Subgroups — C₂×Q₈×AGL(1,𝔽₅) #66261

Order 320.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₅	C₂	1	1	yes	maximal, characteristic
2	Q₈×AGL(1,𝔽₅)	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊AGL(1,𝔽₅)	C₂	1	3	yes	maximal
4	C₂×C₄×AGL(1,𝔽₅)	C₂	1	3	yes	maximal
5	C₂²×AGL(1,𝔽₅)	C₂²	1	3	yes	—
6	C₂²×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
7	C₂×C₅×Q₈	C₄	1	1	yes	characteristic, complemented
8	Q₈×D₅	C₄	1	4	yes	complemented
9	Q₈×D₅	C₂²	1	4	yes	—
10	C₂×C₄×D₅	C₂²	1	3	yes	—
11	C₂×Dic₁₀	C₄	1	3	yes	complemented
12	C₄⋊AGL(1,𝔽₅)	C₂²	1	12	yes	—
13	C₄×AGL(1,𝔽₅)	C₂²	1	12	yes	—
14	C₂×C₄×Q₈	—	5	1	no	maximal, Sylow
15	C₂²×D₅	C₂³	1	1	yes	characteristic
16	C₂×AGL(1,𝔽₅)	—	2	6	no	—
17	C₂×AGL(1,𝔽₅)	Q₈	1	4	yes	direct product factor
18	C₂×AGL(1,𝔽₅)	C₂³	1	6	yes	—
19	C₂×AGL(1,𝔽₅)	C₂³	1	2	yes	—
20	C₅×Q₈	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
22	C₂×Dic₅	C₂×C₄	1	3	yes	—
23	C₄×D₅	C₂×C₄	1	6	yes	—
24	C₄×D₅	C₂³	1	6	yes	—
25	Dic₁₀	C₂×C₄	1	12	yes	complemented
26	C₂²×Q₈	—	5	1	no	—
27	C₄×Q₈	—	5	8	no	—
28	C₂×C₄⋊C₄	—	5	3	no	—
29	C₂×C₄²	—	5	3	no	—
30	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₅	C₂²×C₄	1	1	yes	characteristic
32	C₂×D₅	C₂×Q₈	1	2	yes	—
33	C₂×D₅	#41	1	2	yes	—
34	C₂×D₅	C₂⁴	1	1	yes	characteristic
35	AGL(1,𝔽₅)	—	2	6	no	—
36	AGL(1,𝔽₅)	C₂×Q₈	1	4	yes	direct product factor
37	C₄×C₅	C₂²×C₄	1	6	yes	—
38	Dic₅	C₂²×C₄	1	6	yes	—
39	C₂×Q₈	—	5	4	no	—
40	C₂×Q₈	—	5	4	no	—
41	C₂×Q₈	—	5	3	no	—
42	C₂×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
43	C₂²×C₄	—	5	1	no	—
44	C₂²×C₄	—	5	3	no	—
45	C₂²×C₄	—	5	3	no	—
46	C₄⋊C₄	—	5	12	no	—
47	C₄²	—	5	12	no	—
48	C₂×C₅	C₄×Q₈	1	2	yes	complemented
49	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
50	D₅	C₄×Q₈	1	2	yes	complemented
51	D₅	C₂²×Q₈	1	1	yes	characteristic
52	D₅	C₂×#41	1	1	yes	characteristic
53	C₂³	—	5	1	no	—
54	Q₈	—	5	12	no	—
55	Q₈	C₂×AGL(1,𝔽₅)	1	4	yes	direct product factor
56	C₂×C₄	—	5	2	no	—
57	C₂×C₄	—	5	4	no	—
58	C₂×C₄	—	5	3	no	—
59	C₂×C₄	—	5	6	no	—
60	C₂×C₄	—	5	6	no	—
61	C₂×C₄	—	5	6	no	—
62	C₂×C₄	—	10	6	no	—
63	C₂×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
64	C₅	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	5	2	no	—
66	C₂²	—	5	2	no	—
67	C₂²	—	5	1	no	—
68	C₂²	—	5	1	no	—
69	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₄	—	5	4	no	—
71	C₄	—	5	6	no	—
72	C₄	—	10	6	no	—
73	C₄	C₂²×AGL(1,𝔽₅)	1	6	yes	—
74	C₂	—	5	1	no	—
75	C₂	—	5	1	no	—
76	C₂	—	5	2	no	—
77	C₂	Q₈×AGL(1,𝔽₅)	1	2	yes	direct product factor
78	C₂	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₄
Frattini	C₂	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₅×Q₈	C₄
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂²×D₅	C₂³
cosocle	C₂×D₅	C₂⁴
hypercenter	C₂×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₄×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×AGL(1,𝔽₅)	C₂	direct (N × H)
7. C₂×C₅×Q₈	C₄	semidirect
8. Q₈×D₅	C₄	semidirect
11. C₂×Dic₁₀	C₄	semidirect
17. C₂×AGL(1,𝔽₅)	Q₈	direct (N × H)
20. C₅×Q₈	C₂×C₄	semidirect
25. Dic₁₀	C₂×C₄	semidirect
36. AGL(1,𝔽₅)	C₂×Q₈	direct (N × H)
42. C₂×Q₈	AGL(1,𝔽₅)	direct (N × H)
48. C₂×C₅	C₄×Q₈	semidirect
50. D₅	C₄×Q₈	semidirect
55. Q₈	C₂×AGL(1,𝔽₅)	direct (N × H)
64. C₅	C₂×C₄×Q₈	semidirect
77. C₂	Q₈×AGL(1,𝔽₅)	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×Q₈×D₅	2
Q₈×AGL(1,𝔽₅)	2
C₂×C₄⋊AGL(1,𝔽₅)	2
C₂×C₄×AGL(1,𝔽₅)	2
C₂×C₄×Q₈	5

Subgroups — C2×Q8×AGL(1,𝔽5) #66261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×AGL(1,𝔽₅) #66261