Subgroups — C₂×C₂²⋊#170 #65797

Order 320.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×#170	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#170	C₂	1	4	yes	maximal, direct product factor
4	C₂⁴×C₅	C₄	1	1	yes	characteristic
5	C₂²×Dic₅	—	2	4	no	—
6	C₂²×Dic₅	C₄	1	2	yes	—
7	C₂²×Dic₅	C₄	1	1	yes	characteristic
8	C₂²×Dic₅	C₂²	1	2	yes	—
9	C₂²×Dic₅	C₂²	1	1	yes	characteristic
10	C₂×#170	—	2	4	no	—
11	C₂×#170	C₂²	1	4	yes	complemented
12	C₂×C₂²⋊C₈	—	5	1	no	maximal, Sylow
13	C₂³×C₅	—	2	2	no	—
14	C₂³×C₅	—	2	2	no	—
15	C₂³×C₅	C₈	1	4	yes	complemented
16	C₂³×C₅	C₂×C₄	1	2	yes	—
17	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₅	—	2	2	no	—
19	C₂×Dic₅	—	2	4	no	—
20	C₂×Dic₅	—	2	4	no	—
21	C₂×Dic₅	C₂×C₄	1	2	yes	—
22	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₅	D₄	1	4	yes	—
24	C₂×Dic₅	C₂³	1	1	yes	characteristic
25	#170	—	2	4	no	—
26	C₂³×C₄	—	5	1	no	—
27	C₂²×C₈	—	5	2	no	—
28	C₂²⋊C₈	—	5	4	no	—
29	C₂²×C₅	—	2	8	no	—
30	C₂²×C₅	—	2	2	no	—
31	C₂²×C₅	—	2	2	no	—
32	C₂²×C₅	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₅	C₂×C₈	1	2	yes	—
34	C₂²×C₅	C₂×C₈	1	4	yes	complemented
35	C₂²×C₅	M₁₆	1	2	yes	—
36	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
37	Dic₅	—	2	2	no	—
38	Dic₅	C₂²⋊C₄	1	2	yes	—
39	Dic₅	C₂×D₄	1	2	yes	—
40	C₂⁴	AGL(1,𝔽₅)	1	1	yes	characteristic
41	C₂²×C₄	—	5	1	no	—
42	C₂²×C₄	—	5	2	no	—
43	C₂²×C₄	—	5	2	no	—
44	C₂²×C₄	—	5	1	no	—
45	C₂²×C₄	—	10	4	no	—
46	C₂×C₈	—	5	4	no	—
47	C₂×C₈	—	10	4	no	—
48	C₂×C₅	—	2	4	no	—
49	C₂×C₅	C₂²⋊C₈	1	4	yes	complemented
50	C₂×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₅	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
53	C₂³	—	2	2	no	—
54	C₂³	—	2	2	no	—
55	C₂³	#170	1	4	yes	complemented
56	C₂³	C₂×AGL(1,𝔽₅)	1	2	yes	—
57	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂×C₄	—	5	4	no	—
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	5	1	no	—
61	C₂×C₄	—	5	2	no	—
62	C₂×C₄	—	10	4	no	—
63	C₂×C₄	—	10	4	no	—
64	C₂×C₄	—	10	2	no	—
65	C₈	—	10	4	no	—
66	C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	—	2	8	no	—
70	C₂²	C₂×#170	1	2	yes	—
71	C₂²	C₂×#170	1	4	yes	complemented
72	C₂²	#704	1	2	yes	—
73	C₂²	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
74	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
75	C₄	—	5	2	no	—
76	C₄	—	5	2	no	—
77	C₄	—	10	2	no	—
78	C₂	—	2	4	no	—
79	C₂	C₂²⋊#170	1	4	yes	direct product factor
80	C₂	C₂²×#170	1	1	yes	characteristic
81	C₂	C₂×#704	1	1	yes	characteristic
82	C₂	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₈
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting, prime radical	C₂⁴×C₅	C₄
socle	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂⁴	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂²×#170	C₂	semidirect
3. C₂²⋊#170	C₂	direct (N × H)
11. C₂×#170	C₂²	semidirect
15. C₂³×C₅	C₈	semidirect
34. C₂²×C₅	C₂×C₈	semidirect
49. C₂×C₅	C₂²⋊C₈	semidirect
55. C₂³	#170	semidirect
66. C₅	C₂×C₂²⋊C₈	semidirect
71. C₂²	C₂×#170	semidirect
79. C₂	C₂²⋊#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
C₂³×Dic₅	2
C₂²×#170	2
C₂²⋊#170	2
C₂×C₂²⋊C₈	5

Subgroups — C2×C22⋊#170 #65797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊#170 #65797