Subgroups — C₂×C₄×#170 #65746

Order 320.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂²×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₄×C₈
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle, prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄×#170	C₂	direct (N × H)
7. C₂×#170	C₄	direct (N × H)
13. C₂×C₄×C₅	C₈	semidirect
15. C₂×Dic₅	C₈	semidirect
20. #170	C₂×C₄	direct (N × H)
28. C₄×C₅	C₂×C₈	semidirect
30. Dic₅	C₂×C₈	semidirect
39. C₂×C₅	C₄×C₈	semidirect
49. C₂×C₄	#170	direct (N × H)
52. C₅	C₂×C₄×C₈	semidirect
60. C₄	C₂×#170	direct (N × H)
61. C₂	C₄×#170	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#170	2
C₂×C₄×Dic₅	2
C₄×#170	2
C₂×C₄×C₈	5

Subgroups — C2×C4×#170 #65746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#170 #65746