Subgroups — #49089897

Order 35868.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31822459	C₂	1	2	yes	maximal, complemented
2	C₆₁×#64425	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₇⋊D₇)⋊D₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₆₁×#3862	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₆₁×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇×D₄₂₇	C₂×C₃	1	2	yes	complemented
7	AGL(1,𝔽₇)×D₆₁	—	7	2	no	maximal
8	C₇²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	ΓL(1,𝔽₈)⋊D₆₁	—	7	2	no	—
10	AGL(1,𝔽₇)×C₆₁	—	7	2	no	—
11	ΓL(1,𝔽₈)×D₆₁	—	7	2	no	—
12	D₇×D₆₁	—	7	2	no	—
13	ΓL(1,𝔽₈)×C₆₁	—	7	2	no	—
14	D₄₂₇	AGL(1,𝔽₇)	1	2	yes	complemented
15	C₇×D₆₁	—	7	2	no	—
16	D₇×C₆₁	—	7	2	no	—
17	D₇×C₆₁	—	42	1	no	—
18	C₂×C₃×D₆₁	—	49	1	no	Hall
19	#64422⋊C₂	—	61	1	no	maximal, Hall
20	C₇×C₆₁	—	6	1	no	—
21	C₇×C₆₁	C₂×AGL(1,𝔽₇)	1	2	yes	complemented
22	C₂×C₃×C₆₁	—	49	1	no	—
23	C₃×D₆₁	—	49	2	no	—
24	#64425	D₆₁	1	1	yes	characteristic, complemented
25	#64422	—	61	2	no	—
26	C₂×D₆₁	—	49	1	no	Hall
27	D₇²	—	61	1	no	Hall
28	C₃×C₆₁	—	49	1	no	Hall
29	#3862	C₂×D₆₁	1	1	yes	characteristic, complemented, Hall
30	C₂×C₆₁	—	49	1	no	—
31	D₆₁	—	7	2	no	—
32	C₇⋊D₇	C₃×D₆₁	1	1	yes	characteristic, complemented
33	C₇×D₇	—	61	2	no	—
34	C₂×AGL(1,𝔽₇)	—	427	2	no	—
35	C₆₁	#64422⋊C₂	1	1	yes	characteristic, complemented, Sylow
36	C₇²	C₂×C₃×D₆₁	1	1	yes	characteristic, complemented, Sylow
37	C₂×ΓL(1,𝔽₈)	—	427	2	no	—
38	AGL(1,𝔽₇)	—	7	2	no	—
39	AGL(1,𝔽₇)	—	427	2	no	—
40	C₂×D₇	—	427	2	no	—
41	ΓL(1,𝔽₈)	—	7	2	no	—
42	C₂×C₇	—	427	2	no	—
43	D₇	—	7	2	no	—
44	D₇	—	42	1	no	—
45	D₇	—	61	2	no	—
46	C₂²×C₃	—	2989	1	no	Hall
47	C₇	—	6	1	no	—
48	C₇	AGL(1,𝔽₇)×D₆₁	1	2	yes	complemented
49	C₂×C₃	—	49	1	no	—
50	C₂×C₃	—	2989	2	no	—
51	C₂²	—	2989	1	no	Sylow
52	C₃	—	49	1	no	Sylow
53	C₂	—	49	1	no	—
54	C₂	—	427	2	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. #31822459	C₂	semidirect
2. C₆₁×#64425	C₂	semidirect
3. (C₇⋊D₇)⋊D₆₁	C₃	semidirect
4. C₆₁×#3862	C₂²	semidirect
5. C₆₁×C₇⋊D₇	C₂×C₃	semidirect
6. C₇×D₄₂₇	C₂×C₃	semidirect
8. C₇²×C₆₁	C₂²×C₃	semidirect
14. D₄₂₇	AGL(1,𝔽₇)	semidirect
21. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
24. #64425	D₆₁	semidirect
29. #3862	C₂×D₆₁	semidirect
32. C₇⋊D₇	C₃×D₆₁	semidirect
35. C₆₁	#64422⋊C₂	semidirect
36. C₇²	C₂×C₃×D₆₁	semidirect
48. C₇	AGL(1,𝔽₇)×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#31822459	2
C₆₁×#64425	2
(C₇⋊D₇)⋊D₆₁	3
AGL(1,𝔽₇)×D₆₁	7
#64422⋊C₂	61