Subgroups — 2₋¹⁺⁴⋊ΓL(1,𝔽₈) #10621616

Order 672.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini	C₂	C₂²×C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇×Q₈	C₂²×C₃
Fitting	C₇×2₋¹⁺⁴	C₃
socle	C₂×C₇	C₂²×A₄
hypercenter	D₄	C₂²⋊ΓL(1,𝔽₈)
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #41⋊ΓL(1,𝔽₈)	C₂	semidirect
2. C₂×C₇⋊SL(2,𝔽₃)	C₂	semidirect
3. C₇×2₋¹⁺⁴	C₃	semidirect
8. C₇×#41	C₂×C₃	semidirect
10. C₂×C₇×Q₈	C₂×C₃	semidirect
27. 2₋¹⁺⁴	ΓL(1,𝔽₈)	semidirect
38. #41	C₂×ΓL(1,𝔽₈)	semidirect
40. C₂×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
55. C₇	2₋¹⁺⁴⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#41⋊ΓL(1,𝔽₈)	2
C₂×C₇⋊SL(2,𝔽₃)	2
C₇×2₋¹⁺⁴	3
D₄×ΓL(1,𝔽₈)	4
2₋¹⁺⁴⋊C₃	7

Subgroups — 2−1+4⋊ΓL(1,𝔽8) #10621616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 2₋¹⁺⁴⋊ΓL(1,𝔽₈) #10621616