₁₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#63270	C₂	semidirect
2. #16838470	C₂	direct (N × H)
3. C₂×C₁₃×C₇⋊D₇	C₃	semidirect
4. C₇×#63270	C₂²	semidirect
5. C₂×C₇²×C₁₃	C₂×C₃	semidirect
6. C₁₃×C₇⋊D₇	C₂×C₃	semidirect
9. C₇²×C₁₃	C₂²×C₃	semidirect
11. C₂×#63270	D₇	semidirect
20. #63270	C₂×D₇	semidirect
22. C₂×C₇⋊D₇	C₁₃⋊C₃	semidirect
24. C₂×C₇×C₁₃	AGL(1,𝔽₇)	semidirect
25. C₂×C₇×C₁₃	C₃×D₇	semidirect
31. C₂×C₇²	C₂×C₁₃⋊C₃	semidirect
32. C₇⋊D₇	C₂×C₁₃⋊C₃	semidirect
34. C₇×C₁₃	C₂×AGL(1,𝔽₇)	semidirect
35. C₇×C₁₃	C₂×C₃×D₇	semidirect
41. C₇²	C₂²×C₁₃⋊C₃	semidirect
51. C₂×C₁₃	ΓL(1,𝔽₈)⋊D₇	semidirect
55. C₂×C₇	D₇×C₁₃⋊C₃	semidirect
56. C₂×C₇	#10587720	semidirect
60. C₁₃	C₂×ΓL(1,𝔽₈)⋊D₇	semidirect
63. C₇	C₂×D₇×C₁₃⋊C₃	semidirect
64. C₇	C₂×#10587720	semidirect
70. C₂	#16838470	direct (N × H)

Subgroups — C2×#16838470 #21439727

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16838470 #21439727