Subgroups — D₆₁×#64422 #49089890

Order 35868.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁×#3862	C₂	1	1	yes	maximal, characteristic, complemented
2	#31822459	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#64422	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₇×D₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₆₁×#3862	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₄₂₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₇×C₆₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇²×D₆₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×ΓL(1,𝔽₈)×D₆₁	—	7	1	no	maximal
10	AGL(1,𝔽₇)×D₆₁	—	7	1	no	maximal
11	C₇²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)⋊D₆₁	—	7	1	no	—
13	AGL(1,𝔽₇)×C₆₁	—	7	1	no	—
14	ΓL(1,𝔽₈)×D₆₁	—	7	1	no	—
15	ΓL(1,𝔽₈)×D₆₁	—	7	1	no	—
16	ΓL(1,𝔽₈)×D₆₁	—	7	1	no	—
17	C₂×ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
18	C₂×C₇×D₆₁	—	7	1	no	—
19	D₇×D₆₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
20	ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
21	ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
22	C₂×C₇×C₆₁	—	7	1	no	—
23	D₄₂₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₇×D₆₁	—	6	1	no	—
25	C₇×D₆₁	—	7	1	no	—
26	C₇×D₆₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
27	C₇×D₆₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	D₇×C₆₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
29	C₂×C₃×D₆₁	—	49	1	no	Hall
30	C₂×#64422	—	61	1	no	maximal, Hall
31	C₇×C₆₁	—	6	1	no	—
32	C₇×C₆₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
33	C₇×C₆₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₂×C₃×C₆₁	—	49	1	no	—
35	C₃×D₆₁	—	49	1	no	—
36	C₃×D₆₁	—	49	1	no	—
37	C₂×#3862	—	61	1	no	—
38	#64422	—	61	1	no	—
39	#64422	D₆₁	1	1	yes	characteristic, direct product factor
40	C₂×D₆₁	—	7	1	no	Hall
41	C₂×C₇×D₇	—	61	1	no	Hall
42	C₃×C₆₁	—	49	1	no	Hall
43	#3862	C₂×D₆₁	1	1	yes	characteristic, complemented, Hall
44	C₂×C₆₁	—	7	1	no	—
45	D₆₁	—	7	1	no	—
46	D₆₁	#64422	1	1	yes	characteristic, direct product factor
47	C₂×C₇²	—	61	1	no	—
48	C₇×D₇	—	61	1	no	—
49	C₇×D₇	C₃×D₆₁	1	1	yes	characteristic, complemented
50	C₂²×ΓL(1,𝔽₈)	—	427	1	no	—
51	C₂×AGL(1,𝔽₇)	—	427	1	no	—
52	C₆₁	C₂×#64422	1	1	yes	characteristic, complemented, Sylow
53	C₇²	C₂×C₃×D₆₁	1	1	yes	characteristic, complemented, Sylow
54	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
55	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
56	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
58	AGL(1,𝔽₇)	—	7	1	no	—
59	AGL(1,𝔽₇)	—	427	1	no	—
60	C₂²×C₇	—	427	1	no	—
61	C₂×D₇	—	61	1	no	—
62	ΓL(1,𝔽₈)	—	7	1	no	—
63	ΓL(1,𝔽₈)	—	7	1	no	—
64	C₂×C₇	—	7	1	no	—
65	C₂×C₇	—	61	1	no	—
66	C₂×C₇	—	61	1	no	—
67	C₂×C₇	—	366	1	no	—
68	C₂×C₇	—	427	1	no	—
69	D₇	—	61	1	no	—
70	D₇	ΓL(1,𝔽₈)×D₆₁	1	1	yes	characteristic, complemented
71	C₂²×C₃	—	2989	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	AGL(1,𝔽₇)×D₆₁	1	1	yes	characteristic, complemented
74	C₇	C₂×ΓL(1,𝔽₈)×D₆₁	1	1	yes	characteristic, complemented
75	C₂×C₃	—	49	1	no	—
76	C₂×C₃	—	2989	1	no	—
77	C₂×C₃	—	2989	1	no	—
78	C₂²	—	427	1	no	Sylow
79	C₃	—	49	1	no	Sylow
80	C₂	—	7	1	no	—
81	C₂	—	61	1	no	—
82	C₂	—	427	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₆₁	C₂²×C₃
LCS	G ▷ C₇²×C₆₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. D₆₁×#3862	C₂	semidirect
2. #31822459	C₂	semidirect
3. C₆₁×#64422	C₂	semidirect
4. C₇×D₇×D₆₁	C₃	semidirect
5. C₆₁×#3862	C₂²	semidirect
6. C₇×D₄₂₇	C₂×C₃	semidirect
7. C₇×D₇×C₆₁	C₂×C₃	semidirect
8. C₇²×D₆₁	C₂×C₃	semidirect
11. C₇²×C₆₁	C₂²×C₃	semidirect
19. D₇×D₆₁	ΓL(1,𝔽₈)	semidirect
23. D₄₂₇	C₂×ΓL(1,𝔽₈)	semidirect
26. C₇×D₆₁	AGL(1,𝔽₇)	semidirect
27. C₇×D₆₁	C₂×ΓL(1,𝔽₈)	semidirect
28. D₇×C₆₁	C₂×ΓL(1,𝔽₈)	semidirect
32. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
33. C₇×C₆₁	C₂²×ΓL(1,𝔽₈)	semidirect
39. #64422	D₆₁	direct (N × H)
43. #3862	C₂×D₆₁	semidirect
46. D₆₁	#64422	direct (N × H)
49. C₇×D₇	C₃×D₆₁	semidirect
52. C₆₁	C₂×#64422	semidirect
53. C₇²	C₂×C₃×D₆₁	semidirect
70. D₇	ΓL(1,𝔽₈)×D₆₁	semidirect
73. C₇	AGL(1,𝔽₇)×D₆₁	semidirect
74. C₇	C₂×ΓL(1,𝔽₈)×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
D₆₁×#3862	2
#31822459	2
C₆₁×#64422	2
C₇×D₇×D₆₁	3
C₂×ΓL(1,𝔽₈)×D₆₁	7
AGL(1,𝔽₇)×D₆₁	7
C₂×#64422	61

Subgroups — D61×#64422 #49089890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#64422 #49089890