Subgroups — C₂×ΓL(1,𝔽₈)⋊C₁₆ #10620439

Order 672.

56 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	ΓL(1,𝔽₈)⋊C₁₆	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇⋊C₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×C₄×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	C₈×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
6	C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₈	C₂×C₃	1	1	yes	characteristic
8	C₇⋊C₁₆	C₂×C₃	1	2	yes	complemented
9	C₂×C₃×C₁₆	—	7	1	no	maximal, Hall
10	C₂²×ΓL(1,𝔽₈)	C₈	1	1	yes	characteristic
11	C₄×ΓL(1,𝔽₈)	C₈	1	1	yes	characteristic
12	C₄×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₇	C₃×C₄	1	1	yes	characteristic
14	C₇×C₈	C₃×C₄	1	1	yes	characteristic
15	C₇×C₈	C₂²×C₃	1	1	yes	characteristic
16	C₂×C₃×C₈	—	7	1	no	—
17	C₃×C₁₆	—	7	2	no	—
18	C₂×ΓL(1,𝔽₈)	C₁₆	1	2	yes	complemented
19	C₂×ΓL(1,𝔽₈)	C₂×C₈	1	1	yes	characteristic
20	C₂×C₁₆	—	7	1	no	Sylow
21	C₂²×C₇	C₃×C₈	1	1	yes	characteristic
22	C₄×C₇	C₃×C₈	1	1	yes	characteristic
23	C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄	—	7	1	no	—
25	C₃×C₈	—	7	1	no	—
26	C₃×C₈	—	7	1	no	—
27	ΓL(1,𝔽₈)	C₂×C₁₆	1	1	yes	characteristic, complemented, Hall
28	C₂×C₈	AGL(1,𝔽₇)	1	1	yes	characteristic
29	C₁₆	—	7	2	no	—
30	C₂×C₇	C₃×C₁₆	1	2	yes	complemented
31	C₂×C₇	C₂×C₃×C₈	1	1	yes	characteristic
32	C₂²×C₃	—	7	1	no	—
33	C₃×C₄	—	7	1	no	—
34	C₃×C₄	—	7	1	no	—
35	C₂×C₄	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
36	C₈	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
37	C₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
38	C₇	C₂×C₃×C₁₆	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃	—	7	2	no	—
40	C₂×C₃	—	7	1	no	—
41	C₂²	ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic
42	C₄	ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic
43	C₄	C₂×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
44	C₃	—	7	1	no	Sylow
45	C₂	ΓL(1,𝔽₈)⋊C₁₆	1	2	yes	direct product factor
46	C₂	C₂×ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂×C₃×C₁₆
Frattini	C₈	C₂×AGL(1,𝔽₇)
Fitting	C₂×C₇×C₈	C₂×C₃
socle	C₂²×C₇	C₃×C₈
prime radical	C₂²×ΓL(1,𝔽₈)	C₈
cosocle	C₇×C₈	C₂²×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. ΓL(1,𝔽₈)⋊C₁₆	C₂	direct (N × H)
3. C₂×C₇⋊C₁₆	C₃	semidirect
8. C₇⋊C₁₆	C₂×C₃	semidirect
18. C₂×ΓL(1,𝔽₈)	C₁₆	semidirect
27. ΓL(1,𝔽₈)	C₂×C₁₆	semidirect
30. C₂×C₇	C₃×C₁₆	semidirect
38. C₇	C₂×C₃×C₁₆	semidirect
45. C₂	ΓL(1,𝔽₈)⋊C₁₆	direct (N × H)

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
ΓL(1,𝔽₈)⋊C₁₆	2
C₂×C₇⋊C₁₆	3
C₂×C₃×C₁₆	7

Subgroups — C2×ΓL(1,𝔽8)⋊C16 #10620439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)⋊C₁₆ #10620439