Subgroups — #6401⋊C₃ #10620455

Order 672.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	Dic₂₈⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#6401	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	D₄×ΓL(1,𝔽₈)	—	2	1	no	—
6	ΓL(1,𝔽₈)⋊Q₈	—	2	1	no	—
7	C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	C₇×D₈	C₂×C₃	1	1	yes	characteristic
9	Dic₂₈	C₂×C₃	1	1	yes	characteristic, complemented
10	C₇⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×SD₃₂	—	7	1	no	maximal, Hall
12	C₂²×ΓL(1,𝔽₈)	—	4	1	no	—
13	C₄×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
14	ΓL(1,𝔽₈)⋊C₄	—	4	1	no	—
15	C₇×D₄	—	2	1	no	—
16	Dic₁₄	—	2	1	no	—
17	C₇×C₈	C₂²×C₃	1	1	yes	characteristic
18	C₃×Q₁₆	—	7	1	no	—
19	C₃×D₈	—	14	1	no	—
20	C₃×C₁₆	—	7	1	no	—
21	C₂×ΓL(1,𝔽₈)	—	8	1	no	—
22	C₂×ΓL(1,𝔽₈)	D₈	1	1	yes	characteristic
23	SD₃₂	—	7	1	no	Sylow
24	C₂²×C₇	—	4	1	no	—
25	C₄×C₇	C₃×D₄	1	1	yes	characteristic
26	Dic₇	—	4	1	no	—
27	C₃×Q₈	—	14	1	no	—
28	C₃×D₄	—	14	1	no	—
29	C₃×C₈	—	14	1	no	—
30	ΓL(1,𝔽₈)	SD₃₂	1	1	yes	characteristic, complemented, Hall
31	Q₁₆	—	7	1	no	—
32	D₈	AGL(1,𝔽₇)	1	1	yes	characteristic
33	C₁₆	—	7	1	no	—
34	C₂×C₇	—	8	1	no	—
35	C₂×C₇	C₃×D₈	1	1	yes	characteristic
36	C₂²×C₃	—	28	1	no	—
37	C₃×C₄	—	14	1	no	—
38	C₃×C₄	—	28	1	no	—
39	Q₈	—	14	1	no	—
40	D₄	—	2	1	no	—
41	C₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
42	C₇	C₃×SD₃₂	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	14	1	no	—
44	C₂×C₃	—	56	1	no	—
45	C₂²	—	4	1	no	—
46	C₄	—	28	1	no	—
47	C₄	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic
48	C₃	—	14	1	no	Sylow
49	C₂	—	8	1	no	—
50	C₂	#1141⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#1141⋊C₃
commutator, cosocle	C₇×C₈	C₂²×C₃
Frattini	C₈	C₂×AGL(1,𝔽₇)
Fitting	C₇×D₈	C₂×C₃
socle	C₂×C₇	C₃×D₈
prime radical	D₈×ΓL(1,𝔽₈)	C₂
hypercenter	D₈	AGL(1,𝔽₇)
hypocenter	C₇	C₃×SD₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
2. Dic₂₈⋊C₃	C₂	semidirect
3. ΓL(1,𝔽₈)⋊C₁₆	C₂	semidirect
4. #6401	C₃	semidirect
9. Dic₂₈	C₂×C₃	semidirect
10. C₇⋊C₁₆	C₂×C₃	semidirect
30. ΓL(1,𝔽₈)	SD₃₂	semidirect
42. C₇	C₃×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
D₈×ΓL(1,𝔽₈)	2
Dic₂₈⋊C₃	2
ΓL(1,𝔽₈)⋊C₁₆	2
#6401	3
C₃×SD₃₂	7

Subgroups — #6401⋊C3 #10620455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #6401⋊C₃ #10620455