Subgroups — C₅×#510 #65673

Order 320.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Q₃₂	C₂	1	2	yes	maximal, complemented
4	C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
5	C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#41	—	2	1	no	—
7	C₂×C₅×Q₈	—	2	1	no	—
8	C₅×Q₁₆	—	2	1	no	—
9	C₅×Q₁₆	C₂²	1	2	yes	—
10	C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₅×SD₁₆	—	2	1	no	—
12	C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₁₆	C₂²	1	2	yes	—
15	#510	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₅×Q₈	—	2	1	no	—
17	C₅×Q₈	—	2	2	no	—
18	C₅×Q₈	—	4	1	no	—
19	C₅×D₄	—	2	1	no	—
20	C₅×D₄	—	4	1	no	—
21	C₂×C₄×C₅	—	4	1	no	—
22	C₂×C₄×C₅	—	4	1	no	—
23	C₂×C₄×C₅	D₄	1	1	yes	characteristic
24	C₅×C₈	D₄	1	1	yes	characteristic
25	C₅×C₈	C₂³	1	1	yes	characteristic
26	#135	C₂×C₅	1	1	yes	characteristic
27	C₂×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
28	Q₃₂	C₂×C₅	1	2	yes	complemented
29	SD₃₂	C₂×C₅	1	2	yes	complemented
30	M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
31	C₂²×C₅	—	4	1	no	—
32	C₂²×C₅	D₈	1	1	yes	characteristic
33	C₄×C₅	—	4	1	no	—
34	C₄×C₅	—	4	2	no	—
35	C₄×C₅	D₈	1	1	yes	characteristic
36	C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	#41	—	2	1	no	—
38	C₂×Q₈	—	2	1	no	—
39	Q₁₆	—	2	1	no	—
40	Q₁₆	C₂²×C₅	1	2	yes	—
41	Q₁₆	C₂²×C₅	1	1	yes	characteristic
42	SD₁₆	—	2	1	no	—
43	D₈	C₂²×C₅	1	1	yes	characteristic
44	C₂×C₈	C₂²×C₅	1	1	yes	characteristic
45	C₁₆	C₂²×C₅	1	2	yes	—
46	C₂×C₅	—	2	1	no	—
47	C₂×C₅	—	8	1	no	—
48	C₂×C₅	C₂×D₈	1	1	yes	characteristic
49	Q₈	—	2	1	no	—
50	Q₈	—	2	2	no	—
51	Q₈	—	4	1	no	—
52	D₄	—	2	1	no	—
53	D₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	C₅×D₄	1	1	yes	characteristic
57	C₈	C₅×D₄	1	1	yes	characteristic
58	C₈	C₂³×C₅	1	1	yes	characteristic
59	C₅	#510	1	1	yes	characteristic, direct product factor, Sylow
60	C₂²	—	4	1	no	—
61	C₂²	C₅×D₈	1	1	yes	characteristic
62	C₄	—	4	1	no	—
63	C₄	—	4	2	no	—
64	C₄	C₅×D₈	1	1	yes	characteristic
65	C₄	C₂×C₅×D₄	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	8	1	no	—
68	C₂	C₂×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂³×C₅
prime radical	C₅×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×Q₁₆	C₂	semidirect
3. C₅×Q₃₂	C₂	semidirect
4. C₅×SD₃₂	C₂	semidirect
5. C₅×M₃₂	C₂	semidirect
15. #510	C₅	direct (N × H)
27. C₂×Q₁₆	C₂×C₅	semidirect
28. Q₃₂	C₂×C₅	semidirect
29. SD₃₂	C₂×C₅	semidirect
30. M₃₂	C₂×C₅	semidirect
59. C₅	#510	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
C₂×C₅×Q₁₆	2
C₅×Q₃₂	2
C₅×SD₃₂	2
C₅×M₃₂	2
#510	5

Subgroups — C5×#510 #65673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#510 #65673