Subgroups — D₆₁×#64423 #49089872

Order 35868.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁×#3861	C₂	1	1	yes	maximal, characteristic, complemented
2	#31822453	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#64423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₇×D₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₆₁×#3861	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	ΓL(1,𝔽₈)×D₆₁	—	14	3	no	—
18	C₂×ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
19	C₂×C₇×D₆₁	—	7	1	no	—
20	D₇×D₆₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
22	ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
23	ΓL(1,𝔽₈)×C₆₁	—	14	3	no	—
24	C₂×C₇×C₆₁	—	7	1	no	—
25	D₄₂₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
26	C₇×D₆₁	—	2	3	no	—
27	C₇×D₆₁	—	7	1	no	—
28	C₇×D₆₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₇×D₆₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	D₇×C₆₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₂×C₃×D₆₁	—	49	1	no	Hall
32	C₂×#64423	—	61	1	no	maximal, Hall
33	C₇×C₆₁	—	2	3	no	—
34	C₇×C₆₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
35	C₇×C₆₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
36	C₂×C₃×C₆₁	—	49	1	no	—
37	C₃×D₆₁	—	49	1	no	—
38	C₃×D₆₁	—	49	1	no	—
39	C₂×#3861	—	61	1	no	—
40	#64423	—	61	1	no	—
41	#64423	D₆₁	1	1	yes	characteristic, direct product factor
42	C₂×D₆₁	—	7	1	no	Hall
43	C₂×C₇×D₇	—	61	1	no	Hall
44	C₃×C₆₁	—	49	1	no	Hall
45	#3861	C₂×D₆₁	1	1	yes	characteristic, complemented, Hall
46	C₂×C₆₁	—	7	1	no	—
47	D₆₁	—	7	1	no	—
48	D₆₁	#64423	1	1	yes	characteristic, direct product factor
49	C₂×C₇²	—	61	1	no	—
50	C₇×D₇	—	61	1	no	—
51	C₇×D₇	C₃×D₆₁	1	1	yes	characteristic, complemented
52	C₂²×ΓL(1,𝔽₈)	—	427	1	no	—
53	C₂×AGL(1,𝔽₇)	—	427	1	no	—
54	C₆₁	C₂×#64423	1	1	yes	characteristic, complemented, Sylow
55	C₇²	C₂×C₃×D₆₁	1	1	yes	characteristic, complemented, Sylow
56	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	854	3	no	—
58	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
59	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
60	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
61	AGL(1,𝔽₇)	—	7	1	no	—
62	AGL(1,𝔽₇)	—	427	1	no	—
63	C₂²×C₇	—	427	1	no	—
64	C₂×D₇	—	61	1	no	—
65	ΓL(1,𝔽₈)	—	7	1	no	—
66	ΓL(1,𝔽₈)	—	7	1	no	—
67	ΓL(1,𝔽₈)	—	14	3	no	—
68	C₂×C₇	—	7	1	no	—
69	C₂×C₇	—	61	1	no	—
70	C₂×C₇	—	61	1	no	—
71	C₂×C₇	—	122	3	no	—
72	C₂×C₇	—	427	1	no	—
73	D₇	—	61	1	no	—
74	D₇	ΓL(1,𝔽₈)×D₆₁	1	1	yes	characteristic, complemented
75	C₂²×C₃	—	2989	1	no	Hall
76	C₇	—	2	3	no	—
77	C₇	AGL(1,𝔽₇)×D₆₁	1	1	yes	characteristic, complemented
78	C₇	C₂×ΓL(1,𝔽₈)×D₆₁	1	1	yes	characteristic, complemented
79	C₂×C₃	—	49	1	no	—
80	C₂×C₃	—	2989	1	no	—
81	C₂×C₃	—	2989	1	no	—
82	C₂²	—	427	1	no	Sylow
83	C₃	—	49	1	no	Sylow
84	C₂	—	7	1	no	—
85	C₂	—	61	1	no	—
86	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×#3861	C₂	semidirect
2. #31822453	C₂	semidirect
3. C₆₁×#64423	C₂	semidirect
4. C₇×D₇×D₆₁	C₃	semidirect
5. C₆₁×#3861	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
20. D₇×D₆₁	ΓL(1,𝔽₈)	semidirect
25. D₄₂₇	C₂×ΓL(1,𝔽₈)	semidirect
28. C₇×D₆₁	AGL(1,𝔽₇)	semidirect
29. C₇×D₆₁	C₂×ΓL(1,𝔽₈)	semidirect
30. D₇×C₆₁	C₂×ΓL(1,𝔽₈)	semidirect
34. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
35. C₇×C₆₁	C₂²×ΓL(1,𝔽₈)	semidirect
41. #64423	D₆₁	direct (N × H)
45. #3861	C₂×D₆₁	semidirect
48. D₆₁	#64423	direct (N × H)
51. C₇×D₇	C₃×D₆₁	semidirect
54. C₆₁	C₂×#64423	semidirect
55. C₇²	C₂×C₃×D₆₁	semidirect
74. D₇	ΓL(1,𝔽₈)×D₆₁	semidirect
77. C₇	AGL(1,𝔽₇)×D₆₁	semidirect
78. 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×#3861	2
#31822453	2
C₆₁×#64423	2
C₇×D₇×D₆₁	3
C₂×ΓL(1,𝔽₈)×D₆₁	7
AGL(1,𝔽₇)×D₆₁	7
C₂×#64423	61

Subgroups — D61×#64423 #49089872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#64423 #49089872