Subgroups — C₅×#809 #91052

Order 480.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#102	—	3	1	no	maximal, Hall
5	C₂²×C₅×D₃	—	2	1	no	—
6	C₂×C₃×C₄×C₅	—	2	1	no	—
7	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₅×D₁₂	—	2	1	no	—
9	C₅×D₁₂	C₄	1	2	yes	complemented
10	C₅×C₃⋊C₈	—	2	1	no	—
11	#809	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₅×D₄	—	3	1	no	—
13	C₂×C₅×C₈	—	3	1	no	—
14	C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
15	C₂²×C₃×C₅	D₄	1	1	yes	characteristic
16	C₂×C₅×D₃	—	2	2	no	—
17	C₂×C₅×D₃	—	4	1	no	—
18	C₂×C₅×D₃	—	4	1	no	—
19	C₃×C₄×C₅	—	4	1	no	—
20	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₃×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂×D₁₂	C₂×C₅	1	1	yes	characteristic
23	C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
24	C₂×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
25	C₂³×C₅	—	6	1	no	—
26	C₅×D₄	—	3	2	no	—
27	C₅×D₄	—	6	1	no	—
28	C₂×C₄×C₅	—	2	1	no	—
29	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
30	C₅×C₈	—	6	1	no	—
31	#102	—	3	1	no	Sylow
32	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃×C₅	D₈	1	1	yes	characteristic
34	C₂×C₃×C₅	SD₁₆	1	1	yes	characteristic
35	C₅×D₃	—	4	2	no	—
36	C₂²×D₃	—	2	1	no	—
37	C₂×C₃×C₄	—	2	1	no	—
38	C₂×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
39	D₁₂	—	2	1	no	—
40	D₁₂	C₄×C₅	1	2	yes	complemented
41	C₃⋊C₈	—	2	1	no	—
42	C₂²×C₅	—	6	2	no	—
43	C₂²×C₅	—	12	1	no	—
44	C₂²×C₅	—	12	1	no	—
45	C₂²×C₅	#67	1	1	yes	characteristic
46	C₄×C₅	—	4	1	no	—
47	C₄×C₅	C₄×D₃	1	1	yes	characteristic
48	C₄×C₅	D₁₂	1	1	yes	characteristic
49	C₂×D₄	—	3	1	no	—
50	C₂×C₈	—	3	1	no	—
51	C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
52	C₃×C₅	#102	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₃	C₅×D₄	1	1	yes	characteristic
54	C₂×D₃	—	2	2	no	—
55	C₂×D₃	—	4	1	no	—
56	C₂×D₃	—	4	1	no	—
57	C₃×C₄	—	4	1	no	—
58	C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
59	C₃×C₄	C₅×D₄	1	1	yes	characteristic
60	C₂×C₅	—	12	2	no	—
61	C₂×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
62	C₂×C₅	#213	1	1	yes	characteristic
63	C₂×C₅	#215	1	1	yes	characteristic
64	C₂³	—	6	1	no	—
65	D₄	—	3	2	no	—
66	D₄	—	6	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	C₂×C₅×D₃	1	1	yes	characteristic
69	C₈	—	6	1	no	—
70	C₂×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃	C₅×D₈	1	1	yes	characteristic
72	C₂×C₃	C₅×SD₁₆	1	1	yes	characteristic
73	D₃	—	4	2	no	—
74	C₅	#809	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	—	6	2	no	—
76	C₂²	—	12	1	no	—
77	C₂²	—	12	1	no	—
78	C₂²	C₅×#67	1	1	yes	characteristic
79	C₄	—	4	1	no	—
80	C₄	C₄×C₅×D₃	1	1	yes	characteristic
81	C₄	C₅×D₁₂	1	1	yes	characteristic
82	C₃	C₅×#102	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	12	2	no	—
84	C₂	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
85	C₂	C₅×#213	1	1	yes	characteristic
86	C₂	C₅×#215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#67
commutator	C₃×C₄	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₄	C₂
socle	C₂²×C₃×C₅	D₄
prime radical	C₂×C₅×D₁₂	C₂
cosocle	C₂×C₃×C₄	C₂²×C₅
hypercenter	C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₅×#102
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₄⋊C₄	C₂	semidirect
3. C₂×C₅×C₃⋊C₈	C₂	semidirect
9. C₅×D₁₂	C₄	semidirect
11. #809	C₅	direct (N × H)
14. C₅×C₄⋊C₄	D₃	semidirect
23. C₃×C₄⋊C₄	C₂×C₅	semidirect
24. C₂×C₃⋊C₈	C₂×C₅	semidirect
40. D₁₂	C₄×C₅	semidirect
51. C₄⋊C₄	C₅×D₃	semidirect
52. C₃×C₅	#102	semidirect
74. C₅	#809	direct (N × H)
82. C₃	C₅×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₁₂	2
C₃×C₅×C₄⋊C₄	2
C₂×C₅×C₃⋊C₈	2
C₅×#102	3
#809	5

Subgroups — C5×#809 #91052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#809 #91052