Subgroups — C₃²×C₄×#170 #13086252

Order 1440.

264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#170	C₂	1	2	yes	maximal
2	C₃²×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#170	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂×C₃²×Dic₅	C₄	1	1	yes	characteristic
6	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
7	C₃²×#170	C₄	1	4	yes	direct product factor
8	C₃²×C₄×C₈	—	5	1	no	maximal, Hall
9	C₂×C₃×#170	C₂×C₃	1	8	yes	—
10	C₃×C₄×Dic₅	C₂×C₃	1	4	yes	—
11	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
12	C₃²×C₄×C₅	C₈	1	2	yes	complemented
13	C₃²×Dic₅	C₈	1	2	yes	complemented
14	C₃²×Dic₅	C₂×C₄	1	2	yes	—
15	C₄×#170	C₃²	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₃²×C₈	—	5	2	no	—
17	C₃²×C₄²	—	5	1	no	—
18	C₂×C₃×C₄×C₅	C₃×C₄	1	4	yes	—
19	C₂×C₃×Dic₅	C₃×C₄	1	4	yes	—
20	C₂×C₃×Dic₅	C₂²×C₃	1	4	yes	—
21	C₃×#170	C₃×C₄	1	16	yes	direct product factor
22	C₃×C₄×C₈	—	5	4	no	—
23	C₂×C₃²×C₅	C₄²	1	1	yes	characteristic
24	C₂×C₃²×C₅	C₂×C₈	1	1	yes	characteristic
25	C₂×C₃²×C₅	C₂×C₈	1	1	yes	characteristic
26	C₂×#170	C₂×C₃²	1	2	yes	—
27	C₄×Dic₅	C₂×C₃²	1	1	yes	characteristic
28	C₂×C₃²×C₄	—	5	1	no	—
29	C₂×C₃²×C₄	—	5	1	no	—
30	C₂×C₃²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
31	C₃²×C₈	—	5	4	no	—
32	C₂²×C₃×C₅	C₂×C₃×C₄	1	4	yes	—
33	C₃×C₄×C₅	C₃×C₈	1	8	yes	complemented
34	C₃×Dic₅	C₃×C₈	1	8	yes	complemented
35	C₃×Dic₅	C₂×C₃×C₄	1	8	yes	—
36	C₂×C₃×C₈	—	5	8	no	—
37	C₃×C₄²	—	5	4	no	—
38	C₃²×C₅	C₄×C₈	1	1	yes	characteristic, complemented, Hall
39	C₂×C₄×C₅	C₃²×C₄	1	1	yes	characteristic
40	C₂×Dic₅	C₃²×C₄	1	1	yes	characteristic
41	C₂×Dic₅	C₂²×C₃²	1	1	yes	characteristic
42	#170	C₃²×C₄	1	4	yes	direct product factor
43	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₃²×C₄	—	5	2	no	—
45	C₃²×C₄	—	5	2	no	—
46	C₃²×C₄	#170	1	2	yes	direct product factor
47	C₄×C₈	—	5	1	no	Sylow
48	C₂×C₃×C₅	C₃×C₄²	1	4	yes	—
49	C₂×C₃×C₅	C₂×C₃×C₈	1	4	yes	—
50	C₂×C₃×C₅	C₂×C₃×C₈	1	4	yes	—
51	C₂×C₃×C₄	—	5	4	no	—
52	C₂×C₃×C₄	—	5	4	no	—
53	C₂×C₃×C₄	C₃×AGL(1,𝔽₅)	1	4	yes	—
54	C₃×C₈	—	5	16	no	—
55	C₂²×C₅	C₂×C₃²×C₄	1	1	yes	characteristic
56	C₄×C₅	C₃²×C₈	1	2	yes	complemented
57	Dic₅	C₃²×C₈	1	2	yes	complemented
58	Dic₅	C₂×C₃²×C₄	1	2	yes	—
59	C₂×C₃²	D₅⋊C₈	1	1	yes	characteristic
60	C₂×C₃²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₂×C₃²	C₂×#170	1	1	yes	characteristic
62	C₂×C₈	—	5	2	no	—
63	C₄²	—	5	1	no	—
64	C₃×C₅	C₃×C₄×C₈	1	4	yes	complemented
65	C₂²×C₃	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
66	C₃×C₄	—	5	8	no	—
67	C₃×C₄	—	5	8	no	—
68	C₃×C₄	C₃×#170	1	8	yes	direct product factor
69	C₂×C₅	C₃²×C₄²	1	1	yes	characteristic
70	C₂×C₅	C₂×C₃²×C₈	1	1	yes	characteristic
71	C₂×C₅	C₂×C₃²×C₈	1	1	yes	characteristic
72	C₃²	C₄×#170	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	—	5	1	no	—
75	C₂×C₄	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₈	—	5	4	no	—
77	C₂×C₃	C₃×D₅⋊C₈	1	4	yes	—
78	C₂×C₃	C₃×C₄×AGL(1,𝔽₅)	1	4	yes	—
79	C₂×C₃	C₂×C₃×#170	1	4	yes	—
80	C₅	C₃²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
81	C₂²	C₂×C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₄	—	5	2	no	—
83	C₄	—	5	2	no	—
84	C₄	C₃²×#170	1	2	yes	direct product factor
85	C₃	C₃×C₄×#170	1	4	yes	direct product factor
86	C₂	C₃²×D₅⋊C₈	1	1	yes	characteristic
87	C₂	C₃²×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₂	C₂×C₃²×#170	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃²×C₄×C₈
Frattini	C₂²	C₂×C₃²×AGL(1,𝔽₅)
Fitting	C₂×C₃²×C₄×C₅	C₄
socle, prime radical	C₂²×C₃²×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂²×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃²×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×#170	C₃	direct (N × H)
7. C₃²×#170	C₄	direct (N × H)
12. C₃²×C₄×C₅	C₈	semidirect
13. C₃²×Dic₅	C₈	semidirect
15. C₄×#170	C₃²	direct (N × H)
21. C₃×#170	C₃×C₄	direct (N × H)
33. C₃×C₄×C₅	C₃×C₈	semidirect
34. C₃×Dic₅	C₃×C₈	semidirect
38. C₃²×C₅	C₄×C₈	semidirect
42. #170	C₃²×C₄	direct (N × H)
46. C₃²×C₄	#170	direct (N × H)
56. C₄×C₅	C₃²×C₈	semidirect
57. Dic₅	C₃²×C₈	semidirect
64. C₃×C₅	C₃×C₄×C₈	semidirect
68. C₃×C₄	C₃×#170	direct (N × H)
72. C₃²	C₄×#170	direct (N × H)
80. C₅	C₃²×C₄×C₈	semidirect
84. C₄	C₃²×#170	direct (N × H)
85. C₃	C₃×C₄×#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
C₂×C₃²×#170	2
C₃²×C₄×Dic₅	2
C₃×C₄×#170	3
C₃²×C₄×C₈	5

Subgroups — C32×C4×#170 #13086252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×#170 #13086252