Subgroups — C₂×C₃×#700 #91972

Order 480.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#170	C₂	1	2	yes	maximal, complemented
3	C₃×#700	C₂	1	4	yes	maximal, direct product factor
4	C₂×#700	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×D₅	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄×C₅	C₄	1	1	yes	characteristic
7	C₂×C₃×Dic₅	C₂²	1	1	yes	characteristic
8	C₃×C₄×D₅	C₄	1	2	yes	—
9	C₃×C₄×D₅	C₂²	1	2	yes	—
10	C₃×#170	C₂²	1	4	yes	complemented
11	C₂×C₃×M₁₆	—	5	1	no	maximal, Hall
12	C₂×C₄×D₅	C₂×C₃	1	1	yes	characteristic
13	C₂×#170	C₂×C₃	1	2	yes	complemented
14	#700	C₂×C₃	1	4	yes	direct product factor
15	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×D₅	—	2	2	no	—
17	C₂×C₃×D₅	C₂×C₄	1	2	yes	—
18	C₃×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
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	C₂²×D₅	C₃×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅	C₃×C₄	1	1	yes	characteristic
26	C₂×Dic₅	C₂²×C₃	1	1	yes	characteristic
27	C₄×D₅	C₃×C₄	1	2	yes	—
28	C₄×D₅	C₂²×C₃	1	2	yes	—
29	#170	C₂²×C₃	1	4	yes	complemented
30	C₂×M₁₆	—	5	1	no	Sylow
31	C₂×C₃×C₅	M₁₆	1	2	yes	complemented
32	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₃×D₅	—	2	2	no	—
34	C₂³×C₃	—	5	1	no	—
35	C₂×C₃×C₄	—	5	1	no	—
36	C₂×C₃×C₄	—	5	2	no	—
37	C₂×C₃×C₄	—	5	2	no	—
38	C₂×C₃×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₃×C₈	—	5	4	no	—
40	C₂²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
41	C₂×D₅	—	2	2	no	—
42	C₂×D₅	C₂×C₃×C₄	1	2	yes	—
43	C₄×C₅	C₂×C₃×C₄	1	2	yes	—
44	Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
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	—	5	2	no	—
51	C₂²×C₃	—	10	2	no	—
52	C₂²×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₃×C₄	—	5	1	no	—
54	C₃×C₄	—	5	1	no	—
55	C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
56	C₂×C₅	C₃×M₁₆	1	2	yes	complemented
57	C₂×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
58	D₅	—	2	2	no	—
59	C₂³	—	5	1	no	—
60	C₂×C₄	—	5	1	no	—
61	C₂×C₄	—	5	2	no	—
62	C₂×C₄	—	5	2	no	—
63	C₂×C₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₈	—	5	4	no	—
65	C₂×C₃	—	10	2	no	—
66	C₂×C₃	#700	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	—	5	2	no	—
70	C₂²	—	10	2	no	—
71	C₂²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄	—	5	1	no	—
73	C₄	—	5	1	no	—
74	C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
75	C₃	C₂×#700	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	—	10	2	no	—
77	C₂	C₃×#700	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	C₂×C₃×C₄×C₅	C₄
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₄
cosocle	Dic₅	C₂³×C₃
hypercenter	C₂×C₃×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₃×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#170	C₂	semidirect
3. C₃×#700	C₂	direct (N × H)
4. C₂×#700	C₃	direct (N × H)
10. C₃×#170	C₂²	semidirect
13. C₂×#170	C₂×C₃	semidirect
14. #700	C₂×C₃	direct (N × H)
29. #170	C₂²×C₃	semidirect
31. C₂×C₃×C₅	M₁₆	semidirect
49. C₃×C₅	C₂×M₁₆	semidirect
56. C₂×C₅	C₃×M₁₆	semidirect
66. C₂×C₃	#700	direct (N × H)
68. C₅	C₂×C₃×M₁₆	semidirect
75. C₃	C₂×#700	direct (N × H)
77. C₂	C₃×#700	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×C₃×C₄×D₅	2
C₂×C₃×#170	2
C₃×#700	2
C₂×#700	3
C₂×C₃×M₁₆	5

Subgroups — C2×C3×#700 #91972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#700 #91972