Subgroups — C₁₆×C₂²⋊ΓL(1,𝔽₈) #13055077

Order 1344.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₂²⋊ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₂²⋊ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
4	C₁₆×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
6	C₂×C₇×C₁₆	—	3	1	no	—
7	C₂×C₇×C₁₆	—	3	1	no	—
8	A₄×C₁₆	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	C₈	1	1	yes	characteristic
10	C₈×ΓL(1,𝔽₈)	—	4	1	no	—
11	C₂²×C₄×C₇	C₃×C₄	1	1	yes	characteristic
12	C₂×C₇×C₈	—	3	1	no	—
13	C₂×C₇×C₈	—	3	1	no	—
14	C₇×C₁₆	—	3	1	no	—
15	C₇×C₁₆	A₄	1	1	yes	characteristic, complemented
16	C₈×A₄	—	7	1	no	—
17	C₂²⋊ΓL(1,𝔽₈)	C₁₆	1	1	yes	characteristic, direct product factor
18	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
19	C₂²×C₁₆	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
20	C₂³×C₇	C₃×C₈	1	1	yes	characteristic
21	C₂×C₄×C₇	—	3	1	no	—
22	C₂×C₄×C₇	—	3	1	no	—
23	C₇×C₈	—	3	1	no	—
24	C₇×C₈	C₂×A₄	1	1	yes	characteristic
25	C₄×A₄	—	7	1	no	—
26	C₃×C₁₆	—	28	1	no	—
27	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
28	C₂²×C₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₂×C₁₆	—	3	1	no	—
30	C₂×C₁₆	—	3	1	no	—
31	C₂²×C₇	—	3	1	no	—
32	C₂²×C₇	—	3	1	no	—
33	C₂²×C₇	C₃×C₁₆	1	1	yes	characteristic, complemented
34	C₄×C₇	—	3	1	no	—
35	C₄×C₇	C₄×A₄	1	1	yes	characteristic
36	C₂×A₄	—	7	1	no	—
37	C₃×C₈	—	28	1	no	—
38	ΓL(1,𝔽₈)	—	4	1	no	Hall
39	C₂²×C₄	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
40	C₂×C₈	—	3	1	no	—
41	C₂×C₈	—	3	1	no	—
42	C₁₆	—	3	1	no	—
43	C₁₆	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
44	C₂×C₇	—	3	1	no	—
45	C₂×C₇	—	3	1	no	—
46	C₂×C₇	C₈×A₄	1	1	yes	characteristic
47	A₄	—	7	1	no	—
48	C₃×C₄	—	28	1	no	—
49	C₂³	C₈×ΓL(1,𝔽₈)	1	1	yes	characteristic
50	C₂×C₄	—	3	1	no	—
51	C₂×C₄	—	3	1	no	—
52	C₈	—	3	1	no	—
53	C₈	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
54	C₇	A₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	—	28	1	no	—
56	C₂²	—	3	1	no	—
57	C₂²	—	3	1	no	—
58	C₂²	C₁₆×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
59	C₄	—	3	1	no	—
60	C₄	C₄×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
61	C₃	—	28	1	no	Sylow
62	C₂	—	3	1	no	—
63	C₂	—	3	1	no	—
64	C₂	C₈×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₁₆	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₂²×C₇	C₃×C₁₆
Frattini	C₈	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₂²×C₇×C₁₆	C₃
socle	C₂³×C₇	C₃×C₈
prime radical	C₂×C₂²⋊ΓL(1,𝔽₈)	C₈
cosocle	C₂²×C₇×C₈	C₂×C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₁₆	C₃	semidirect
15. C₇×C₁₆	A₄	semidirect
17. C₂²⋊ΓL(1,𝔽₈)	C₁₆	direct (N × H)
19. C₂²×C₁₆	ΓL(1,𝔽₈)	semidirect
33. C₂²×C₇	C₃×C₁₆	semidirect
43. C₁₆	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
54. C₇	A₄×C₁₆	semidirect
58. C₂²	C₁₆×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₂²⋊ΓL(1,𝔽₈)	2
C₂²×C₇×C₁₆	3
C₁₆×ΓL(1,𝔽₈)	4
A₄×C₁₆	7

Subgroups — C16×C22⋊ΓL(1,𝔽8) #13055077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₂²⋊ΓL(1,𝔽₈) #13055077