Subgroups — C₅×#454 #65617

Order 320.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#104	C₂	1	2	yes	maximal, complemented
5	C₅×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#41	—	2	2	no	—
7	C₅×#41	—	2	2	no	—
8	C₅×#41	C₂²	1	2	yes	—
9	C₂×C₅×Q₈	—	2	2	no	—
10	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₅×SD₁₆	—	2	2	no	—
13	C₅×D₈	—	2	2	no	—
14	C₅×M₁₆	C₂²	1	2	yes	—
15	C₅×C₂²⋊C₄	—	2	2	no	—
16	C₄²×C₅	C₂²	1	1	yes	characteristic
17	#454	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₅	—	2	1	no	—
19	C₅×Q₈	—	2	2	no	—
20	C₅×Q₈	—	2	2	no	—
21	C₅×Q₈	D₄	1	2	yes	—
22	C₅×D₄	—	2	2	no	—
23	C₅×D₄	—	2	2	no	—
24	C₅×D₄	—	2	2	no	—
25	C₅×D₄	D₄	1	2	yes	—
26	C₂×C₄×C₅	—	2	4	no	—
27	C₂×C₄×C₅	—	2	2	no	—
28	C₂×C₄×C₅	—	2	1	no	—
29	C₂×C₄×C₅	D₄	1	2	yes	—
30	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₅×C₈	—	2	2	no	—
32	2₋¹⁺⁴	C₂×C₅	1	1	yes	characteristic, complemented
33	AGL(1,ℤ/8)	C₂×C₅	1	2	yes	complemented
34	#124	C₂×C₅	1	1	yes	characteristic, complemented
35	#104	C₂×C₅	1	2	yes	complemented
36	#101	C₂×C₅	1	1	yes	characteristic, complemented
37	C₂²×C₅	—	2	2	no	—
38	C₂²×C₅	—	4	2	no	—
39	C₂²×C₅	—	4	1	no	—
40	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
41	C₄×C₅	—	2	2	no	—
42	C₄×C₅	—	2	2	no	—
43	C₄×C₅	—	4	1	no	—
44	C₄×C₅	C₂×D₄	1	2	yes	—
45	#41	—	2	2	no	—
46	#41	—	2	2	no	—
47	#41	C₂²×C₅	1	2	yes	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	C₂²×C₅	1	1	yes	characteristic
50	C₂×D₄	C₂²×C₅	1	1	yes	characteristic
51	SD₁₆	—	2	2	no	—
52	D₈	—	2	2	no	—
53	M₁₆	C₂²×C₅	1	2	yes	—
54	C₂²⋊C₄	—	2	2	no	—
55	C₄²	C₂²×C₅	1	1	yes	characteristic
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	—	4	2	no	—
58	C₂×C₅	—	8	1	no	—
59	C₂×C₅	#120	1	1	yes	characteristic
60	C₂³	—	2	1	no	—
61	Q₈	—	2	2	no	—
62	Q₈	—	2	2	no	—
63	Q₈	C₅×D₄	1	2	yes	—
64	D₄	—	2	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	2	2	no	—
67	D₄	C₅×D₄	1	2	yes	—
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	C₅×D₄	1	2	yes	—
72	C₂×C₄	C₂³×C₅	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₅	#454	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	—	2	2	no	—
76	C₂²	—	4	2	no	—
77	C₂²	—	4	1	no	—
78	C₂²	C₂×C₅×D₄	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	C₂×C₅×D₄	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	4	2	no	—
85	C₂	—	8	1	no	—
86	C₂	C₅×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×2₋¹⁺⁴	C₂	semidirect
2. C₅×AGL(1,ℤ/8)	C₂	semidirect
3. C₅×#124	C₂	semidirect
4. C₅×#104	C₂	semidirect
5. C₅×#101	C₂	semidirect
17. #454	C₅	direct (N × H)
32. 2₋¹⁺⁴	C₂×C₅	semidirect
33. AGL(1,ℤ/8)	C₂×C₅	semidirect
34. #124	C₂×C₅	semidirect
35. #104	C₂×C₅	semidirect
36. #101	C₂×C₅	semidirect
74. C₅	#454	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×2₋¹⁺⁴	2
C₅×AGL(1,ℤ/8)	2
C₅×#124	2
C₅×#104	2
C₅×#101	2
#454	5

Subgroups — C5×#454 #65617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#454 #65617