Subgroups — ΓL(1,𝔽₈)×H(3,𝔽₄) #13060722

Order 1344.

520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×#124	C₂	1	9	yes	maximal, complemented
2	ΓL(1,𝔽₈)×#120	C₂	1	6	yes	maximal, complemented
3	C₇×H(3,𝔽₄)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂⁴×ΓL(1,𝔽₈)	C₂²	1	2	yes	complemented
5	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	3	yes	complemented
6	C₂×D₄×ΓL(1,𝔽₈)	C₂²	1	9	yes	—
7	C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	1	18	yes	complemented
8	C₄²×ΓL(1,𝔽₈)	C₂²	1	3	yes	complemented
9	C₇×#124	C₂×C₃	1	9	yes	complemented
10	C₇×#120	C₂×C₃	1	6	yes	complemented
11	C₃×H(3,𝔽₄)	—	7	1	no	maximal, Hall
12	C₂³×ΓL(1,𝔽₈)	—	4	6	no	—
13	C₂³×ΓL(1,𝔽₈)	C₂³	1	6	yes	—
14	Q₈×ΓL(1,𝔽₈)	—	4	3	no	—
15	D₄×ΓL(1,𝔽₈)	—	4	9	no	—
16	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	9	yes	—
17	C₂⁴×C₇	C₂²×C₃	1	2	yes	complemented
18	C₂×C₇×Q₈	C₂²×C₃	1	3	yes	complemented
19	C₂×C₇×D₄	C₂²×C₃	1	9	yes	—
20	C₇×C₂²⋊C₄	C₂²×C₃	1	18	yes	complemented
21	C₄²×C₇	C₂²×C₃	1	3	yes	complemented
22	C₃×#124	—	7	9	no	—
23	C₃×#120	—	7	6	no	—
24	C₂²×ΓL(1,𝔽₈)	—	2	18	no	—
25	C₂²×ΓL(1,𝔽₈)	—	4	8	no	—
26	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
27	C₄×ΓL(1,𝔽₈)	—	2	9	no	—
28	H(3,𝔽₄)	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
29	C₂³×C₇	—	4	6	no	—
30	C₂³×C₇	C₂³×C₃	1	6	yes	—
31	C₇×Q₈	—	4	3	no	—
32	C₇×D₄	—	4	9	no	—
33	C₂×C₄×C₇	C₂³×C₃	1	9	yes	—
34	C₂⁴×C₃	—	7	2	no	—
35	C₂×C₃×Q₈	—	7	3	no	—
36	C₂×C₃×D₄	—	7	9	no	—
37	C₃×C₂²⋊C₄	—	7	18	no	—
38	C₃×C₄²	—	7	3	no	—
39	C₂×ΓL(1,𝔽₈)	—	4	6	no	—
40	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	3	yes	—
41	#124	C₂×ΓL(1,𝔽₈)	1	9	yes	complemented
42	#120	C₂×ΓL(1,𝔽₈)	1	6	yes	complemented
43	C₂²×C₇	—	2	18	no	—
44	C₂²×C₇	—	4	8	no	—
45	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
46	C₄×C₇	—	2	9	no	—
47	C₂³×C₃	—	7	6	no	—
48	C₂³×C₃	—	28	6	no	—
49	C₃×Q₈	—	28	3	no	—
50	C₃×D₄	—	28	9	no	—
51	C₂×C₃×C₄	—	7	9	no	—
52	ΓL(1,𝔽₈)	H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Hall
53	C₂⁴	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
54	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	3	yes	complemented
55	C₂×D₄	C₂²×ΓL(1,𝔽₈)	1	9	yes	—
56	C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	1	18	yes	complemented
57	C₄²	C₂²×ΓL(1,𝔽₈)	1	3	yes	complemented
58	C₂×C₇	—	4	6	no	—
59	C₂×C₇	C₃×H(5,𝔽₂)	1	3	yes	—
60	C₂²×C₃	—	7	1	no	—
61	C₂²×C₃	—	14	18	no	—
62	C₂²×C₃	—	28	8	no	—
63	C₃×C₄	—	14	9	no	—
64	C₂³	—	4	6	no	—
65	C₂³	C₂³×ΓL(1,𝔽₈)	1	6	yes	—
66	Q₈	—	4	3	no	—
67	D₄	—	4	9	no	—
68	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	9	yes	—
69	C₇	C₃×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	3	no	—
71	C₂×C₃	—	28	6	no	—
72	C₂²	—	2	18	no	—
73	C₂²	—	4	8	no	—
74	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
75	C₄	—	2	9	no	—
76	C₃	—	7	1	no	Sylow
77	C₂	—	4	6	no	—
78	C₂	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×H(3,𝔽₄)	C₃
hypercenter	H(3,𝔽₄)	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×H(3,𝔽₄)
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#124	C₂	semidirect
2. ΓL(1,𝔽₈)×#120	C₂	semidirect
3. C₇×H(3,𝔽₄)	C₃	semidirect
4. C₂⁴×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₂×Q₈×ΓL(1,𝔽₈)	C₂²	semidirect
7. C₂²⋊C₄×ΓL(1,𝔽₈)	C₂²	semidirect
8. C₄²×ΓL(1,𝔽₈)	C₂²	semidirect
9. C₇×#124	C₂×C₃	semidirect
10. C₇×#120	C₂×C₃	semidirect
17. C₂⁴×C₇	C₂²×C₃	semidirect
18. C₂×C₇×Q₈	C₂²×C₃	semidirect
20. C₇×C₂²⋊C₄	C₂²×C₃	semidirect
21. C₄²×C₇	C₂²×C₃	semidirect
28. H(3,𝔽₄)	ΓL(1,𝔽₈)	direct (N × H)
41. #124	C₂×ΓL(1,𝔽₈)	semidirect
42. #120	C₂×ΓL(1,𝔽₈)	semidirect
52. ΓL(1,𝔽₈)	H(3,𝔽₄)	direct (N × H)
53. C₂⁴	C₂²×ΓL(1,𝔽₈)	semidirect
54. C₂×Q₈	C₂²×ΓL(1,𝔽₈)	semidirect
56. C₂²⋊C₄	C₂²×ΓL(1,𝔽₈)	semidirect
57. C₄²	C₂²×ΓL(1,𝔽₈)	semidirect
69. C₇	C₃×H(3,𝔽₄)	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×#124	2
ΓL(1,𝔽₈)×#120	2
C₇×H(3,𝔽₄)	3
C₃×H(3,𝔽₄)	7

Subgroups — ΓL(1,𝔽8)×H(3,𝔽4) #13060722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×H(3,𝔽₄) #13060722