Subgroups — #6377⋊C₃ #10620431

Order 672.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×ΓL(1,𝔽₈)⋊C₈	C₂	1	2	yes	maximal
3	#6377	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×C₄×ΓL(1,𝔽₈)	C₄	1	2	yes	—
5	C₂×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	ΓL(1,𝔽₈)⋊C₈	C₄	1	4	yes	complemented
7	C₄²×C₇	C₂×C₃	1	1	yes	characteristic
8	C₂×C₇⋊C₈	C₂×C₃	1	2	yes	—
9	C₃×#97	—	7	1	no	maximal, Hall
10	C₂²×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
11	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
12	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₇	C₃×C₄	1	2	yes	—
14	C₂×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
15	C₇⋊C₈	C₃×C₄	1	4	yes	complemented
16	C₂×C₃×C₈	—	7	2	no	—
17	C₃×C₄²	—	7	1	no	—
18	C₂×ΓL(1,𝔽₈)	C₄²	1	1	yes	characteristic
19	C₂×ΓL(1,𝔽₈)	M₁₆	1	2	yes	—
20	#97	—	7	1	no	Sylow
21	C₂²×C₇	C₂×C₃×C₄	1	1	yes	characteristic
22	C₄×C₇	—	2	2	no	—
23	C₄×C₇	C₂×C₃×C₄	1	2	yes	—
24	C₂×C₃×C₄	—	7	2	no	—
25	C₂×C₃×C₄	—	7	1	no	—
26	C₃×C₈	—	7	4	no	—
27	ΓL(1,𝔽₈)	#97	1	1	yes	characteristic, complemented, Hall
28	C₂×C₈	—	7	2	no	—
29	C₄²	AGL(1,𝔽₇)	1	1	yes	characteristic
30	C₂×C₇	C₃×C₄²	1	1	yes	characteristic
31	C₂×C₇	C₃×M₁₆	1	2	yes	—
32	C₂²×C₃	—	7	1	no	—
33	C₃×C₄	—	7	2	no	—
34	C₃×C₄	—	14	2	no	—
35	C₂×C₄	ΓL(1,𝔽₈)⋊C₄	1	2	yes	—
36	C₂×C₄	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
37	C₈	—	7	4	no	—
38	C₇	C₃×#97	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃	—	7	2	no	—
40	C₂×C₃	—	7	1	no	—
41	C₂²	C₂×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	C₄×AGL(1,𝔽₇)	1	2	yes	—
44	C₃	—	7	1	no	Sylow
45	C₂	#1136⋊C₃	1	2	yes	—
46	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	C₄²×C₇	C₂×C₃
socle	C₂²×C₇	C₂×C₃×C₄
prime radical	C₂²×ΓL(1,𝔽₈)	C₂×C₄
cosocle	C₂×C₄×C₇	C₂²×C₃
hypercenter	C₄²	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#97
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
3. #6377	C₃	semidirect
6. ΓL(1,𝔽₈)⋊C₈	C₄	semidirect
15. C₇⋊C₈	C₃×C₄	semidirect
27. ΓL(1,𝔽₈)	#97	semidirect
38. 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
C₄²×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)⋊C₈	2
#6377	3
C₃×#97	7

Subgroups — #6377⋊C3 #10620431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #6377⋊C₃ #10620431