Subgroups — C₂³×C₃×#170 #11756975

Order 960.

676 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#170	C₂	1	14	yes	maximal, direct product factor
3	C₂³×#170	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂⁴×C₃×C₅	C₄	1	1	yes	characteristic
5	C₂²×C₃×Dic₅	C₄	1	7	yes	—
6	C₂²×C₃×Dic₅	C₂²	1	7	yes	—
7	C₂×C₃×#170	C₂²	1	28	yes	direct product factor
8	C₂³×C₃×C₈	—	5	1	no	maximal, Hall
9	C₂³×Dic₅	C₂×C₃	1	1	yes	characteristic
10	C₂²×#170	C₂×C₃	1	14	yes	direct product factor
11	C₂³×C₃×C₅	C₈	1	8	yes	complemented
12	C₂³×C₃×C₅	C₂×C₄	1	7	yes	—
13	C₂×C₃×Dic₅	C₂×C₄	1	21	yes	—
14	C₂×C₃×Dic₅	C₂³	1	7	yes	—
15	C₃×#170	C₂³	1	8	yes	direct product factor
16	C₂³×C₃×C₄	—	5	1	no	—
17	C₂²×C₃×C₈	—	5	14	no	—
18	C₂⁴×C₅	C₃×C₄	1	1	yes	characteristic
19	C₂²×Dic₅	C₃×C₄	1	7	yes	—
20	C₂²×Dic₅	C₂²×C₃	1	7	yes	—
21	C₂×#170	C₂²×C₃	1	28	yes	direct product factor
22	C₂³×C₈	—	5	1	no	Sylow
23	C₂²×C₃×C₅	C₂×C₈	1	28	yes	complemented
24	C₂²×C₃×C₅	C₂²×C₄	1	7	yes	—
25	C₃×Dic₅	C₂²×C₄	1	7	yes	—
26	C₃×Dic₅	C₂⁴	1	1	yes	characteristic
27	C₂⁴×C₃	AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₂²×C₃×C₄	—	5	7	no	—
29	C₂²×C₃×C₄	—	5	7	no	—
30	C₂×C₃×C₈	—	5	28	no	—
31	C₂³×C₅	C₃×C₈	1	8	yes	complemented
32	C₂³×C₅	C₂×C₃×C₄	1	7	yes	—
33	C₂×Dic₅	C₂×C₃×C₄	1	21	yes	—
34	C₂×Dic₅	C₂³×C₃	1	7	yes	—
35	#170	C₂³×C₃	1	8	yes	direct product factor
36	C₂³×C₄	—	5	1	no	—
37	C₂²×C₈	—	5	14	no	—
38	C₂×C₃×C₅	C₂²×C₈	1	14	yes	complemented
39	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
40	C₂³×C₃	#170	1	8	yes	direct product factor
41	C₂³×C₃	C₂×AGL(1,𝔽₅)	1	7	yes	—
42	C₂×C₃×C₄	—	5	21	no	—
43	C₂×C₃×C₄	—	5	7	no	—
44	C₃×C₈	—	5	8	no	—
45	C₂²×C₅	C₂×C₃×C₈	1	28	yes	complemented
46	C₂²×C₅	C₂²×C₃×C₄	1	7	yes	—
47	Dic₅	C₂²×C₃×C₄	1	7	yes	—
48	Dic₅	C₂⁴×C₃	1	1	yes	characteristic
49	C₂⁴	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₂²×C₄	—	5	7	no	—
51	C₂²×C₄	—	5	7	no	—
52	C₂×C₈	—	5	28	no	—
53	C₃×C₅	C₂³×C₈	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₃	C₂×#170	1	28	yes	direct product factor
55	C₂²×C₃	C₂²×AGL(1,𝔽₅)	1	7	yes	—
56	C₃×C₄	—	5	7	no	—
57	C₃×C₄	—	5	1	no	—
58	C₂×C₅	C₂²×C₃×C₈	1	14	yes	complemented
59	C₂×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
60	C₂³	C₃×#170	1	8	yes	direct product factor
61	C₂³	C₂×C₃×AGL(1,𝔽₅)	1	7	yes	—
62	C₂×C₄	—	5	7	no	—
63	C₂×C₄	—	5	21	no	—
64	C₈	—	5	8	no	—
65	C₂×C₃	C₂²×#170	1	14	yes	direct product factor
66	C₂×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₅	C₂³×C₃×C₈	1	1	yes	characteristic, complemented, Sylow
68	C₂²	C₂×C₃×#170	1	28	yes	direct product factor
69	C₂²	C₂²×C₃×AGL(1,𝔽₅)	1	7	yes	—
70	C₄	—	5	1	no	—
71	C₄	—	5	7	no	—
72	C₃	C₂³×#170	1	1	yes	characteristic, direct product factor, Sylow
73	C₂	C₂²×C₃×#170	1	14	yes	direct product factor
74	C₂	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₃	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂³×C₃×C₈
Frattini	C₂	C₂³×C₃×AGL(1,𝔽₅)
Fitting, socle, prime radical	C₂⁴×C₃×C₅	C₄
cosocle	Dic₅	C₂⁴×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂⁴×C₃

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×#170	C₂	direct (N × H)
3. C₂³×#170	C₃	direct (N × H)
7. C₂×C₃×#170	C₂²	direct (N × H)
10. C₂²×#170	C₂×C₃	direct (N × H)
11. C₂³×C₃×C₅	C₈	semidirect
15. C₃×#170	C₂³	direct (N × H)
21. C₂×#170	C₂²×C₃	direct (N × H)
23. C₂²×C₃×C₅	C₂×C₈	semidirect
31. C₂³×C₅	C₃×C₈	semidirect
35. #170	C₂³×C₃	direct (N × H)
38. C₂×C₃×C₅	C₂²×C₈	semidirect
40. C₂³×C₃	#170	direct (N × H)
45. C₂²×C₅	C₂×C₃×C₈	semidirect
53. C₃×C₅	C₂³×C₈	semidirect
54. C₂²×C₃	C₂×#170	direct (N × H)
58. C₂×C₅	C₂²×C₃×C₈	semidirect
60. C₂³	C₃×#170	direct (N × H)
65. C₂×C₃	C₂²×#170	direct (N × H)
67. C₅	C₂³×C₃×C₈	semidirect
68. C₂²	C₂×C₃×#170	direct (N × H)
72. C₃	C₂³×#170	direct (N × H)
73. C₂	C₂²×C₃×#170	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×Dic₅	2
C₂²×C₃×#170	2
C₂³×#170	3
C₂³×C₃×C₈	5

Subgroups — C23×C3×#170 #11756975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×#170 #11756975