Subgroups — #49089889

Order 35868.

52 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
2	C₇×C₇⋊#6923	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₆₁×#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇²×D₆₁	C₂×C₃	1	1	yes	characteristic
5	ΓL(1,𝔽₈)×#6923	—	7	1	no	maximal
6	ΓL(1,𝔽₈)⋊#6923	—	7	1	no	maximal
7	C₇²×C₆₁	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	ΓL(1,𝔽₈)×D₆₁	—	7	1	no	—
9	ΓL(1,𝔽₈)×D₆₁	—	7	1	no	—
10	C₇⋊#6923	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
11	C₇×#6923	—	7	1	no	—
12	ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
13	ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
14	C₇×D₆₁	—	6	1	no	—
15	C₇×D₆₁	AGL(1,𝔽₇)	1	1	yes	characteristic
16	C₇×D₆₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	C₃×#6923	—	49	1	no	Hall
18	#10596872	—	61	1	no	maximal, Hall
19	C₇×C₆₁	—	6	1	no	—
20	C₇×C₆₁	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
21	C₇×C₆₁	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₃×D₆₁	—	49	1	no	—
23	C₂×#3862	—	61	1	no	—
24	#6923	—	7	1	no	Hall
25	C₇×Dic₇	—	61	1	no	Hall
26	C₃×C₆₁	—	49	1	no	Hall
27	#3862	#6923	1	1	yes	characteristic, complemented, Hall
28	D₆₁	#64422	1	1	yes	characteristic
29	C₂×C₇²	—	61	1	no	—
30	C₄×ΓL(1,𝔽₈)	—	427	1	no	—
31	ΓL(1,𝔽₈)⋊C₄	—	427	1	no	—
32	C₆₁	#10596872	1	1	yes	characteristic, complemented, Sylow
33	C₇²	C₃×#6923	1	1	yes	characteristic, complemented, Sylow
34	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
35	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
36	C₄×C₇	—	427	1	no	—
37	Dic₇	—	61	1	no	—
38	ΓL(1,𝔽₈)	—	7	1	no	—
39	ΓL(1,𝔽₈)	—	7	1	no	—
40	C₂×C₇	—	61	1	no	—
41	C₂×C₇	—	61	1	no	—
42	C₂×C₇	—	366	1	no	—
43	C₃×C₄	—	2989	1	no	Hall
44	C₇	—	6	1	no	—
45	C₇	ΓL(1,𝔽₈)⋊#6923	1	1	yes	characteristic, complemented
46	C₇	ΓL(1,𝔽₈)×#6923	1	1	yes	characteristic, complemented
47	C₂×C₃	—	2989	1	no	—
48	C₄	—	427	1	no	Sylow
49	C₃	—	49	1	no	Sylow
50	C₂	—	61	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²×C₆₁	C₃×C₄
prime radical	D₆₁×#3862	C₂
cosocle	C₇²×D₆₁	C₂×C₃
LCS	G ▷ C₇²×C₆₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×C₇⋊#6923	C₃	semidirect
3. C₆₁×#3862	C₄	semidirect
7. C₇²×C₆₁	C₃×C₄	semidirect
10. C₇⋊#6923	ΓL(1,𝔽₈)	semidirect
20. C₇×C₆₁	ΓL(1,𝔽₈)⋊C₄	semidirect
21. C₇×C₆₁	C₄×ΓL(1,𝔽₈)	semidirect
27. #3862	#6923	semidirect
32. C₆₁	#10596872	semidirect
33. C₇²	C₃×#6923	semidirect
45. C₇	ΓL(1,𝔽₈)⋊#6923	semidirect
46. C₇	ΓL(1,𝔽₈)×#6923	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
C₇×C₇⋊#6923	3
ΓL(1,𝔽₈)×#6923	7
ΓL(1,𝔽₈)⋊#6923	7
#10596872	61