Subgroups — Q₈²×ΓL(1,𝔽₈) #13060719

Order 1344.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₄⋊Q₈	C₂	1	9	yes	maximal
2	C₄×Q₈×ΓL(1,𝔽₈)	C₂	1	6	yes	maximal
3	C₇×Q₈²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	6	yes	—
5	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
6	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	18	yes	—
7	C₄²×ΓL(1,𝔽₈)	C₂²	1	9	yes	—
8	C₇×C₄⋊Q₈	C₂×C₃	1	9	yes	—
9	C₄×C₇×Q₈	C₂×C₃	1	6	yes	—
10	C₃×Q₈²	—	7	1	no	maximal, Hall
11	Q₈×ΓL(1,𝔽₈)	—	4	6	no	—
12	Q₈×ΓL(1,𝔽₈)	Q₈	1	8	yes	direct product factor
13	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	9	yes	—
14	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	6	yes	—
15	C₂×C₇×Q₈	C₂²×C₃	1	6	yes	—
16	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
17	C₇×C₄⋊C₄	C₂²×C₃	1	18	yes	—
18	C₄²×C₇	C₂²×C₃	1	9	yes	—
19	C₃×C₄⋊Q₈	—	7	9	no	—
20	C₃×C₄×Q₈	—	7	6	no	—
21	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
22	C₄×ΓL(1,𝔽₈)	—	2	9	no	—
23	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	12	yes	—
24	Q₈²	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
25	C₇×Q₈	—	4	6	no	—
26	C₇×Q₈	C₃×Q₈	1	8	yes	complemented
27	C₂×C₄×C₇	C₂³×C₃	1	9	yes	—
28	C₂×C₄×C₇	C₂³×C₃	1	6	yes	—
29	C₂×C₃×Q₈	—	7	2	no	—
30	C₂×C₃×Q₈	—	7	6	no	—
31	C₃×C₄⋊C₄	—	7	18	no	—
32	C₃×C₄²	—	7	9	no	—
33	C₂×ΓL(1,𝔽₈)	C₂²×Q₈	1	2	yes	—
34	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	1	yes	characteristic
35	C₄⋊Q₈	C₂×ΓL(1,𝔽₈)	1	9	yes	—
36	C₄×Q₈	C₂×ΓL(1,𝔽₈)	1	6	yes	—
37	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
38	C₄×C₇	—	2	9	no	—
39	C₄×C₇	C₂×C₃×Q₈	1	12	yes	—
40	C₃×Q₈	—	7	8	no	—
41	C₃×Q₈	—	28	6	no	—
42	C₂×C₃×C₄	—	7	6	no	—
43	C₂×C₃×C₄	—	7	9	no	—
44	ΓL(1,𝔽₈)	Q₈²	1	1	yes	characteristic, direct product factor, Hall
45	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	6	yes	—
46	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
47	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	18	yes	—
48	C₄²	C₂²×ΓL(1,𝔽₈)	1	9	yes	—
49	C₂×C₇	C₂²×C₃×Q₈	1	2	yes	—
50	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₃	—	7	1	no	—
52	C₃×C₄	—	7	12	no	—
53	C₃×C₄	—	14	9	no	—
54	Q₈	—	4	6	no	—
55	Q₈	Q₈×ΓL(1,𝔽₈)	1	8	yes	direct product factor
56	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	9	yes	—
57	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	6	yes	—
58	C₇	C₃×Q₈²	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₃	—	7	2	no	—
60	C₂×C₃	—	7	1	no	—
61	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic
62	C₄	—	2	9	no	—
63	C₄	C₂×Q₈×ΓL(1,𝔽₈)	1	12	yes	—
64	C₃	—	7	1	no	Sylow
65	C₂	C₂²×Q₈×ΓL(1,𝔽₈)	1	2	yes	—
66	C₂	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×Q₈²	C₃
prime radical	C₂²×ΓL(1,𝔽₈)	C₂⁴
hypercenter	Q₈²	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×Q₈²
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ Q₈²

Complements

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

N	H ≅ G/N	type
3. C₇×Q₈²	C₃	semidirect
12. Q₈×ΓL(1,𝔽₈)	Q₈	direct (N × H)
24. Q₈²	ΓL(1,𝔽₈)	direct (N × H)
26. C₇×Q₈	C₃×Q₈	semidirect
44. ΓL(1,𝔽₈)	Q₈²	direct (N × H)
55. Q₈	Q₈×ΓL(1,𝔽₈)	direct (N × H)
58. C₇	C₃×Q₈²	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)×C₄⋊Q₈	2
C₄×Q₈×ΓL(1,𝔽₈)	2
C₇×Q₈²	3
C₃×Q₈²	7

Subgroups — Q82×ΓL(1,𝔽8) #13060719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈²×ΓL(1,𝔽₈) #13060719