Subgroups — C₂×#4004 #65752

Order 320.

138 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	#4004	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	—	2	4	no	—
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	—	2	2	no	—
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	2	yes	—
18	C₂×Dic₅	D₄	1	2	yes	—
19	C₂×Dic₅	Q₈	1	2	yes	—
20	C₂×Dic₅	C₂³	1	1	yes	characteristic
21	#170	—	2	4	no	—
22	C₂²×C₈	—	5	2	no	—
23	C₂×C₄²	—	5	1	no	—
24	C₄⋊C₈	—	5	4	no	—
25	C₂²×C₅	C₄⋊C₄	1	2	yes	—
26	C₂²×C₅	C₂×C₈	1	2	yes	—
27	C₂²×C₅	M₁₆	1	2	yes	—
28	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₅	—	2	2	no	—
30	Dic₅	C₄⋊C₄	1	2	yes	—
31	Dic₅	C₂×C₈	1	4	yes	complemented
32	Dic₅	C₂×D₄	1	1	yes	characteristic
33	Dic₅	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄	—	5	1	no	—
35	C₂²×C₄	—	5	1	no	—
36	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
37	C₂×C₈	—	5	4	no	—
38	C₂×C₈	—	10	4	no	—
39	C₄²	—	5	2	no	—
40	C₄²	—	5	2	no	—
41	C₂×C₅	C₄⋊C₈	1	4	yes	complemented
42	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₅	C₂²×C₈	1	1	yes	characteristic
44	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
45	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	5	4	no	—
48	C₂×C₄	—	5	2	no	—
49	C₂×C₄	—	5	2	no	—
50	C₂×C₄	—	5	1	no	—
51	C₂×C₄	—	5	2	no	—
52	C₂×C₄	—	5	1	no	—
53	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
54	C₈	—	10	4	no	—
55	C₅	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
56	C₂²	D₅⋊C₈	1	2	yes	—
57	C₂²	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
58	C₂²	#704	1	2	yes	—
59	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	5	4	no	—
62	C₄	—	5	1	no	—
63	C₄	—	5	2	no	—
64	C₄	—	5	1	no	—
65	C₂	#4004	1	4	yes	direct product factor
66	C₂	C₂×D₅⋊C₈	1	1	yes	characteristic
67	C₂	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₂	C₂×#704	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	C₂²×C₄×C₅	C₄
socle, prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. #4004	C₂	direct (N × H)
15. C₂×Dic₅	C₈	semidirect
31. Dic₅	C₂×C₈	semidirect
41. C₂×C₅	C₄⋊C₈	semidirect
55. C₅	C₂×C₄⋊C₈	semidirect
65. C₂	#4004	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
#4004	2
C₂×C₄⋊C₈	5

Subgroups — C2×#4004 #65752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4004 #65752