Subgroups — C₇×ΓL(1,𝔽₈)×C₆₁⋊C₃ #41002017

Order 26901.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₆₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×ΓL(1,𝔽₈)×C₆₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×#13046197	C₃	1	1	yes	maximal, characteristic, complemented
4	C₇×#13046196	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₇×C₆₁⋊C₃	—	7	1	no	maximal
6	ΓL(1,𝔽₈)×C₆₁⋊C₃	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₇²×C₆₁	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇×C₆₁	—	7	1	no	—
9	#13046197	C₃×C₇	1	1	yes	characteristic, complemented
10	#13046196	C₃×C₇	1	1	yes	characteristic, complemented
11	C₇×C₆₁⋊C₃	—	3	2	no	—
12	C₇×C₆₁⋊C₃	—	7	1	no	—
13	C₇×C₆₁⋊C₃	—	7	1	no	—
14	C₇×C₆₁⋊C₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₇×C₆₁⋊C₃	C₃×C₇	1	1	yes	characteristic, complemented
16	ΓL(1,𝔽₈)×C₆₁	C₃×C₇	1	1	yes	characteristic, complemented
17	C₃×C₆₁⋊C₃	—	7	1	no	Hall
18	C₃×C₇×ΓL(1,𝔽₈)	—	61	1	no	maximal, Hall
19	C₇×C₆₁	—	3	2	no	—
20	C₇×C₆₁	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₇×C₆₁	C₃²×C₇	1	1	yes	characteristic, complemented
22	C₃×C₆₁	—	7	1	no	—
23	C₆₁⋊C₃	—	7	1	no	—
24	C₆₁⋊C₃	—	7	1	no	—
25	C₆₁⋊C₃	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
26	C₃×C₇²	—	61	1	no	—
27	C₇×ΓL(1,𝔽₈)	—	61	1	no	—
28	C₇×ΓL(1,𝔽₈)	—	61	1	no	—
29	C₇×ΓL(1,𝔽₈)	C₆₁⋊C₃	1	1	yes	characteristic, direct product factor
30	C₃²×C₇	—	427	1	no	—
31	C₃×ΓL(1,𝔽₈)	—	61	1	no	—
32	C₆₁	C₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
33	C₇²	C₃×C₆₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
34	C₃×C₇	—	7	1	no	—
35	C₃×C₇	—	61	1	no	—
36	C₃×C₇	—	61	1	no	—
37	C₃×C₇	—	427	1	no	—
38	C₃×C₇	—	427	1	no	—
39	C₃×C₇	—	183	2	no	—
40	ΓL(1,𝔽₈)	—	61	1	no	—
41	ΓL(1,𝔽₈)	—	61	1	no	—
42	ΓL(1,𝔽₈)	C₇×C₆₁⋊C₃	1	1	yes	characteristic, direct product factor
43	C₃²	—	427	1	no	Sylow
44	C₇	—	3	2	no	—
45	C₇	ΓL(1,𝔽₈)×C₆₁⋊C₃	1	1	yes	characteristic, direct product factor
46	C₇	C₃×C₇×C₆₁⋊C₃	1	1	yes	characteristic, complemented
47	C₃	—	7	1	no	—
48	C₃	—	61	1	no	—
49	C₃	—	427	1	no	—
50	C₃	—	427	1	no	—

	subgroup	G/H
center, hypercenter	C₇	ΓL(1,𝔽₈)×C₆₁⋊C₃
commutator, cosocle, hypocenter	C₇×C₆₁	C₃²×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₆₁	C₃²
LCS	G ▷ C₇×C₆₁
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇²×C₆₁⋊C₃	C₃	semidirect
2. C₇×ΓL(1,𝔽₈)×C₆₁	C₃	semidirect
3. C₇×#13046197	C₃	semidirect
4. C₇×#13046196	C₃	semidirect
6. ΓL(1,𝔽₈)×C₆₁⋊C₃	C₇	direct (N × H)
7. C₇²×C₆₁	C₃²	semidirect
9. #13046197	C₃×C₇	semidirect
10. #13046196	C₃×C₇	semidirect
14. C₇×C₆₁⋊C₃	ΓL(1,𝔽₈)	direct (N × H)
15. C₇×C₆₁⋊C₃	C₃×C₇	semidirect
16. ΓL(1,𝔽₈)×C₆₁	C₃×C₇	semidirect
20. C₇×C₆₁	C₃×ΓL(1,𝔽₈)	semidirect
21. C₇×C₆₁	C₃²×C₇	semidirect
25. C₆₁⋊C₃	C₇×ΓL(1,𝔽₈)	direct (N × H)
29. C₇×ΓL(1,𝔽₈)	C₆₁⋊C₃	direct (N × H)
32. C₆₁	C₃×C₇×ΓL(1,𝔽₈)	semidirect
33. C₇²	C₃×C₆₁⋊C₃	semidirect
42. ΓL(1,𝔽₈)	C₇×C₆₁⋊C₃	direct (N × H)
45. C₇	ΓL(1,𝔽₈)×C₆₁⋊C₃	direct (N × H)
46. C₇	C₃×C₇×C₆₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₇²×C₆₁⋊C₃	3
C₇×ΓL(1,𝔽₈)×C₆₁	3
C₇×#13046197	3
C₇×#13046196	3
C₃×C₇×C₆₁⋊C₃	7
ΓL(1,𝔽₈)×C₆₁⋊C₃	7
C₃×C₇×ΓL(1,𝔽₈)	61

Subgroups — C7×ΓL(1,𝔽8)×C61⋊C3 #41002017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₈)×C₆₁⋊C₃ #41002017