Subgroups — C₅×C₄⋊C₄×ΓL(1,𝔽₈) #13132249

Order 1680.

104 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	2	yes	maximal
2	C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
3	C₅×C₇×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
5	C₄×C₅×ΓL(1,𝔽₈)	—	2	2	no	—
6	C₄×C₅×ΓL(1,𝔽₈)	C₄	1	2	yes	complemented
7	C₄⋊C₄×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₄×C₅×C₇	C₂×C₃	1	2	yes	—
9	C₂×C₄×C₅×C₇	C₂×C₃	1	1	yes	characteristic
10	C₃×C₅×C₄⋊C₄	—	7	1	no	maximal, Hall
11	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
12	C₂×C₅×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
13	C₂×C₅×ΓL(1,𝔽₈)	Q₈	1	1	yes	characteristic
14	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₅	1	2	yes	—
15	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic
16	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
17	C₄×C₅×C₇	—	2	2	no	—
18	C₄×C₅×C₇	C₃×C₄	1	2	yes	complemented
19	C₂×C₃×C₄×C₅	—	7	1	no	—
20	C₂×C₃×C₄×C₅	—	7	2	no	—
21	C₇×C₄⋊C₄	C₃×C₅	1	1	yes	characteristic, complemented, Hall
22	C₅×ΓL(1,𝔽₈)	C₄⋊C₄	1	1	yes	characteristic, direct product factor, Hall
23	C₂²×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
24	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
25	C₄×ΓL(1,𝔽₈)	C₄×C₅	1	2	yes	complemented
26	C₅×C₄⋊C₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
27	C₂×C₅×C₇	C₂×C₃×C₄	1	1	yes	characteristic
28	C₂×C₅×C₇	C₃×D₄	1	1	yes	characteristic
29	C₂×C₅×C₇	C₃×Q₈	1	1	yes	characteristic
30	C₂²×C₃×C₅	—	7	1	no	—
31	C₃×C₄×C₅	—	7	2	no	—
32	C₃×C₄×C₅	—	14	2	no	—
33	C₂×C₄×C₇	C₂×C₃×C₅	1	2	yes	—
34	C₂×C₄×C₇	C₂×C₃×C₅	1	1	yes	characteristic
35	C₃×C₄⋊C₄	—	7	1	no	Hall
36	C₂×ΓL(1,𝔽₈)	C₂×C₄×C₅	1	1	yes	characteristic
37	C₂×ΓL(1,𝔽₈)	C₅×D₄	1	1	yes	characteristic
38	C₂×ΓL(1,𝔽₈)	C₅×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅	C₂×ΓL(1,𝔽₈)	1	2	yes	—
40	C₂×C₄×C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
41	C₅×C₇	C₃×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
42	C₂×C₃×C₅	—	7	1	no	—
43	C₂×C₃×C₅	—	7	1	no	—
44	C₂×C₃×C₅	—	7	1	no	—
45	C₂²×C₇	C₂²×C₃×C₅	1	1	yes	characteristic
46	C₄×C₇	—	2	2	no	—
47	C₄×C₇	C₃×C₄×C₅	1	2	yes	complemented
48	C₂×C₃×C₄	—	7	1	no	—
49	C₂×C₃×C₄	—	7	2	no	—
50	ΓL(1,𝔽₈)	C₅×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Hall
51	C₂²×C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
52	C₄×C₅	—	2	2	no	—
53	C₄×C₅	C₄×ΓL(1,𝔽₈)	1	2	yes	complemented
54	C₄⋊C₄	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
55	C₃×C₅	—	7	1	no	Hall
56	C₂×C₇	C₂×C₃×C₄×C₅	1	1	yes	characteristic
57	C₂×C₇	C₃×C₅×D₄	1	1	yes	characteristic
58	C₂×C₇	C₃×C₅×Q₈	1	1	yes	characteristic
59	C₂²×C₃	—	7	1	no	—
60	C₃×C₄	—	7	2	no	—
61	C₃×C₄	—	14	2	no	—
62	C₂×C₅	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
63	C₂×C₅	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
64	C₂×C₅	Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic
65	C₂×C₄	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	—
66	C₂×C₄	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
67	C₇	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₃	—	7	1	no	—
69	C₂×C₃	—	7	1	no	—
70	C₂×C₃	—	7	1	no	—
71	C₅	C₄⋊C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	C₄×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
75	C₃	—	7	1	no	Sylow
76	C₂	C₂×C₄×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
77	C₂	C₅×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
78	C₂	C₅×Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅×C₇×C₄⋊C₄	C₃	semidirect
6. C₄×C₅×ΓL(1,𝔽₈)	C₄	semidirect
7. C₄⋊C₄×ΓL(1,𝔽₈)	C₅	direct (N × H)
18. C₄×C₅×C₇	C₃×C₄	semidirect
21. C₇×C₄⋊C₄	C₃×C₅	semidirect
22. C₅×ΓL(1,𝔽₈)	C₄⋊C₄	direct (N × H)
25. C₄×ΓL(1,𝔽₈)	C₄×C₅	semidirect
26. C₅×C₄⋊C₄	ΓL(1,𝔽₈)	direct (N × H)
41. C₅×C₇	C₃×C₄⋊C₄	semidirect
47. C₄×C₇	C₃×C₄×C₅	semidirect
50. ΓL(1,𝔽₈)	C₅×C₄⋊C₄	direct (N × H)
53. C₄×C₅	C₄×ΓL(1,𝔽₈)	semidirect
54. C₄⋊C₄	C₅×ΓL(1,𝔽₈)	direct (N × H)
67. C₇	C₃×C₅×C₄⋊C₄	semidirect
71. C₅	C₄⋊C₄×ΓL(1,𝔽₈)	direct (N × H)
74. C₄	C₄×C₅×ΓL(1,𝔽₈)	semidirect

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×C₇×C₄⋊C₄	3
C₄⋊C₄×ΓL(1,𝔽₈)	5
C₃×C₅×C₄⋊C₄	7

Subgroups — C5×C4⋊C4×ΓL(1,𝔽8) #13132249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₄⋊C₄×ΓL(1,𝔽₈) #13132249