Subgroups — C₂×C₃×#704 #91982

Order 480.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₃×#704	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×#170	C₂	1	2	yes	maximal, complemented
4	C₂×#704	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₅	C₄	1	1	yes	characteristic
6	C₂×C₃×Dic₅	C₄	1	2	yes	—
7	C₂×C₃×Dic₅	C₄	1	1	yes	characteristic
8	C₂×C₃×Dic₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×Dic₅	C₂²	1	2	yes	—
10	C₃×#170	C₂²	1	4	yes	complemented
11	C₂×C₃×M₁₆	—	5	1	no	maximal, Hall
12	C₂²×Dic₅	C₂×C₃	1	1	yes	characteristic
13	#704	C₂×C₃	1	4	yes	direct product factor
14	C₂×#170	C₂×C₃	1	2	yes	complemented
15	C₂²×C₃×C₅	—	2	2	no	—
16	C₂²×C₃×C₅	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
19	C₃×Dic₅	C₂×C₄	1	2	yes	—
20	C₃×Dic₅	C₂³	1	1	yes	characteristic
21	C₂²×C₃×C₄	—	5	1	no	—
22	C₃×M₁₆	—	5	4	no	—
23	C₂×C₃×C₈	—	5	2	no	—
24	C₂³×C₅	C₃×C₄	1	1	yes	characteristic
25	C₂×Dic₅	C₃×C₄	1	2	yes	—
26	C₂×Dic₅	C₃×C₄	1	1	yes	characteristic
27	C₂×Dic₅	C₂²×C₃	1	1	yes	characteristic
28	C₂×Dic₅	C₂²×C₃	1	2	yes	—
29	#170	C₂²×C₃	1	4	yes	complemented
30	C₂×M₁₆	—	5	1	no	Sylow
31	C₂×C₃×C₅	—	2	2	no	—
32	C₂×C₃×C₅	M₁₆	1	2	yes	complemented
33	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂³×C₃	AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂×C₃×C₄	—	5	2	no	—
36	C₂×C₃×C₄	—	5	2	no	—
37	C₂×C₃×C₄	—	5	1	no	—
38	C₂×C₃×C₄	—	5	1	no	—
39	C₃×C₈	—	5	4	no	—
40	C₂²×C₅	—	2	2	no	—
41	C₂²×C₅	C₂×C₃×C₄	1	2	yes	—
42	C₂²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
43	Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
44	Dic₅	C₂×C₃×C₄	1	2	yes	—
45	Dic₅	C₂³×C₃	1	1	yes	characteristic
46	C₂²×C₄	—	5	1	no	—
47	M₁₆	—	5	4	no	—
48	C₂×C₈	—	5	2	no	—
49	C₃×C₅	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₃	—	2	2	no	—
51	C₂²×C₃	C₂×AGL(1,𝔽₅)	1	2	yes	—
52	C₂²×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₃×C₄	—	5	1	no	—
54	C₃×C₄	—	5	2	no	—
55	C₃×C₄	—	5	1	no	—
56	C₂×C₅	—	2	2	no	—
57	C₂×C₅	C₃×M₁₆	1	2	yes	complemented
58	C₂×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
59	C₂³	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₂×C₄	—	5	1	no	—
61	C₂×C₄	—	5	2	no	—
62	C₂×C₄	—	5	2	no	—
63	C₂×C₄	—	5	1	no	—
64	C₈	—	5	4	no	—
65	C₂×C₃	—	2	2	no	—
66	C₂×C₃	#704	1	2	yes	direct product factor
67	C₂×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₅	C₂×C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	2	2	no	—
70	C₂²	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
71	C₂²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄	—	5	1	no	—
73	C₄	—	5	1	no	—
74	C₄	—	5	2	no	—
75	C₃	C₂×#704	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	—	2	2	no	—
77	C₂	C₃×#704	1	2	yes	direct product factor
78	C₂	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₃×C₄
Frattini	C₂	C₂²×C₃×AGL(1,𝔽₅)
Fitting, prime radical	C₂³×C₃×C₅	C₄
socle	C₂²×C₃×C₅	C₂×C₄
cosocle	Dic₅	C₂³×C₃
hypercenter	C₂³×C₃	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. C₃×#704	C₂	direct (N × H)
3. C₂×C₃×#170	C₂	semidirect
4. C₂×#704	C₃	direct (N × H)
10. C₃×#170	C₂²	semidirect
13. #704	C₂×C₃	direct (N × H)
14. C₂×#170	C₂×C₃	semidirect
29. #170	C₂²×C₃	semidirect
32. C₂×C₃×C₅	M₁₆	semidirect
49. C₃×C₅	C₂×M₁₆	semidirect
57. C₂×C₅	C₃×M₁₆	semidirect
66. C₂×C₃	#704	direct (N × H)
68. C₅	C₂×C₃×M₁₆	semidirect
75. C₃	C₂×#704	direct (N × H)
77. C₂	C₃×#704	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×C₃×Dic₅	2
C₃×#704	2
C₂×C₃×#170	2
C₂×#704	3
C₂×C₃×M₁₆	5

Subgroups — C2×C3×#704 #91982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#704 #91982