Subgroups — #49089878

Order 35868.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31822452	C₂	1	1	yes	maximal, characteristic, complemented
2	#31822450	C₂	1	1	yes	maximal, characteristic, complemented
3	#31822440	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₄₂₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	#22582225	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇⋊D₄₂₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₄₂₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₇×C₆₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×#14579428	—	7	1	no	maximal
10	#14579420⋊C₂	—	7	1	no	maximal
11	C₇²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	#14579428	—	7	1	no	—
13	#14579428	—	7	1	no	—
14	#14579428	—	7	1	no	—
15	#14579428	—	14	3	no	—
16	#14579424	—	7	1	no	—
17	#14579420	—	7	1	no	—
18	C₂×#13046197	—	7	1	no	—
19	C₂×D₄₂₇	—	7	1	no	—
20	D₇×D₆₁	—	7	1	no	—
21	#13046197	—	7	1	no	—
22	#13046197	—	7	1	no	—
23	#13046197	—	14	3	no	—
24	C₂×C₇×C₆₁	—	7	1	no	—
25	D₄₂₇	—	7	1	no	—
26	D₄₂₇	—	7	1	no	—
27	D₄₂₇	—	14	3	no	—
28	D₄₂₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₇×D₆₁	—	7	1	no	—
30	D₇×C₆₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
31	C₂×D₆₁⋊C₃	—	49	1	no	Hall
32	#64423⋊C₂	—	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	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
36	C₂×C₆₁⋊C₃	—	49	1	no	—
37	D₆₁⋊C₃	—	49	1	no	—
38	D₆₁⋊C₃	—	49	1	no	—
39	#64424	—	61	1	no	—
40	#64423	—	61	1	no	—
41	#64423	—	61	1	no	—
42	C₂×D₆₁	—	49	1	no	Hall
43	D₇²	—	61	1	no	Hall
44	C₆₁⋊C₃	—	49	1	no	Hall
45	#3861	—	61	1	no	Hall
46	C₂×C₆₁	—	7	1	no	—
47	D₆₁	—	7	1	no	—
48	D₆₁	—	49	1	no	—
49	C₇⋊D₇	—	61	1	no	—
50	C₇×D₇	—	61	1	no	—
51	C₇×D₇	D₆₁⋊C₃	1	1	yes	characteristic, complemented
52	C₂×AGL(1,𝔽₇)	—	427	1	no	—
53	C₂×AGL(1,𝔽₇)	—	427	1	no	—
54	C₆₁	#64423⋊C₂	1	1	yes	characteristic, complemented, Sylow
55	C₇²	C₂×D₆₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
58	AGL(1,𝔽₇)	—	854	3	no	—
59	AGL(1,𝔽₇)	—	427	1	no	—
60	AGL(1,𝔽₇)	—	427	1	no	—
61	AGL(1,𝔽₇)	—	427	1	no	—
62	AGL(1,𝔽₇)	—	427	1	no	—
63	C₂×D₇	—	427	1	no	—
64	C₂×D₇	—	427	1	no	—
65	ΓL(1,𝔽₈)	—	854	3	no	—
66	ΓL(1,𝔽₈)	—	427	1	no	—
67	ΓL(1,𝔽₈)	—	427	1	no	—
68	C₂×C₇	—	7	1	no	—
69	C₂×C₇	—	427	1	no	—
70	D₇	—	61	1	no	—
71	D₇	—	854	3	no	—
72	D₇	—	427	1	no	—
73	D₇	—	427	1	no	—
74	D₇	#14579428	1	1	yes	characteristic, complemented
75	C₂²×C₃	—	2989	1	no	Hall
76	C₇	—	2	3	no	—
77	C₇	#14579420⋊C₂	1	1	yes	characteristic, complemented
78	C₇	C₂×#14579428	1	1	yes	characteristic, complemented
79	C₂×C₃	—	2989	1	no	—
80	C₂×C₃	—	2989	1	no	—
81	C₂×C₃	—	2989	1	no	—
82	C₂²	—	2989	1	no	Sylow
83	C₃	—	2989	1	no	Sylow
84	C₂	—	7	1	no	—
85	C₂	—	427	1	no	—
86	C₂	—	2989	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. #31822452	C₂	semidirect
2. #31822450	C₂	semidirect
3. #31822440	C₂	semidirect
4. D₇×D₄₂₇	C₃	semidirect
5. #22582225	C₂²	semidirect
6. C₇⋊D₄₂₇	C₂×C₃	semidirect
7. C₇×D₄₂₇	C₂×C₃	semidirect
8. C₇×D₇×C₆₁	C₂×C₃	semidirect
11. C₇²×C₆₁	C₂²×C₃	semidirect
28. D₄₂₇	AGL(1,𝔽₇)	semidirect
30. D₇×C₆₁	AGL(1,𝔽₇)	semidirect
34. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
35. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
51. C₇×D₇	D₆₁⋊C₃	semidirect
54. C₆₁	#64423⋊C₂	semidirect
55. C₇²	C₂×D₆₁⋊C₃	semidirect
74. D₇	#14579428	semidirect
77. C₇	#14579420⋊C₂	semidirect
78. C₇	C₂×#14579428	semidirect

Maximal subgroups

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

subgroup	index
#31822452	2
#31822450	2
#31822440	2
D₇×D₄₂₇	3
C₂×#14579428	7
#14579420⋊C₂	7
#64423⋊C₂	61