Subgroups — C₇×ΓL(1,𝔽₈)⋊#6157 #44528396

Order 31164.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×ΓL(1,𝔽₈)×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#6157	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×ΓL(1,𝔽₈)×C₅₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇²×D₅₃	C₂×C₃	1	1	yes	characteristic
5	C₃×C₇×#6157	—	7	1	no	maximal
6	ΓL(1,𝔽₈)⋊#6157	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₇²×C₅₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	ΓL(1,𝔽₈)×D₅₃	C₂×C₇	1	1	yes	characteristic
9	C₃×C₇×D₅₃	—	7	1	no	—
10	C₇⋊#6157	C₃×C₇	1	1	yes	characteristic, complemented
11	C₇×#6157	—	7	1	no	—
12	C₃×C₇×C₅₃	—	7	1	no	—
13	ΓL(1,𝔽₈)×C₅₃	C₄×C₇	1	1	yes	characteristic, complemented
14	C₇×D₅₃	—	6	1	no	—
15	C₇×D₅₃	AGL(1,𝔽₇)	1	1	yes	characteristic
16	C₇×D₅₃	C₂×C₃×C₇	1	1	yes	characteristic
17	C₃×#6157	—	7	1	no	Hall
18	C₇×ΓL(1,𝔽₈)⋊C₄	—	53	1	no	maximal, Hall
19	C₇×C₅₃	—	6	1	no	—
20	C₇×C₅₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
21	C₇×C₅₃	C₃×C₄×C₇	1	1	yes	characteristic, complemented
22	C₃×D₅₃	—	7	1	no	—
23	C₂×C₇×ΓL(1,𝔽₈)	—	53	1	no	—
24	#6157	—	7	1	no	Hall
25	C₇×Dic₇	—	53	1	no	Hall
26	C₃×C₅₃	—	7	1	no	Hall
27	C₇×ΓL(1,𝔽₈)	#6157	1	1	yes	characteristic, complemented, Hall
28	D₅₃	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic
29	C₂×C₇²	—	53	1	no	—
30	C₃×C₄×C₇	—	371	1	no	—
31	ΓL(1,𝔽₈)⋊C₄	—	53	1	no	—
32	C₅₃	C₇×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Sylow
33	C₇²	C₃×#6157	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₇	—	371	1	no	—
35	C₂×ΓL(1,𝔽₈)	—	53	1	no	—
36	C₄×C₇	—	371	1	no	—
37	Dic₇	—	53	1	no	—
38	C₃×C₇	—	7	1	no	—
39	ΓL(1,𝔽₈)	C₇×#6157	1	1	yes	characteristic, complemented
40	C₂×C₇	—	53	1	no	—
41	C₂×C₇	—	53	1	no	—
42	C₂×C₇	—	318	1	no	—
43	C₃×C₄	—	371	1	no	Hall
44	C₇	—	6	1	no	—
45	C₇	ΓL(1,𝔽₈)⋊#6157	1	1	yes	characteristic, direct product factor
46	C₇	C₃×C₇×#6157	1	1	yes	characteristic, complemented
47	C₂×C₃	—	371	1	no	—
48	C₄	—	371	1	no	Sylow
49	C₃	—	7	1	no	Sylow
50	C₂	—	53	1	no	—

	subgroup	G/H
center, hypercenter	C₇	ΓL(1,𝔽₈)⋊#6157
commutator, hypocenter	C₇×C₅₃	C₃×C₄×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₅₃	C₃×C₄
prime radical	C₇×ΓL(1,𝔽₈)×D₅₃	C₂
cosocle	C₇×D₅₃	C₂×C₃×C₇
LCS	G ▷ C₇×C₅₃
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
2. C₇×C₇⋊#6157	C₃	semidirect
3. C₇×ΓL(1,𝔽₈)×C₅₃	C₄	semidirect
6. ΓL(1,𝔽₈)⋊#6157	C₇	direct (N × H)
7. C₇²×C₅₃	C₃×C₄	semidirect
10. C₇⋊#6157	C₃×C₇	semidirect
13. ΓL(1,𝔽₈)×C₅₃	C₄×C₇	semidirect
20. C₇×C₅₃	ΓL(1,𝔽₈)⋊C₄	semidirect
21. C₇×C₅₃	C₃×C₄×C₇	semidirect
27. C₇×ΓL(1,𝔽₈)	#6157	semidirect
32. C₅₃	C₇×ΓL(1,𝔽₈)⋊C₄	semidirect
33. C₇²	C₃×#6157	semidirect
39. ΓL(1,𝔽₈)	C₇×#6157	semidirect
45. C₇	ΓL(1,𝔽₈)⋊#6157	direct (N × H)
46. C₇	C₃×C₇×#6157	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)×D₅₃	2
C₇×C₇⋊#6157	3
C₃×C₇×#6157	7
ΓL(1,𝔽₈)⋊#6157	7
C₇×ΓL(1,𝔽₈)⋊C₄	53

Subgroups — C7×ΓL(1,𝔽8)⋊#6157 #44528396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₈)⋊#6157 #44528396