Subgroups — C₅×#568 #66232

Order 320.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×#131	C₂	1	8	yes	maximal, complemented
3	C₂×C₅×M₁₆	C₂	1	6	yes	maximal, complemented
4	C₅×#41	C₄	1	4	yes	—
5	C₅×#41	C₂²	1	4	yes	—
6	C₂×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₅×D₄	C₄	1	3	yes	—
8	C₂²×C₄×C₅	C₂²	1	3	yes	—
9	C₅×M₁₆	C₂²	1	4	yes	—
10	C₅×M₁₆	C₂²	1	12	yes	complemented
11	C₂×C₅×C₈	C₂²	1	12	yes	complemented
12	#568	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₅	C₂×C₄	1	3	yes	—
14	C₅×Q₈	C₂×C₄	1	4	yes	—
15	C₅×D₄	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₅	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₅	C₂³	1	6	yes	—
19	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₅×C₈	C₂³	1	8	yes	—
21	C₂×#41	C₂×C₅	1	1	yes	characteristic
22	#131	C₂×C₅	1	8	yes	complemented
23	C₂×M₁₆	C₂×C₅	1	6	yes	complemented
24	C₂²×C₅	—	4	3	no	—
25	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₅	C₂²×C₄	1	6	yes	—
27	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₅	C₂²×C₄	1	6	yes	—
29	C₄×C₅	C₂⁴	1	1	yes	characteristic
30	#41	C₄×C₅	1	4	yes	—
31	#41	C₂²×C₅	1	4	yes	—
32	C₂×Q₈	C₄×C₅	1	1	yes	characteristic
33	C₂×D₄	C₄×C₅	1	3	yes	—
34	C₂²×C₄	C₂²×C₅	1	3	yes	—
35	M₁₆	C₂²×C₅	1	4	yes	—
36	M₁₆	C₂²×C₅	1	12	yes	complemented
37	C₂×C₈	C₂²×C₅	1	12	yes	complemented
38	C₂×C₅	—	2	1	no	—
39	C₂×C₅	—	2	6	no	—
40	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
41	C₂³	C₂×C₄×C₅	1	3	yes	—
42	Q₈	C₂×C₄×C₅	1	4	yes	—
43	D₄	C₂×C₄×C₅	1	12	yes	—
44	C₂×C₄	C₂×C₄×C₅	1	3	yes	—
45	C₂×C₄	C₂×C₄×C₅	1	6	yes	—
46	C₂×C₄	C₂³×C₅	1	6	yes	—
47	C₂×C₄	C₂³×C₅	1	1	yes	characteristic
48	C₈	C₂³×C₅	1	8	yes	—
49	C₅	#568	1	1	yes	characteristic, direct product factor, Sylow
50	C₂²	—	4	3	no	—
51	C₂²	C₂²×C₄×C₅	1	1	yes	characteristic
52	C₂²	C₂²×C₄×C₅	1	6	yes	—
53	C₄	C₂²×C₄×C₅	1	1	yes	characteristic
54	C₄	C₂²×C₄×C₅	1	6	yes	—
55	C₄	C₂⁴×C₅	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×C₄×C₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂⁴
commutator	C₂	C₂³×C₄×C₅
Frattini, cosocle	C₄	C₂⁴×C₅
socle	C₂×C₅	C₂³×C₄
prime radical	C₂×C₅×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#131	C₂	semidirect
3. C₂×C₅×M₁₆	C₂	semidirect
10. C₅×M₁₆	C₂²	semidirect
11. C₂×C₅×C₈	C₂²	semidirect
12. #568	C₅	direct (N × H)
22. #131	C₂×C₅	semidirect
23. C₂×M₁₆	C₂×C₅	semidirect
36. M₁₆	C₂²×C₅	semidirect
37. C₂×C₈	C₂²×C₅	semidirect
49. C₅	#568	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×C₅×#41	2
C₅×#131	2
C₂×C₅×M₁₆	2
#568	5

Subgroups — C5×#568 #66232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#568 #66232