Subgroups — #6372⋊C₃ #10620426

Order 672.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic
2	C₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#6372	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₄×AGL(1,𝔽₇)	C₄	1	1	yes	characteristic
6	C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	ΓL(1,𝔽₈)⋊C₈	C₄	1	1	yes	characteristic
8	C₈×D₇	C₂×C₃	1	1	yes	characteristic
9	C₇×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
10	C₇⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×M₃₂	—	7	1	no	maximal, Hall
12	C₂×AGL(1,𝔽₇)	C₈	1	1	yes	characteristic
13	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
14	ΓL(1,𝔽₈)⋊C₄	C₈	1	1	yes	characteristic
15	C₄×D₇	C₃×C₄	1	1	yes	characteristic
16	C₇×C₈	C₂²×C₃	1	1	yes	characteristic
17	C₇⋊C₈	C₃×C₄	1	1	yes	characteristic
18	C₂×C₃×C₈	—	7	1	no	—
19	C₃×C₁₆	—	7	1	no	—
20	C₃×C₁₆	—	14	1	no	—
21	C₂×ΓL(1,𝔽₈)	C₂×C₈	1	1	yes	characteristic
22	AGL(1,𝔽₇)	—	2	1	no	—
23	M₃₂	—	7	1	no	Sylow
24	C₂×D₇	C₃×C₈	1	1	yes	characteristic
25	C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
26	Dic₇	C₃×C₈	1	1	yes	characteristic
27	C₂×C₃×C₄	—	7	1	no	—
28	C₃×C₈	—	7	1	no	—
29	C₃×C₈	—	14	1	no	—
30	ΓL(1,𝔽₈)	M₃₂	1	1	yes	characteristic, complemented, Hall
31	C₂×C₈	—	7	1	no	—
32	C₁₆	—	7	1	no	—
33	C₁₆	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
34	C₂×C₇	C₂×C₃×C₈	1	1	yes	characteristic
35	D₇	—	2	1	no	—
36	C₂²×C₃	—	7	1	no	—
37	C₃×C₄	—	7	1	no	—
38	C₃×C₄	—	14	1	no	—
39	C₂×C₄	—	7	1	no	—
40	C₈	—	7	1	no	—
41	C₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
42	C₇	C₃×M₃₂	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	14	1	no	—
44	C₂×C₃	—	14	1	no	—
45	C₂²	—	7	1	no	—
46	C₄	—	7	1	no	—
47	C₄	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
48	C₃	—	14	1	no	Sylow
49	C₂	—	14	1	no	—
50	C₂	C₈×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×AGL(1,𝔽₇)
commutator, socle	C₂×C₇	C₂×C₃×C₈
Fitting	C₇×C₁₆	C₂×C₃
prime radical	C₂×AGL(1,𝔽₇)	C₈
cosocle	C₇×C₈	C₂²×C₃
hypercenter	C₁₆	AGL(1,𝔽₇)
hypocenter	C₇	C₃×M₃₂
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
2. C₁₆×ΓL(1,𝔽₈)	C₂	semidirect
3. ΓL(1,𝔽₈)⋊C₁₆	C₂	semidirect
4. #6372	C₃	semidirect
9. C₇×C₁₆	C₂×C₃	semidirect
10. C₇⋊C₁₆	C₂×C₃	semidirect
30. ΓL(1,𝔽₈)	M₃₂	semidirect
33. C₁₆	AGL(1,𝔽₇)	semidirect
42. C₇	C₃×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×AGL(1,𝔽₇)	2
C₁₆×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)⋊C₁₆	2
#6372	3
C₃×M₃₂	7

Subgroups — #6372⋊C3 #10620426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #6372⋊C₃ #10620426