Subgroups — C₂×#3992 #65717

Order 320.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	#3992	C₂	1	4	yes	maximal, direct product factor
5	C₂²×AGL(1,𝔽₅)	C₄	1	2	yes	—
6	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₂×#170	C₄	1	2	yes	complemented
8	C₄×AGL(1,𝔽₅)	C₄	1	2	yes	—
9	C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
10	D₅⋊C₈	C₄	1	2	yes	complemented
11	D₅⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₈	C₄	1	1	yes	characteristic, complemented
13	C₂×#168	C₄	1	1	yes	characteristic, complemented
14	C₈×D₅	C₄	1	2	yes	complemented
15	C₈×D₅	C₂²	1	2	yes	—
16	C₂×#97	—	5	1	no	maximal, Sylow
17	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
18	C₂×AGL(1,𝔽₅)	—	2	4	no	—
19	C₂×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₄×D₅	C₂³	1	1	yes	characteristic
26	#170	C₂×C₄	1	4	yes	complemented
27	C₅×C₈	C₂×C₄	1	2	yes	complemented
28	#168	C₂×C₄	1	2	yes	complemented
29	C₂²×C₈	—	5	1	no	—
30	C₂²×C₈	—	5	1	no	—
31	C₂×C₄²	—	5	1	no	—
32	#97	—	5	4	no	—
33	C₂²×C₅	C₄²	1	1	yes	characteristic
34	C₂×D₅	C₄²	1	1	yes	characteristic
35	C₂×D₅	M₁₆	1	4	yes	—
36	C₂×D₅	C₂²×C₄	1	1	yes	characteristic
37	AGL(1,𝔽₅)	—	2	4	no	—
38	C₄×C₅	C₄²	1	1	yes	characteristic
39	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	Dic₅	C₄²	1	1	yes	characteristic
41	Dic₅	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄	—	5	1	no	—
43	C₂²×C₄	—	5	2	no	—
44	C₂×C₈	—	5	2	no	—
45	C₂×C₈	—	5	2	no	—
46	C₂×C₈	—	5	2	no	—
47	C₂×C₈	—	5	1	no	—
48	C₂×C₈	—	5	2	no	—
49	C₂×C₈	—	5	2	no	—
50	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
51	C₄²	—	5	2	no	—
52	C₄²	—	5	2	no	—
53	C₂×C₅	#97	1	2	yes	complemented
54	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
55	D₅	#97	1	2	yes	complemented
56	D₅	C₂×M₁₆	1	2	yes	—
57	C₂³	—	5	1	no	—
58	C₂×C₄	—	5	1	no	—
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	5	1	no	—
61	C₂×C₄	—	5	1	no	—
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	5	4	no	—
64	C₂×C₄	—	10	4	no	—
65	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₈	—	5	4	no	—
67	C₈	—	5	2	no	—
68	C₈	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
69	C₅	C₂×#97	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	5	1	no	—
71	C₂²	—	5	1	no	—
72	C₂²	—	5	4	no	—
73	C₂²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₄	—	5	1	no	—
75	C₄	—	5	1	no	—
76	C₄	—	10	4	no	—
77	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₂	—	5	2	no	—
80	C₂	—	5	2	no	—
81	C₂	#3992	1	2	yes	direct product factor
82	C₂	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₄²
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₄×D₅	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#97
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. #3992	C₂	direct (N × H)
7. C₂×#170	C₄	semidirect
10. D₅⋊C₈	C₄	semidirect
12. C₂×C₅×C₈	C₄	semidirect
13. C₂×#168	C₄	semidirect
14. C₈×D₅	C₄	semidirect
26. #170	C₂×C₄	semidirect
27. C₅×C₈	C₂×C₄	semidirect
28. #168	C₂×C₄	semidirect
50. C₂×C₈	AGL(1,𝔽₅)	semidirect
53. C₂×C₅	#97	semidirect
55. D₅	#97	semidirect
68. C₈	C₂×AGL(1,𝔽₅)	semidirect
69. C₅	C₂×#97	semidirect
81. C₂	#3992	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×AGL(1,𝔽₅)	2
C₂×D₅⋊C₈	2
C₂×C₈×D₅	2
#3992	2
C₂×#97	5

Subgroups — C2×#3992 #65717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3992 #65717