Subgroups — C₂²×Dic₅×ΓL(1,𝔽₈) #13132573

Order 1680.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅×ΓL(1,𝔽₈)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₇×Dic₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₅×ΓL(1,𝔽₈)	C₄	1	4	yes	complemented
5	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
6	Dic₅×ΓL(1,𝔽₈)	C₂²	1	4	yes	direct product factor
7	C₂²×C₄×ΓL(1,𝔽₈)	—	5	1	no	maximal, Hall
8	C₂³×C₅×C₇	C₂×C₃	1	1	yes	characteristic
9	C₂×C₇×Dic₅	C₂×C₃	1	6	yes	complemented
10	C₂²×C₃×Dic₅	—	7	1	no	maximal, Hall
11	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₄	1	6	yes	complemented
12	C₂×C₅×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
13	C₂³×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic
14	C₂×C₄×ΓL(1,𝔽₈)	—	5	6	no	—
15	C₂²×C₅×C₇	C₃×C₄	1	4	yes	complemented
16	C₂²×C₅×C₇	C₂²×C₃	1	3	yes	—
17	C₇×Dic₅	C₂²×C₃	1	4	yes	complemented
18	C₂³×C₃×C₅	—	7	1	no	—
19	C₂×C₃×Dic₅	—	7	6	no	—
20	C₂²×C₄×C₇	—	5	1	no	Hall
21	C₅×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
22	C₂²×ΓL(1,𝔽₈)	Dic₅	1	4	yes	direct product factor
23	C₂²×ΓL(1,𝔽₈)	C₂×D₅	1	3	yes	—
24	C₄×ΓL(1,𝔽₈)	—	5	4	no	—
25	C₂²×Dic₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₅×C₇	C₂×C₃×C₄	1	6	yes	complemented
27	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
28	C₂²×C₃×C₅	—	7	4	no	—
29	C₂²×C₃×C₅	—	7	3	no	—
30	C₃×Dic₅	—	7	4	no	—
31	C₂³×C₇	C₃×D₅	1	1	yes	characteristic
32	C₂×C₄×C₇	—	5	6	no	—
33	C₂²×C₃×C₄	—	35	1	no	Hall
34	C₂×ΓL(1,𝔽₈)	C₂×Dic₅	1	6	yes	direct product factor
35	C₂×ΓL(1,𝔽₈)	C₂²×D₅	1	1	yes	characteristic
36	C₂³×C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₂×Dic₅	C₂×ΓL(1,𝔽₈)	1	6	yes	direct product factor
38	C₅×C₇	C₂²×C₃×C₄	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₅	—	7	6	no	—
40	C₂×C₃×C₅	—	7	1	no	—
41	C₂²×C₇	C₃×Dic₅	1	4	yes	complemented
42	C₂²×C₇	C₂×C₃×D₅	1	3	yes	—
43	C₄×C₇	—	5	4	no	—
44	C₂³×C₃	—	7	1	no	—
45	C₂×C₃×C₄	—	35	6	no	—
46	ΓL(1,𝔽₈)	C₂²×Dic₅	1	1	yes	characteristic, direct product factor, Hall
47	C₂²×C₅	C₄×ΓL(1,𝔽₈)	1	4	yes	complemented
48	C₂²×C₅	C₂²×ΓL(1,𝔽₈)	1	3	yes	—
49	Dic₅	C₂²×ΓL(1,𝔽₈)	1	4	yes	direct product factor
50	C₂²×C₄	—	5	1	no	Sylow
51	C₃×C₅	—	7	1	no	Hall
52	C₂×C₇	C₂×C₃×Dic₅	1	6	yes	complemented
53	C₂×C₇	C₂²×C₃×D₅	1	1	yes	characteristic
54	C₂²×C₃	—	7	4	no	—
55	C₂²×C₃	—	7	3	no	—
56	C₃×C₄	—	35	4	no	—
57	C₂×C₅	C₂×C₄×ΓL(1,𝔽₈)	1	6	yes	complemented
58	C₂×C₅	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
59	C₂³	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
60	C₂×C₄	—	5	6	no	—
61	C₇	C₂²×C₃×Dic₅	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	—	7	6	no	—
63	C₂×C₃	—	7	1	no	—
64	C₅	C₂²×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
65	C₂²	Dic₅×ΓL(1,𝔽₈)	1	4	yes	direct product factor
66	C₂²	C₂×D₅×ΓL(1,𝔽₈)	1	3	yes	—
67	C₄	—	5	4	no	—
68	C₃	—	7	1	no	Sylow
69	C₂	C₂×Dic₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
70	C₂	C₂²×D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅×ΓL(1,𝔽₈)
commutator, hypocenter	C₅×C₇	C₂²×C₃×C₄
Frattini	C₂	C₂²×D₅×ΓL(1,𝔽₈)
Fitting, socle	C₂³×C₅×C₇	C₂×C₃
prime radical	C₂³×C₅×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₅×C₇	C₂³×C₃
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₅×ΓL(1,𝔽₈)	C₂	direct (N × H)
3. C₂²×C₇×Dic₅	C₃	semidirect
4. C₂²×C₅×ΓL(1,𝔽₈)	C₄	semidirect
6. Dic₅×ΓL(1,𝔽₈)	C₂²	direct (N × H)
9. C₂×C₇×Dic₅	C₂×C₃	semidirect
11. C₂×C₅×ΓL(1,𝔽₈)	C₂×C₄	semidirect
15. C₂²×C₅×C₇	C₃×C₄	semidirect
17. C₇×Dic₅	C₂²×C₃	semidirect
21. C₅×ΓL(1,𝔽₈)	C₂²×C₄	semidirect
22. C₂²×ΓL(1,𝔽₈)	Dic₅	direct (N × H)
25. C₂²×Dic₅	ΓL(1,𝔽₈)	direct (N × H)
26. C₂×C₅×C₇	C₂×C₃×C₄	semidirect
34. C₂×ΓL(1,𝔽₈)	C₂×Dic₅	direct (N × H)
37. C₂×Dic₅	C₂×ΓL(1,𝔽₈)	direct (N × H)
38. C₅×C₇	C₂²×C₃×C₄	semidirect
41. C₂²×C₇	C₃×Dic₅	semidirect
46. ΓL(1,𝔽₈)	C₂²×Dic₅	direct (N × H)
47. C₂²×C₅	C₄×ΓL(1,𝔽₈)	semidirect
49. Dic₅	C₂²×ΓL(1,𝔽₈)	direct (N × H)
52. C₂×C₇	C₂×C₃×Dic₅	semidirect
57. C₂×C₅	C₂×C₄×ΓL(1,𝔽₈)	semidirect
61. C₇	C₂²×C₃×Dic₅	semidirect
64. C₅	C₂²×C₄×ΓL(1,𝔽₈)	semidirect
65. C₂²	Dic₅×ΓL(1,𝔽₈)	direct (N × H)
69. C₂	C₂×Dic₅×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
C₂³×C₅×ΓL(1,𝔽₈)	2
C₂×Dic₅×ΓL(1,𝔽₈)	2
C₂²×C₇×Dic₅	3
C₂²×C₄×ΓL(1,𝔽₈)	5
C₂²×C₃×Dic₅	7

Subgroups — C22×Dic5×ΓL(1,𝔽8) #13132573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₅×ΓL(1,𝔽₈) #13132573