Subgroups — #504⋊ΓL(1,𝔽₈) #13055078

Order 1344.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131⋊ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₇×#504	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	#41⋊ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
4	C₁₆×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₇×#131	C₂×C₃	1	1	yes	characteristic
6	C₇×M₃₂	—	3	1	no	—
7	C₂×C₇×C₁₆	—	3	1	no	—
8	#504⋊C₃	—	7	1	no	maximal, Hall
9	C₇⋊SL(2,𝔽₃)	C₈	1	1	yes	characteristic
10	C₈×ΓL(1,𝔽₈)	—	4	1	no	—
11	C₇×#41	C₃×C₄	1	1	yes	characteristic
12	C₇×M₁₆	—	3	1	no	—
13	C₂×C₇×C₈	—	3	1	no	—
14	C₇×C₁₆	—	3	1	no	—
15	C₇×C₁₆	A₄	1	1	yes	characteristic
16	#131⋊C₃	—	7	1	no	—
17	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
18	#504	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
19	C₇×Q₈	C₃×C₈	1	1	yes	characteristic
20	C₇×D₄	—	3	1	no	—
21	C₂×C₄×C₇	—	3	1	no	—
22	C₇×C₈	—	3	1	no	—
23	C₇×C₈	C₂×A₄	1	1	yes	characteristic
24	#41⋊C₃	—	7	1	no	—
25	C₃×C₁₆	—	28	1	no	—
26	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
27	#131	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	M₃₂	—	3	1	no	—
29	C₂×C₁₆	—	3	1	no	—
30	C₂²×C₇	—	3	1	no	—
31	C₄×C₇	—	3	1	no	—
32	C₄×C₇	C₄×A₄	1	1	yes	characteristic
33	SL(2,𝔽₃)	—	7	1	no	—
34	C₃×C₈	—	28	1	no	—
35	ΓL(1,𝔽₈)	—	4	1	no	Hall
36	#41	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	M₁₆	—	3	1	no	—
38	C₂×C₈	—	3	1	no	—
39	C₁₆	—	3	1	no	—
40	C₁₆	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
41	C₂×C₇	—	6	1	no	—
42	C₂×C₇	C₈×A₄	1	1	yes	characteristic
43	C₃×C₄	—	28	1	no	—
44	Q₈	C₈×ΓL(1,𝔽₈)	1	1	yes	characteristic
45	D₄	—	3	1	no	—
46	C₂×C₄	—	3	1	no	—
47	C₈	—	3	1	no	—
48	C₈	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
49	C₇	#504⋊C₃	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	—	28	1	no	—
51	C₂²	—	3	1	no	—
52	C₄	—	3	1	no	—
53	C₄	C₄×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
54	C₃	—	28	1	no	Sylow
55	C₂	—	6	1	no	—
56	C₂	C₈×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₁₆	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₇×Q₈	C₃×C₈
Frattini	C₈	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#504	C₃
socle	C₂×C₇	C₈×A₄
prime radical	#41⋊ΓL(1,𝔽₈)	C₄
cosocle	C₇×#131	C₂×C₃
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
2. C₇×#504	C₃	semidirect
18. #504	ΓL(1,𝔽₈)	semidirect
49. C₇	#504⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#131⋊ΓL(1,𝔽₈)	2
C₇×#504	3
C₁₆×ΓL(1,𝔽₈)	4
#504⋊C₃	7

Subgroups — #504⋊ΓL(1,𝔽8) #13055078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #504⋊ΓL(1,𝔽₈) #13055078