Subgroups — C₇×C₆₁×ΓL(1,𝔽₈)⋊C₄ #49089727

Order 35868.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×ΓL(1,𝔽₈)×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₇×Dic₇×C₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×ΓL(1,𝔽₈)×C₆₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇²×C₆₁	C₂×C₃	1	1	yes	characteristic
5	C₆₁×ΓL(1,𝔽₈)⋊C₄	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₃×C₄×C₇×C₆₁	—	7	1	no	maximal
7	C₇²×C₆₁	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₂×ΓL(1,𝔽₈)×C₆₁	C₂×C₇	1	1	yes	characteristic
9	C₂×C₃×C₇×C₆₁	—	7	1	no	—
10	C₄×C₇×C₆₁	—	7	1	no	—
11	Dic₇×C₆₁	C₃×C₇	1	1	yes	characteristic, complemented
12	C₃×C₇×C₆₁	—	7	1	no	—
13	ΓL(1,𝔽₈)×C₆₁	C₄×C₇	1	1	yes	characteristic, complemented
14	C₂×C₇×C₆₁	—	6	1	no	—
15	C₂×C₇×C₆₁	AGL(1,𝔽₇)	1	1	yes	characteristic
16	C₂×C₇×C₆₁	C₂×C₃×C₇	1	1	yes	characteristic
17	C₃×C₄×C₆₁	—	7	1	no	Hall
18	C₇×ΓL(1,𝔽₈)⋊C₄	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₇×C₆₁	—	6	1	no	—
20	C₇×C₆₁	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, direct product factor
21	C₇×C₆₁	C₃×C₄×C₇	1	1	yes	characteristic, complemented
22	C₂×C₃×C₆₁	—	7	1	no	—
23	C₂×C₇×ΓL(1,𝔽₈)	C₂×C₆₁	1	1	yes	characteristic
24	C₄×C₆₁	—	7	1	no	Hall
25	C₇×Dic₇	C₃×C₆₁	1	1	yes	characteristic, complemented, Hall
26	C₃×C₆₁	—	7	1	no	Hall
27	C₇×ΓL(1,𝔽₈)	C₄×C₆₁	1	1	yes	characteristic, complemented, Hall
28	C₂×C₆₁	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic
29	C₂×C₇²	C₂×C₃×C₆₁	1	1	yes	characteristic
30	C₃×C₄×C₇	—	7	1	no	—
31	ΓL(1,𝔽₈)⋊C₄	C₇×C₆₁	1	1	yes	characteristic, direct product factor
32	C₆₁	C₇×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
33	C₇²	C₃×C₄×C₆₁	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₇	—	7	1	no	—
35	C₂×ΓL(1,𝔽₈)	C₂×C₇×C₆₁	1	1	yes	characteristic
36	C₄×C₇	—	7	1	no	—
37	Dic₇	C₃×C₇×C₆₁	1	1	yes	characteristic, complemented
38	C₃×C₇	—	7	1	no	—
39	ΓL(1,𝔽₈)	C₄×C₇×C₆₁	1	1	yes	characteristic, complemented
40	C₂×C₇	—	6	1	no	—
41	C₂×C₇	C₂×C₃×C₇×C₆₁	1	1	yes	characteristic
42	C₂×C₇	AGL(1,𝔽₇)×C₆₁	1	1	yes	characteristic
43	C₃×C₄	—	7	1	no	Hall
44	C₇	—	6	1	no	—
45	C₇	C₃×C₄×C₇×C₆₁	1	1	yes	characteristic, complemented
46	C₇	C₆₁×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, direct product factor
47	C₂×C₃	—	7	1	no	—
48	C₄	—	7	1	no	Sylow
49	C₃	—	7	1	no	Sylow
50	C₂	C₇×AGL(1,𝔽₇)×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₇×C₆₁	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₃×C₄×C₇×C₆₁
Frattini	C₂	C₇×AGL(1,𝔽₇)×C₆₁
Fitting, socle	C₂×C₇²×C₆₁	C₂×C₃
prime radical	C₂×C₇×ΓL(1,𝔽₈)×C₆₁	C₂
cosocle	C₂×C₇	C₂×C₃×C₇×C₆₁
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₇×C₆₁

Complements

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

N	H ≅ G/N	type
2. C₇×Dic₇×C₆₁	C₃	semidirect
3. C₇×ΓL(1,𝔽₈)×C₆₁	C₄	semidirect
5. C₆₁×ΓL(1,𝔽₈)⋊C₄	C₇	direct (N × H)
7. C₇²×C₆₁	C₃×C₄	semidirect
11. Dic₇×C₆₁	C₃×C₇	semidirect
13. ΓL(1,𝔽₈)×C₆₁	C₄×C₇	semidirect
18. C₇×ΓL(1,𝔽₈)⋊C₄	C₆₁	direct (N × H)
20. C₇×C₆₁	ΓL(1,𝔽₈)⋊C₄	direct (N × H)
21. C₇×C₆₁	C₃×C₄×C₇	semidirect
25. C₇×Dic₇	C₃×C₆₁	semidirect
27. C₇×ΓL(1,𝔽₈)	C₄×C₆₁	semidirect
31. ΓL(1,𝔽₈)⋊C₄	C₇×C₆₁	direct (N × H)
32. C₆₁	C₇×ΓL(1,𝔽₈)⋊C₄	direct (N × H)
33. C₇²	C₃×C₄×C₆₁	semidirect
37. Dic₇	C₃×C₇×C₆₁	semidirect
39. ΓL(1,𝔽₈)	C₄×C₇×C₆₁	semidirect
45. C₇	C₃×C₄×C₇×C₆₁	semidirect
46. C₇	C₆₁×ΓL(1,𝔽₈)⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×ΓL(1,𝔽₈)×C₆₁	2
C₇×Dic₇×C₆₁	3
C₆₁×ΓL(1,𝔽₈)⋊C₄	7
C₃×C₄×C₇×C₆₁	7
C₇×ΓL(1,𝔽₈)⋊C₄	61

Subgroups — C7×C61×ΓL(1,𝔽8)⋊C4 #49089727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₆₁×ΓL(1,𝔽₈)⋊C₄ #49089727