Subgroups — #13132582

Order 1680.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊(C₅⋊AGL(1,𝔽₇))	C₂	1	3	yes	maximal, complemented
3	C₄×C₅⋊AGL(1,𝔽₇)	C₂	1	3	yes	maximal, complemented
4	#10587991	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×C₅⋊AGL(1,𝔽₇)	C₂²	1	3	yes	—
6	C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
7	ΓL(1,𝔽₈)⋊Dic₅	C₂²	1	1	yes	characteristic
8	#66659	—	5	1	no	maximal, Hall
9	C₅×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
10	D₁₄₀	C₂×C₃	1	3	yes	complemented
11	C₄×D₃₅	C₂×C₃	1	3	yes	complemented
12	C₃×#713	—	7	1	no	maximal, Hall
13	C₂×C₅×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
14	C₅⋊AGL(1,𝔽₇)	—	2	3	no	—
15	Q₈×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented
16	C₄⋊AGL(1,𝔽₇)	—	5	3	no	—
17	C₄×AGL(1,𝔽₇)	—	5	3	no	—
18	C₂×D₃₅	C₂²×C₃	1	3	yes	—
19	C₄×C₅×C₇	C₂²×C₃	1	3	yes	—
20	Dic₃₅	C₂²×C₃	1	1	yes	characteristic
21	C₃×C₅×Q₈	—	7	1	no	—
22	C₃×D₂₀	—	7	3	no	—
23	C₃×C₄×D₅	—	7	3	no	—
24	#1161	—	5	1	no	Hall
25	C₅×ΓL(1,𝔽₈)	#41	1	1	yes	characteristic, complemented, Hall
26	C₂×AGL(1,𝔽₇)	—	5	3	no	—
27	C₄×ΓL(1,𝔽₈)	C₂×D₅	1	3	yes	—
28	ΓL(1,𝔽₈)⋊C₄	—	5	1	no	—
29	#713	—	7	1	no	Hall
30	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
31	D₃₅	—	2	3	no	—
32	C₂×C₃×D₅	—	7	3	no	—
33	C₃×C₄×C₅	—	7	3	no	—
34	C₃×Dic₅	—	7	1	no	—
35	C₇×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
36	D₂₈	—	5	3	no	—
37	C₄×D₇	—	5	3	no	—
38	C₃×#41	—	35	1	no	Hall
39	C₂×ΓL(1,𝔽₈)	C₂²×D₅	1	1	yes	characteristic
40	AGL(1,𝔽₇)	—	10	3	no	—
41	C₅×Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
42	D₂₀	—	7	3	no	—
43	C₄×D₅	—	7	3	no	—
44	C₅×C₇	C₃×#41	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₅	—	7	1	no	—
46	C₃×D₅	—	14	3	no	—
47	C₂×D₇	—	5	3	no	—
48	C₄×C₇	C₂×C₃×D₅	1	3	yes	—
49	Dic₇	—	5	1	no	—
50	C₃×Q₈	—	7	1	no	—
51	C₃×D₄	—	35	3	no	—
52	C₂×C₃×C₄	—	35	3	no	—
53	ΓL(1,𝔽₈)	#713	1	1	yes	characteristic, complemented, Hall
54	C₂×D₅	—	7	3	no	—
55	C₄×C₅	C₂×AGL(1,𝔽₇)	1	3	yes	—
56	Dic₅	—	7	1	no	—
57	#41	—	35	1	no	Sylow
58	C₃×C₅	—	7	1	no	Hall
59	C₂×C₇	C₂²×C₃×D₅	1	1	yes	characteristic
60	D₇	—	10	3	no	—
61	C₂²×C₃	—	35	3	no	—
62	C₃×C₄	—	7	3	no	—
63	C₃×C₄	—	35	1	no	—
64	C₂×C₅	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
65	D₅	—	14	3	no	—
66	Q₈	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
67	D₄	—	35	3	no	—
68	C₂×C₄	—	35	3	no	—
69	C₇	C₃×#713	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	1	no	—
71	C₂×C₃	—	70	3	no	—
72	C₅	#66659	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	35	3	no	—
74	C₄	—	35	1	no	—
75	C₄	C₂×C₅⋊AGL(1,𝔽₇)	1	3	yes	—
76	C₃	—	7	1	no	Sylow
77	C₂	—	70	3	no	—
78	C₂	C₂²×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×C₅⋊AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₅×C₇	C₂³×C₃
Fitting	C₅×C₇×Q₈	C₂×C₃
hypercenter	Q₈	C₅⋊AGL(1,𝔽₇)
hypocenter	C₅×C₇	C₃×#41
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×ΓL(1,𝔽₈)	C₂	semidirect
2. C₄⋊(C₅⋊AGL(1,𝔽₇))	C₂	semidirect
3. C₄×C₅⋊AGL(1,𝔽₇)	C₂	semidirect
4. #10587991	C₃	semidirect
9. C₅×C₇×Q₈	C₂×C₃	semidirect
10. D₁₄₀	C₂×C₃	semidirect
11. C₄×D₃₅	C₂×C₃	semidirect
15. Q₈×ΓL(1,𝔽₈)	D₅	semidirect
25. C₅×ΓL(1,𝔽₈)	#41	semidirect
35. C₇×Q₈	C₃×D₅	semidirect
41. C₅×Q₈	AGL(1,𝔽₇)	semidirect
44. C₅×C₇	C₃×#41	semidirect
53. ΓL(1,𝔽₈)	#713	semidirect
66. Q₈	C₅⋊AGL(1,𝔽₇)	semidirect
69. C₇	C₃×#713	semidirect
72. C₅	#66659	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×ΓL(1,𝔽₈)	2
C₄⋊(C₅⋊AGL(1,𝔽₇))	2
C₄×C₅⋊AGL(1,𝔽₇)	2
#10587991	3
#66659	5
C₃×#713	7