Subgroups — C₂×C₄₁×#64422 #37563121

Order 24108.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#3862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#64422	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₇×C₄₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₄₁×#3862	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×D₇×C₄₁	C₂×C₃	1	2	yes	complemented
6	C₂×C₇²×C₄₁	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂²×ΓL(1,𝔽₈)×C₄₁	—	7	1	no	maximal
8	C₂×C₄₁×AGL(1,𝔽₇)	—	7	1	no	maximal
9	C₇²×C₄₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×ΓL(1,𝔽₈)×C₄₁	—	7	2	no	—
11	C₂×ΓL(1,𝔽₈)×C₄₁	—	7	1	no	—
12	C₂×ΓL(1,𝔽₈)×C₄₁	—	7	1	no	—
13	C₄₁×AGL(1,𝔽₇)	—	7	2	no	—
14	C₂²×C₇×C₄₁	—	7	1	no	—
15	C₂×D₇×C₄₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	ΓL(1,𝔽₈)×C₄₁	—	7	1	no	—
17	ΓL(1,𝔽₈)×C₄₁	—	7	1	no	—
18	C₂×#64422	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₇×C₄₁	—	6	1	no	—
20	C₂×C₇×C₄₁	—	7	2	no	—
21	C₂×C₇×C₄₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₂×C₇×C₄₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	D₇×C₄₁	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
24	C₂²×C₃×C₄₁	—	49	1	no	Hall
25	C₂×#3862	C₂×C₄₁	1	1	yes	characteristic, complemented
26	#64422	C₂×C₄₁	1	2	yes	direct product factor
27	C₇×C₄₁	—	6	1	no	—
28	C₇×C₄₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₇×C₄₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄₁	—	49	2	no	—
31	C₂×C₃×C₄₁	—	49	1	no	—
32	C₂×C₇×D₇	C₃×C₄₁	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₄₁	—	7	1	no	Hall
34	#3862	C₂²×C₄₁	1	1	yes	characteristic, complemented, Hall
35	C₃×C₄₁	—	49	1	no	Hall
36	C₂×C₇²	C₂×C₃×C₄₁	1	1	yes	characteristic, complemented
37	C₇×D₇	C₂×C₃×C₄₁	1	2	yes	complemented
38	C₂²×ΓL(1,𝔽₈)	—	7	1	no	—
39	C₂×AGL(1,𝔽₇)	—	7	1	no	—
40	C₂×C₄₁	—	7	2	no	—
41	C₂×C₄₁	#64422	1	1	yes	characteristic, direct product factor
42	C₇²	C₂²×C₃×C₄₁	1	1	yes	characteristic, complemented, Sylow
43	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
44	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
45	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
46	AGL(1,𝔽₇)	—	7	2	no	—
47	C₄₁	C₂×#64422	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₇	—	7	1	no	—
49	C₂×D₇	ΓL(1,𝔽₈)×C₄₁	1	1	yes	characteristic, complemented
50	ΓL(1,𝔽₈)	—	7	1	no	—
51	ΓL(1,𝔽₈)	—	7	1	no	—
52	C₂×C₇	—	6	1	no	—
53	C₂×C₇	—	7	2	no	—
54	C₂×C₇	C₄₁×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
55	C₂×C₇	C₂×ΓL(1,𝔽₈)×C₄₁	1	1	yes	characteristic, complemented
56	D₇	C₂×ΓL(1,𝔽₈)×C₄₁	1	2	yes	complemented
57	C₂²×C₃	—	49	1	no	Hall
58	C₇	—	6	1	no	—
59	C₇	C₂×C₄₁×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
60	C₇	C₂²×ΓL(1,𝔽₈)×C₄₁	1	1	yes	characteristic, complemented
61	C₂×C₃	—	49	2	no	—
62	C₂×C₃	—	49	1	no	—
63	C₂²	—	7	1	no	Sylow
64	C₃	—	49	1	no	Sylow
65	C₂	—	7	2	no	—
66	C₂	C₄₁×#64422	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₄₁	#64422
commutator, cosocle, hypocenter	C₇²	C₂²×C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₁	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₄₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#3862	C₂	semidirect
2. C₄₁×#64422	C₂	direct (N × H)
3. C₂×C₇×D₇×C₄₁	C₃	semidirect
4. C₄₁×#3862	C₂²	semidirect
5. C₇×D₇×C₄₁	C₂×C₃	semidirect
6. C₂×C₇²×C₄₁	C₂×C₃	semidirect
9. C₇²×C₄₁	C₂²×C₃	semidirect
15. C₂×D₇×C₄₁	ΓL(1,𝔽₈)	semidirect
18. C₂×#64422	C₄₁	direct (N × H)
21. C₂×C₇×C₄₁	AGL(1,𝔽₇)	semidirect
22. C₂×C₇×C₄₁	C₂×ΓL(1,𝔽₈)	semidirect
23. D₇×C₄₁	C₂×ΓL(1,𝔽₈)	semidirect
25. C₂×#3862	C₂×C₄₁	semidirect
26. #64422	C₂×C₄₁	direct (N × H)
28. C₇×C₄₁	C₂×AGL(1,𝔽₇)	semidirect
29. C₇×C₄₁	C₂²×ΓL(1,𝔽₈)	semidirect
32. C₂×C₇×D₇	C₃×C₄₁	semidirect
34. #3862	C₂²×C₄₁	semidirect
36. C₂×C₇²	C₂×C₃×C₄₁	semidirect
37. C₇×D₇	C₂×C₃×C₄₁	semidirect
41. C₂×C₄₁	#64422	direct (N × H)
42. C₇²	C₂²×C₃×C₄₁	semidirect
47. C₄₁	C₂×#64422	direct (N × H)
49. C₂×D₇	ΓL(1,𝔽₈)×C₄₁	semidirect
54. C₂×C₇	C₄₁×AGL(1,𝔽₇)	semidirect
55. C₂×C₇	C₂×ΓL(1,𝔽₈)×C₄₁	semidirect
56. D₇	C₂×ΓL(1,𝔽₈)×C₄₁	semidirect
59. C₇	C₂×C₄₁×AGL(1,𝔽₇)	semidirect
60. C₇	C₂²×ΓL(1,𝔽₈)×C₄₁	semidirect
66. C₂	C₄₁×#64422	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×C₄₁×#3862	2
C₄₁×#64422	2
C₂×C₇×D₇×C₄₁	3
C₂²×ΓL(1,𝔽₈)×C₄₁	7
C₂×C₄₁×AGL(1,𝔽₇)	7
C₂×#64422	41

Subgroups — C2×C41×#64422 #37563121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#64422 #37563121