Subgroups — #6389⋊C₃ #10620443

Order 672.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×ΓL(1,𝔽₈)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#6389	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₂×ΓL(1,𝔽₈)⋊C₄	C₄	1	2	yes	—
7	C₈×ΓL(1,𝔽₈)	C₄	1	2	yes	complemented
8	ΓL(1,𝔽₈)⋊C₈	C₄	1	2	yes	complemented
9	C₂×C₇×C₈	C₂×C₃	1	1	yes	characteristic
10	C₄×Dic₇	C₂×C₃	1	1	yes	characteristic
11	C₂×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic
12	C₃×#97	—	7	1	no	maximal, Hall
13	C₂²×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
14	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
15	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
16	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
17	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
18	C₂×Dic₇	C₃×C₄	1	2	yes	—
19	C₇×C₈	C₃×C₄	1	2	yes	complemented
20	C₇⋊C₈	C₃×C₄	1	2	yes	complemented
21	C₂×C₃×C₈	—	7	1	no	—
22	C₂×C₃×C₈	—	7	1	no	—
23	C₃×C₄²	—	7	1	no	—
24	C₂×ΓL(1,𝔽₈)	C₄²	1	1	yes	characteristic
25	C₂×ΓL(1,𝔽₈)	M₁₆	1	2	yes	—
26	#97	—	7	1	no	Sylow
27	C₂²×C₇	C₂×C₃×C₄	1	1	yes	characteristic
28	C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
29	C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
30	Dic₇	—	2	2	no	—
31	C₂×C₃×C₄	—	7	2	no	—
32	C₂×C₃×C₄	—	7	1	no	—
33	C₃×C₈	—	7	2	no	—
34	C₃×C₈	—	7	2	no	—
35	ΓL(1,𝔽₈)	#97	1	1	yes	characteristic, complemented, Hall
36	C₂×C₈	—	7	1	no	—
37	C₂×C₈	AGL(1,𝔽₇)	1	1	yes	characteristic
38	C₄²	—	7	1	no	—
39	C₂×C₇	C₃×C₄²	1	1	yes	characteristic
40	C₂×C₇	C₃×M₁₆	1	2	yes	—
41	C₂²×C₃	—	7	1	no	—
42	C₃×C₄	—	7	1	no	—
43	C₃×C₄	—	7	1	no	—
44	C₃×C₄	—	14	2	no	—
45	C₂×C₄	—	7	2	no	—
46	C₂×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
47	C₈	—	7	2	no	—
48	C₈	ΓL(1,𝔽₈)⋊C₄	1	2	yes	complemented
49	C₇	C₃×#97	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	—	7	2	no	—
51	C₂×C₃	—	7	1	no	—
52	C₂²	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
53	C₄	—	14	2	no	—
54	C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
55	C₄	C₂×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
56	C₃	—	7	1	no	Sylow
57	C₂	#1131⋊C₃	1	2	yes	—
58	C₂	C₄×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×AGL(1,𝔽₇)
commutator	C₂×C₇	C₃×C₄²
Fitting	C₂×C₇×C₈	C₂×C₃
socle	C₂²×C₇	C₂×C₃×C₄
prime radical	C₂²×ΓL(1,𝔽₈)	C₂×C₄
cosocle	C₂×C₄×C₇	C₂²×C₃
hypercenter	C₂×C₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#97
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. #6389	C₃	semidirect
7. C₈×ΓL(1,𝔽₈)	C₄	semidirect
8. ΓL(1,𝔽₈)⋊C₈	C₄	semidirect
19. C₇×C₈	C₃×C₄	semidirect
20. C₇⋊C₈	C₃×C₄	semidirect
35. ΓL(1,𝔽₈)	#97	semidirect
48. C₈	ΓL(1,𝔽₈)⋊C₄	semidirect
49. C₇	C₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×ΓL(1,𝔽₈)	2
C₄×ΓL(1,𝔽₈)⋊C₄	2
C₂×ΓL(1,𝔽₈)⋊C₈	2
#6389	3
C₃×#97	7

Subgroups — #6389⋊C3 #10620443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #6389⋊C₃ #10620443