Subgroups — ΓL(1,𝔽₈)×C₇⋊#6923 #49089854

Order 35868.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×ΓL(1,𝔽₈)×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#6923	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×ΓL(1,𝔽₈)×C₆₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇²×D₆₁	C₂×C₃	1	1	yes	characteristic
5	C₃×C₇⋊#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	D₇	1	1	yes	characteristic
9	C₃×C₇×D₆₁	—	7	1	no	—
10	C₇⋊#6923	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
11	C₇×#6923	—	7	1	no	—
12	C₃×C₇×C₆₁	—	7	1	no	—
13	ΓL(1,𝔽₈)×C₆₁	Dic₇	1	1	yes	characteristic, complemented
14	C₇×D₆₁	—	6	1	no	—
15	C₇×D₆₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₇×D₆₁	C₃×D₇	1	1	yes	characteristic
17	C₃×#6923	—	49	1	no	Hall
18	ΓL(1,𝔽₈)×Dic₇	—	61	1	no	maximal, Hall
19	C₇×C₆₁	—	6	1	no	—
20	C₇×C₆₁	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₇×C₆₁	C₃×Dic₇	1	1	yes	characteristic, complemented
22	C₃×D₆₁	—	7	1	no	—
23	C₂×C₇×ΓL(1,𝔽₈)	—	61	1	no	—
24	#6923	—	7	1	no	Hall
25	C₇×Dic₇	—	61	1	no	Hall
26	C₃×C₆₁	—	7	1	no	Hall
27	C₇×ΓL(1,𝔽₈)	#6923	1	1	yes	characteristic, complemented, Hall
28	D₆₁	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₂×C₇²	—	61	1	no	—
30	C₃×Dic₇	—	427	1	no	—
31	C₄×ΓL(1,𝔽₈)	—	427	1	no	—
32	C₆₁	ΓL(1,𝔽₈)×Dic₇	1	1	yes	characteristic, complemented, Sylow
33	C₇²	C₃×#6923	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₇	—	427	1	no	—
35	C₂×ΓL(1,𝔽₈)	—	61	1	no	—
36	C₄×C₇	—	427	1	no	—
37	Dic₇	—	61	1	no	—
38	C₃×C₇	—	7	1	no	—
39	ΓL(1,𝔽₈)	C₇⋊#6923	1	1	yes	characteristic, direct product factor
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	C₃×C₇⋊#6923	1	1	yes	characteristic, complemented
47	C₂×C₃	—	427	1	no	—
48	C₄	—	427	1	no	Sylow
49	C₃	—	7	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	C₇×ΓL(1,𝔽₈)×D₆₁	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. C₇×ΓL(1,𝔽₈)×C₆₁	C₄	semidirect
7. C₇²×C₆₁	C₃×C₄	semidirect
10. C₇⋊#6923	ΓL(1,𝔽₈)	direct (N × H)
13. ΓL(1,𝔽₈)×C₆₁	Dic₇	semidirect
20. C₇×C₆₁	C₄×ΓL(1,𝔽₈)	semidirect
21. C₇×C₆₁	C₃×Dic₇	semidirect
27. C₇×ΓL(1,𝔽₈)	#6923	semidirect
32. C₆₁	ΓL(1,𝔽₈)×Dic₇	semidirect
33. C₇²	C₃×#6923	semidirect
39. ΓL(1,𝔽₈)	C₇⋊#6923	direct (N × H)
45. C₇	ΓL(1,𝔽₈)×#6923	semidirect
46. C₇	C₃×C₇⋊#6923	semidirect

Maximal subgroups

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

subgroup	index
C₇×ΓL(1,𝔽₈)×D₆₁	2
C₇×C₇⋊#6923	3
C₃×C₇⋊#6923	7
ΓL(1,𝔽₈)×#6923	7
ΓL(1,𝔽₈)×Dic₇	61

Subgroups — ΓL(1,𝔽8)×C7⋊#6923 #49089854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₇⋊#6923 #49089854