Subgroups — ΓL(1,𝔽₈)⋊(C₇⋊#1039) #28413157

Order 14700.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₇⋊(C₇⋊#1039)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×C₇×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
4	ΓL(1,𝔽₈)⋊Dic₃₅	—	5	6	no	maximal
5	C₂×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic
6	C₃×C₇⋊#1039	—	7	1	no	maximal
7	ΓL(1,𝔽₈)⋊#1039	—	7	1	no	maximal
8	C₂×C₅×C₇×ΓL(1,𝔽₈)	D₅	1	6	yes	—
9	C₅²×C₇²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₅²×C₇	—	7	1	no	—
11	C₂×C₅²×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic
12	C₇⋊Dic₃₅	—	5	6	no	—
13	C₅×C₇×ΓL(1,𝔽₈)	Dic₅	1	6	yes	complemented
14	C₇⋊#1039	—	7	1	no	—
15	C₇⋊#1039	—	7	1	no	—
16	C₇⋊#1039	—	21	2	no	—
17	ΓL(1,𝔽₈)⋊Dic₇	—	25	1	no	Hall
18	C₃×C₅²×C₇	—	7	1	no	—
19	C₅²×ΓL(1,𝔽₈)	Dic₇	1	1	yes	characteristic, complemented
20	C₂×C₅×C₇²	C₃×D₅	1	6	yes	—
21	C₃×Dic₃₅	—	35	6	no	—
22	ΓL(1,𝔽₈)⋊Dic₅	—	35	6	no	—
23	C₂×C₅²×C₇	—	3	2	no	—
24	C₂×C₅²×C₇	AGL(1,𝔽₇)	1	1	yes	characteristic
25	C₂×C₅²×C₇	C₃×D₇	1	1	yes	characteristic
26	C₃×#1039	—	49	1	no	Hall
27	C₂×C₇×ΓL(1,𝔽₈)	C₅⋊D₅	1	1	yes	characteristic
28	C₅×C₇²	C₃×Dic₅	1	6	yes	complemented
29	C₂×C₃×C₅×C₇	—	7	6	no	—
30	C₂×C₅×ΓL(1,𝔽₈)	D₃₅	1	6	yes	—
31	C₇⋊Dic₇	—	25	1	no	Hall
32	C₅²×C₇	—	3	2	no	—
33	C₅²×C₇	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
34	C₅²×C₇	C₃×Dic₇	1	1	yes	characteristic, complemented
35	C₂×C₃×C₅²	—	7	1	no	—
36	C₇×ΓL(1,𝔽₈)	#1039	1	1	yes	characteristic, complemented, Hall
37	Dic₃₅	—	35	6	no	—
38	Dic₃₅	—	35	6	no	—
39	Dic₃₅	—	105	12	no	—
40	C₃×C₅×C₇	—	7	6	no	—
41	C₅×ΓL(1,𝔽₈)	Dic₃₅	1	6	yes	complemented
42	#1039	—	49	1	no	Hall
43	C₂×C₇²	C₃×C₅⋊D₅	1	1	yes	characteristic
44	C₃×Dic₇	—	175	1	no	—
45	ΓL(1,𝔽₈)⋊C₄	—	175	1	no	—
46	C₃×C₅²	—	7	1	no	Hall
47	C₂×C₅×C₇	—	3	12	no	—
48	C₂×C₅×C₇	C₅⋊AGL(1,𝔽₇)	1	6	yes	—
49	C₂×C₅×C₇	C₃×D₃₅	1	6	yes	—
50	C₃×Dic₅	—	245	6	no	—
51	C₂×C₅²	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic
52	C₇²	C₃×#1039	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₇	—	7	1	no	—
54	C₂×ΓL(1,𝔽₈)	C₅⋊D₃₅	1	1	yes	characteristic
55	C₅×C₇	—	3	12	no	—
56	C₅×C₇	ΓL(1,𝔽₈)⋊Dic₅	1	6	yes	complemented
57	C₅×C₇	C₃×Dic₃₅	1	6	yes	complemented
58	C₂×C₃×C₅	—	7	6	no	—
59	Dic₇	—	525	2	no	—
60	Dic₇	—	175	1	no	—
61	Dic₇	—	175	1	no	—
62	C₅²	ΓL(1,𝔽₈)⋊Dic₇	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₇	—	7	1	no	—
64	ΓL(1,𝔽₈)	C₇⋊#1039	1	1	yes	characteristic, complemented
65	Dic₅	—	245	6	no	—
66	C₃×C₅	—	7	6	no	—
67	C₂×C₇	—	3	2	no	—
68	C₂×C₇	C₅⋊(C₅⋊AGL(1,𝔽₇))	1	1	yes	characteristic
69	C₂×C₇	C₃×C₅⋊D₃₅	1	1	yes	characteristic
70	C₃×C₄	—	1225	1	no	Hall
71	C₂×C₅	C₅⋊(ΓL(1,𝔽₈)⋊D₇)	1	6	yes	—
72	C₇	—	3	2	no	—
73	C₇	ΓL(1,𝔽₈)⋊#1039	1	1	yes	characteristic, complemented
74	C₇	C₃×C₇⋊#1039	1	1	yes	characteristic, complemented
75	C₂×C₃	—	7	1	no	—
76	C₅	ΓL(1,𝔽₈)⋊Dic₃₅	1	6	yes	complemented
77	C₄	—	1225	1	no	Sylow
78	C₃	—	7	1	no	Sylow
79	C₂	C₅⋊(C₅⋊(ΓL(1,𝔽₈)⋊D₇))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊(C₅⋊(ΓL(1,𝔽₈)⋊D₇))
commutator, hypocenter	C₅²×C₇²	C₃×C₄
Fitting, socle, cosocle	C₂×C₅²×C₇²	C₂×C₃
prime radical	C₂×C₅²×C₇×ΓL(1,𝔽₈)	C₂
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇⋊(C₇⋊#1039)	C₃	semidirect
3. C₅²×C₇×ΓL(1,𝔽₈)	C₄	semidirect
9. C₅²×C₇²	C₃×C₄	semidirect
13. C₅×C₇×ΓL(1,𝔽₈)	Dic₅	semidirect
19. C₅²×ΓL(1,𝔽₈)	Dic₇	semidirect
28. C₅×C₇²	C₃×Dic₅	semidirect
33. C₅²×C₇	ΓL(1,𝔽₈)⋊C₄	semidirect
34. C₅²×C₇	C₃×Dic₇	semidirect
36. C₇×ΓL(1,𝔽₈)	#1039	semidirect
41. C₅×ΓL(1,𝔽₈)	Dic₃₅	semidirect
52. C₇²	C₃×#1039	semidirect
56. C₅×C₇	ΓL(1,𝔽₈)⋊Dic₅	semidirect
57. C₅×C₇	C₃×Dic₃₅	semidirect
62. C₅²	ΓL(1,𝔽₈)⋊Dic₇	semidirect
64. ΓL(1,𝔽₈)	C₇⋊#1039	semidirect
73. C₇	ΓL(1,𝔽₈)⋊#1039	semidirect
74. C₇	C₃×C₇⋊#1039	semidirect
76. C₅	ΓL(1,𝔽₈)⋊Dic₃₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×C₇×ΓL(1,𝔽₈)	2
C₇⋊(C₇⋊#1039)	3
ΓL(1,𝔽₈)⋊Dic₃₅	5
C₃×C₇⋊#1039	7
ΓL(1,𝔽₈)⋊#1039	7

Subgroups — ΓL(1,𝔽8)⋊(C7⋊#1039) #28413157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)⋊(C₇⋊#1039) #28413157