Subgroups — #49089892

Order 35868.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31822457	C₂	1	1	yes	maximal, characteristic, complemented
2	#31822456	C₂	1	1	yes	maximal, characteristic, complemented
3	#31822445	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₄₂₇	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×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	#14579421⋊C₂	—	7	1	no	maximal
11	C₇²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	#14579429	—	7	1	no	—
13	#14579428	—	7	1	no	—
14	#14579428	—	7	1	no	—
15	#14579425	—	7	1	no	—
16	#14579421	—	7	1	no	—
17	C₂×#13046197	—	7	1	no	—
18	C₂×D₄₂₇	—	7	1	no	—
19	D₇×D₆₁	—	7	1	no	—
20	#13046197	—	7	1	no	—
21	#13046196	—	7	1	no	—
22	C₂×C₇×C₆₁	—	7	1	no	—
23	D₄₂₇	—	7	1	no	—
24	D₄₂₇	—	7	1	no	—
25	D₄₂₇	—	42	1	no	—
26	D₄₂₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
27	C₇×D₆₁	—	7	1	no	—
28	D₇×C₆₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₂×D₆₁⋊C₃	—	49	1	no	Hall
30	#64422⋊C₂	—	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	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
34	C₂×C₆₁⋊C₃	—	49	1	no	—
35	D₆₁⋊C₃	—	49	1	no	—
36	D₆₁⋊C₃	—	49	1	no	—
37	#64425	—	61	1	no	—
38	#64422	—	61	1	no	—
39	#64422	—	61	1	no	—
40	C₂×D₆₁	—	49	1	no	Hall
41	D₇²	—	61	1	no	Hall
42	C₆₁⋊C₃	—	49	1	no	Hall
43	#3862	—	61	1	no	Hall
44	C₂×C₆₁	—	7	1	no	—
45	D₆₁	—	7	1	no	—
46	D₆₁	—	49	1	no	—
47	C₇⋊D₇	—	61	1	no	—
48	C₇×D₇	—	61	1	no	—
49	C₇×D₇	D₆₁⋊C₃	1	1	yes	characteristic, complemented
50	C₂×AGL(1,𝔽₇)	—	427	1	no	—
51	C₂×AGL(1,𝔽₇)	—	427	1	no	—
52	C₆₁	#64422⋊C₂	1	1	yes	characteristic, complemented, Sylow
53	C₇²	C₂×D₆₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
54	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
55	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
56	AGL(1,𝔽₇)	—	427	1	no	—
57	AGL(1,𝔽₇)	—	427	1	no	—
58	AGL(1,𝔽₇)	—	427	1	no	—
59	AGL(1,𝔽₇)	—	427	1	no	—
60	C₂×D₇	—	427	1	no	—
61	C₂×D₇	—	427	1	no	—
62	ΓL(1,𝔽₈)	—	427	1	no	—
63	ΓL(1,𝔽₈)	—	427	1	no	—
64	C₂×C₇	—	7	1	no	—
65	C₂×C₇	—	427	1	no	—
66	D₇	—	2562	1	no	—
67	D₇	—	61	1	no	—
68	D₇	—	427	1	no	—
69	D₇	—	427	1	no	—
70	D₇	#14579428	1	1	yes	characteristic, complemented
71	C₂²×C₃	—	2989	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	#14579421⋊C₂	1	1	yes	characteristic, complemented
74	C₇	C₂×#14579428	1	1	yes	characteristic, complemented
75	C₂×C₃	—	2989	1	no	—
76	C₂×C₃	—	2989	1	no	—
77	C₂×C₃	—	2989	1	no	—
78	C₂²	—	2989	1	no	Sylow
79	C₃	—	2989	1	no	Sylow
80	C₂	—	7	1	no	—
81	C₂	—	427	1	no	—
82	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. #31822457	C₂	semidirect
2. #31822456	C₂	semidirect
3. #31822445	C₂	semidirect
4. D₇×D₄₂₇	C₃	semidirect
5. C₆₁⋊#3862	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
26. D₄₂₇	AGL(1,𝔽₇)	semidirect
28. D₇×C₆₁	AGL(1,𝔽₇)	semidirect
32. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
33. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
49. C₇×D₇	D₆₁⋊C₃	semidirect
52. C₆₁	#64422⋊C₂	semidirect
53. C₇²	C₂×D₆₁⋊C₃	semidirect
70. D₇	#14579428	semidirect
73. C₇	#14579421⋊C₂	semidirect
74. 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
#31822457	2
#31822456	2
#31822445	2
D₇×D₄₂₇	3
C₂×#14579428	7
#14579421⋊C₂	7
#64422⋊C₂	61