Subgroups — C₃²×C₄×#168 #13084921

Order 1440.

264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#168	C₂	1	2	yes	maximal
3	C₃×C₄×#168	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₅	C₄	1	2	yes	—
5	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₃²×#168	C₄	1	4	yes	direct product factor
7	C₃²×C₄×C₈	—	5	1	no	maximal, Hall
8	C₃×C₄²×C₅	C₂×C₃	1	4	yes	—
9	C₂×C₃×#168	C₂×C₃	1	8	yes	—
10	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
11	C₃²×C₄×C₅	C₈	1	4	yes	complemented
12	C₃²×C₄×C₅	C₂×C₄	1	2	yes	—
13	C₄×#168	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃²×C₈	—	5	2	no	—
15	C₃²×C₄²	D₅	1	1	yes	characteristic
16	C₂×C₃×C₄×C₅	C₃×C₄	1	8	yes	—
17	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
18	C₃×#168	C₃×C₄	1	16	yes	direct product factor
19	C₃×C₄×C₈	—	5	4	no	—
20	C₂×C₃²×C₅	C₄²	1	1	yes	characteristic
21	C₂×C₃²×C₅	C₂×C₈	1	2	yes	—
22	C₄²×C₅	C₂×C₃²	1	1	yes	characteristic
23	C₂×#168	C₂×C₃²	1	2	yes	—
24	C₂×C₃²×C₄	Dic₅	1	2	yes	—
25	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
26	C₃²×C₈	—	5	4	no	—
27	C₂²×C₃×C₅	C₂×C₃×C₄	1	4	yes	—
28	C₃×C₄×C₅	C₃×C₈	1	16	yes	complemented
29	C₃×C₄×C₅	C₂×C₃×C₄	1	8	yes	—
30	C₂×C₃×C₈	—	5	8	no	—
31	C₃×C₄²	C₃×D₅	1	4	yes	—
32	C₃²×C₅	C₄×C₈	1	1	yes	characteristic, complemented, Hall
33	C₂×C₄×C₅	C₃²×C₄	1	2	yes	—
34	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
35	#168	C₃²×C₄	1	4	yes	direct product factor
36	C₂²×C₃²	C₂×Dic₅	1	1	yes	characteristic
37	C₃²×C₄	#168	1	4	yes	direct product factor
38	C₃²×C₄	C₄×D₅	1	2	yes	—
39	C₄×C₈	—	5	1	no	Sylow
40	C₂×C₃×C₅	C₃×C₄²	1	4	yes	—
41	C₂×C₃×C₅	C₂×C₃×C₈	1	8	yes	—
42	C₂×C₃×C₄	C₃×Dic₅	1	8	yes	—
43	C₂×C₃×C₄	C₂×C₃×D₅	1	4	yes	—
44	C₃×C₈	—	5	16	no	—
45	C₂²×C₅	C₂×C₃²×C₄	1	1	yes	characteristic
46	C₄×C₅	C₃²×C₈	1	4	yes	complemented
47	C₄×C₅	C₂×C₃²×C₄	1	2	yes	—
48	C₂×C₃²	C₂×#168	1	2	yes	—
49	C₂×C₃²	C₄×Dic₅	1	1	yes	characteristic
50	C₂×C₈	—	5	2	no	—
51	C₄²	C₃²×D₅	1	1	yes	characteristic
52	C₃×C₅	C₃×C₄×C₈	1	4	yes	complemented
53	C₂²×C₃	C₂×C₃×Dic₅	1	4	yes	—
54	C₃×C₄	C₃×#168	1	16	yes	direct product factor
55	C₃×C₄	C₃×C₄×D₅	1	8	yes	—
56	C₂×C₅	C₃²×C₄²	1	1	yes	characteristic
57	C₂×C₅	C₂×C₃²×C₈	1	2	yes	—
58	C₃²	C₄×#168	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₄	C₃²×Dic₅	1	2	yes	—
60	C₂×C₄	C₂×C₃²×D₅	1	1	yes	characteristic
61	C₈	—	5	4	no	—
62	C₂×C₃	C₂×C₃×#168	1	8	yes	—
63	C₂×C₃	C₃×C₄×Dic₅	1	4	yes	—
64	C₅	C₃²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
65	C₂²	C₂×C₃²×Dic₅	1	1	yes	characteristic
66	C₄	C₃²×#168	1	4	yes	direct product factor
67	C₄	C₃²×C₄×D₅	1	2	yes	—
68	C₃	C₃×C₄×#168	1	4	yes	direct product factor
69	C₂	C₂×C₃²×#168	1	2	yes	—
70	C₂	C₃²×C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₄²	D₅
commutator, hypocenter	C₅	C₃²×C₄×C₈
Frattini	C₂×C₄	C₂×C₃²×D₅
Fitting	C₃²×C₄²×C₅	C₂
socle, prime radical	C₂²×C₃²×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂²×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×#168	C₃	direct (N × H)
6. C₃²×#168	C₄	direct (N × H)
11. C₃²×C₄×C₅	C₈	semidirect
13. C₄×#168	C₃²	direct (N × H)
18. C₃×#168	C₃×C₄	direct (N × H)
28. C₃×C₄×C₅	C₃×C₈	semidirect
32. C₃²×C₅	C₄×C₈	semidirect
35. #168	C₃²×C₄	direct (N × H)
37. C₃²×C₄	#168	direct (N × H)
46. C₄×C₅	C₃²×C₈	semidirect
52. C₃×C₅	C₃×C₄×C₈	semidirect
54. C₃×C₄	C₃×#168	direct (N × H)
58. C₃²	C₄×#168	direct (N × H)
64. C₅	C₃²×C₄×C₈	semidirect
66. C₄	C₃²×#168	direct (N × H)
68. C₃	C₃×C₄×#168	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄²×C₅	2
C₂×C₃²×#168	2
C₃×C₄×#168	3
C₃²×C₄×C₈	5

Subgroups — C32×C4×#168 #13084921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×#168 #13084921