Subgroups — C₅×M₁₆×ΓL(1,𝔽₈) #13132251

Order 1680.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
3	C₅×C₇×M₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₅×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	C₄×C₅×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
6	C₄×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	M₁₆×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₄×C₅×C₇	C₂×C₃	1	1	yes	characteristic
9	C₅×C₇×C₈	C₂×C₃	1	2	yes	complemented
10	C₃×C₅×M₁₆	—	7	1	no	maximal, Hall
11	C₂×C₅×ΓL(1,𝔽₈)	—	2	1	no	—
12	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic
14	C₈×ΓL(1,𝔽₈)	C₂×C₅	1	2	yes	complemented
15	C₂²×C₅×C₇	C₃×C₄	1	1	yes	characteristic
16	C₄×C₅×C₇	C₃×C₄	1	1	yes	characteristic
17	C₄×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
18	C₂×C₃×C₄×C₅	—	7	1	no	—
19	C₃×C₅×C₈	—	7	2	no	—
20	C₇×M₁₆	C₃×C₅	1	1	yes	characteristic, complemented, Hall
21	C₅×ΓL(1,𝔽₈)	M₁₆	1	1	yes	characteristic, direct product factor, Hall
22	C₂²×ΓL(1,𝔽₈)	C₄×C₅	1	1	yes	characteristic
23	C₄×ΓL(1,𝔽₈)	C₄×C₅	1	1	yes	characteristic
24	C₄×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
25	C₅×M₁₆	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₅×C₇	—	2	1	no	—
27	C₂×C₅×C₇	C₂×C₃×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₅	—	7	1	no	—
29	C₃×C₄×C₅	—	7	1	no	—
30	C₃×C₄×C₅	—	7	1	no	—
31	C₂×C₄×C₇	C₂×C₃×C₅	1	1	yes	characteristic
32	C₇×C₈	C₂×C₃×C₅	1	2	yes	complemented
33	C₃×M₁₆	—	7	1	no	Hall
34	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
35	C₂×ΓL(1,𝔽₈)	C₂×C₄×C₅	1	1	yes	characteristic
36	C₂×C₄×C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₅×C₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
38	C₅×C₇	C₃×M₁₆	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₅	—	7	1	no	—
40	C₂×C₃×C₅	—	14	1	no	—
41	C₂²×C₇	C₃×C₄×C₅	1	1	yes	characteristic
42	C₄×C₇	C₃×C₄×C₅	1	1	yes	characteristic
43	C₄×C₇	C₂²×C₃×C₅	1	1	yes	characteristic
44	C₂×C₃×C₄	—	7	1	no	—
45	C₃×C₈	—	7	2	no	—
46	ΓL(1,𝔽₈)	C₅×M₁₆	1	1	yes	characteristic, direct product factor, Hall
47	C₂²×C₅	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
48	C₄×C₅	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
49	C₄×C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
50	M₁₆	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
51	C₃×C₅	—	7	1	no	Hall
52	C₂×C₇	—	2	1	no	—
53	C₂×C₇	C₂×C₃×C₄×C₅	1	1	yes	characteristic
54	C₂²×C₃	—	7	1	no	—
55	C₃×C₄	—	7	1	no	—
56	C₃×C₄	—	7	1	no	—
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
59	C₂×C₄	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
60	C₈	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
61	C₇	C₃×C₅×M₁₆	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	—	7	1	no	—
63	C₂×C₃	—	14	1	no	—
64	C₅	M₁₆×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	C₄×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
66	C₄	C₄×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
67	C₄	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
68	C₃	—	7	1	no	Sylow
69	C₂	—	2	1	no	—
70	C₂	C₂×C₄×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂×C₃×C₄×C₅
Frattini	C₄	C₂²×C₅×ΓL(1,𝔽₈)
Fitting	C₅×C₇×M₁₆	C₃
socle	C₂×C₅×C₇	C₂×C₃×C₄
prime radical	C₂²×C₅×ΓL(1,𝔽₈)	C₄
cosocle	C₄×C₇	C₂²×C₃×C₅
hypercenter	C₅×M₁₆	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×C₅×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₅ ◁ C₅×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×ΓL(1,𝔽₈)	C₂	semidirect
3. C₅×C₇×M₁₆	C₃	semidirect
7. M₁₆×ΓL(1,𝔽₈)	C₅	direct (N × H)
9. C₅×C₇×C₈	C₂×C₃	semidirect
14. C₈×ΓL(1,𝔽₈)	C₂×C₅	semidirect
20. C₇×M₁₆	C₃×C₅	semidirect
21. C₅×ΓL(1,𝔽₈)	M₁₆	direct (N × H)
25. C₅×M₁₆	ΓL(1,𝔽₈)	direct (N × H)
32. C₇×C₈	C₂×C₃×C₅	semidirect
37. C₅×C₈	C₂×ΓL(1,𝔽₈)	semidirect
38. C₅×C₇	C₃×M₁₆	semidirect
46. ΓL(1,𝔽₈)	C₅×M₁₆	direct (N × H)
50. M₁₆	C₅×ΓL(1,𝔽₈)	direct (N × H)
60. C₈	C₂×C₅×ΓL(1,𝔽₈)	semidirect
61. C₇	C₃×C₅×M₁₆	semidirect
64. C₅	M₁₆×ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×ΓL(1,𝔽₈)	2
C₅×C₈×ΓL(1,𝔽₈)	2
C₅×C₇×M₁₆	3
M₁₆×ΓL(1,𝔽₈)	5
C₃×C₅×M₁₆	7

Subgroups — C5×M16×ΓL(1,𝔽8) #13132251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×M₁₆×ΓL(1,𝔽₈) #13132251