Subgroups — C₅×#508 #65671

Order 320.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#135	C₂	1	2	yes	maximal, complemented
2	C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
4	C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#41	—	2	2	no	—
7	C₅×Q₁₆	C₂²	1	2	yes	—
8	C₅×SD₁₆	—	2	2	no	—
9	C₅×D₈	C₂²	1	2	yes	—
10	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₅×C₁₆	C₂²	1	1	yes	characteristic
13	#508	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₅×Q₈	—	2	2	no	—
15	C₅×D₄	—	2	2	no	—
16	C₅×D₄	—	4	2	no	—
17	C₂×C₄×C₅	—	4	2	no	—
18	C₂×C₄×C₅	D₄	1	1	yes	characteristic
19	C₅×C₈	D₄	1	1	yes	characteristic
20	C₅×C₈	C₂³	1	1	yes	characteristic
21	#135	C₂×C₅	1	2	yes	complemented
22	Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
23	SD₃₂	C₂×C₅	1	2	yes	complemented
24	D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
25	C₂×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
26	C₂²×C₅	—	4	2	no	—
27	C₂²×C₅	D₈	1	1	yes	characteristic
28	C₄×C₅	—	4	2	no	—
29	C₄×C₅	D₈	1	1	yes	characteristic
30	C₄×C₅	C₂×D₄	1	1	yes	characteristic
31	#41	—	2	2	no	—
32	Q₁₆	C₂²×C₅	1	2	yes	—
33	SD₁₆	—	2	2	no	—
34	D₈	C₂²×C₅	1	2	yes	—
35	C₂×C₈	C₂²×C₅	1	1	yes	characteristic
36	C₁₆	C₂²×C₅	1	1	yes	characteristic
37	C₁₆	C₂²×C₅	1	1	yes	characteristic
38	C₂×C₅	—	2	1	no	—
39	C₂×C₅	—	8	2	no	—
40	C₂×C₅	C₂×D₈	1	1	yes	characteristic
41	Q₈	—	2	2	no	—
42	D₄	—	2	2	no	—
43	D₄	—	4	2	no	—
44	C₂×C₄	—	4	2	no	—
45	C₂×C₄	C₅×D₄	1	1	yes	characteristic
46	C₈	C₅×D₄	1	1	yes	characteristic
47	C₈	C₂³×C₅	1	1	yes	characteristic
48	C₅	#508	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²	—	4	2	no	—
50	C₂²	C₅×D₈	1	1	yes	characteristic
51	C₄	—	4	2	no	—
52	C₄	C₅×D₈	1	1	yes	characteristic
53	C₄	C₂×C₅×D₄	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	2	no	—
56	C₂	C₂×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	D₈
commutator, Frattini, cosocle	C₈	C₂³×C₅
socle	C₂×C₅	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#135	C₂	semidirect
2. C₅×Q₃₂	C₂	semidirect
3. C₅×SD₃₂	C₂	semidirect
4. C₅×D₁₆	C₂	semidirect
5. C₂×C₅×C₁₆	C₂	semidirect
13. #508	C₅	direct (N × H)
21. #135	C₂×C₅	semidirect
22. Q₃₂	C₂×C₅	semidirect
23. SD₃₂	C₂×C₅	semidirect
24. D₁₆	C₂×C₅	semidirect
25. C₂×C₁₆	C₂×C₅	semidirect
48. C₅	#508	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#135	2
C₅×Q₃₂	2
C₅×SD₃₂	2
C₅×D₁₆	2
C₂×C₅×C₁₆	2
#508	5

Subgroups — C5×#508 #65671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#508 #65671