Subgroups — C₃×D₃×#168 #10623727

Order 720.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#168	—	3	1	no	maximal
5	D₃×#168	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×C₅×D₃	C₄	1	1	yes	characteristic
7	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×Dic₃	C₄	1	1	yes	characteristic
9	C₃×D₃×C₈	—	5	1	no	maximal, Hall
10	C₂×C₃×C₄×C₅	—	3	1	no	—
11	C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
12	C₃⋊#168	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃×#168	—	2	1	no	—
14	C₃×#168	—	3	1	no	—
15	C₃×#168	D₃	1	1	yes	characteristic, direct product factor
16	C₃×#168	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×C₅×D₃	C₈	1	2	yes	complemented
19	C₂×#168	—	3	1	no	Hall
20	C₃×C₄×D₃	D₅	1	1	yes	characteristic
21	C₃²×C₈	—	5	1	no	—
22	C₃×C₃⋊C₈	—	5	1	no	—
23	C₂²×C₃×C₅	—	3	1	no	—
24	C₂×C₅×D₃	C₃×C₄	1	1	yes	characteristic
25	C₃×C₄×C₅	—	2	1	no	—
26	C₃×C₄×C₅	—	3	1	no	—
27	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
28	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
29	C₅×Dic₃	C₃×C₄	1	1	yes	characteristic
30	C₂×C₃×C₈	—	15	1	no	—
31	D₃×C₈	—	5	1	no	—
32	C₃²×C₅	C₂×C₈	1	1	yes	characteristic, complemented, Hall
33	C₂×C₄×C₅	—	3	1	no	—
34	#168	—	3	1	no	—
35	#168	C₃×D₃	1	1	yes	characteristic, direct product factor
36	C₂×C₃×D₃	Dic₅	1	1	yes	characteristic
37	C₃²×C₄	C₂×D₅	1	1	yes	characteristic
38	C₃×Dic₃	Dic₅	1	1	yes	characteristic
39	C₂×C₃×C₅	—	2	1	no	—
40	C₂×C₃×C₅	—	3	2	no	—
41	C₂×C₃×C₅	C₄×D₃	1	1	yes	characteristic
42	C₂×C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic
43	C₅×D₃	C₃×C₈	1	2	yes	complemented
44	C₂×C₃×C₄	—	3	1	no	—
45	C₄×D₃	C₃×D₅	1	1	yes	characteristic
46	C₃×C₈	—	5	1	no	—
47	C₃×C₈	—	5	1	no	—
48	C₃×C₈	—	10	1	no	—
49	C₃×C₈	—	15	1	no	—
50	C₃⋊C₈	—	5	1	no	—
51	C₂²×C₅	—	3	1	no	—
52	C₄×C₅	—	3	1	no	—
53	C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
54	C₂×C₃²	C₂×Dic₅	1	1	yes	characteristic
55	C₃×D₃	#168	1	2	yes	direct product factor
56	C₂×C₈	—	15	1	no	Sylow
57	C₃×C₅	—	2	1	no	—
58	C₃×C₅	D₃×C₈	1	1	yes	characteristic, complemented
59	C₃×C₅	C₂×C₃×C₈	1	1	yes	characteristic, complemented
60	C₂²×C₃	—	3	1	no	—
61	C₂×D₃	C₃×Dic₅	1	1	yes	characteristic
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	3	1	no	—
64	C₃×C₄	D₃×D₅	1	1	yes	characteristic
65	C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
66	Dic₃	C₃×Dic₅	1	1	yes	characteristic
67	C₂×C₅	—	3	2	no	—
68	C₂×C₅	C₃×C₄×D₃	1	1	yes	characteristic
69	C₃²	C₂×#168	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄	—	3	1	no	—
71	C₈	—	5	1	no	—
72	C₈	—	15	1	no	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	3	2	no	—
75	C₂×C₃	D₃×Dic₅	1	1	yes	characteristic
76	C₂×C₃	C₂×C₃×Dic₅	1	1	yes	characteristic
77	D₃	C₃×#168	1	2	yes	direct product factor
78	C₅	C₃×D₃×C₈	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	3	1	no	—
80	C₄	—	3	1	no	—
81	C₄	C₃×D₃×D₅	1	1	yes	characteristic
82	C₃	—	2	1	no	—
83	C₃	D₃×#168	1	1	yes	characteristic, direct product factor
84	C₃	C₂×C₃×#168	1	1	yes	characteristic, complemented
85	C₂	—	3	2	no	—
86	C₂	C₃×D₃×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂×C₃×C₈
Frattini	C₄	C₃×D₃×D₅
Fitting	C₃²×C₄×C₅	C₂²
socle	C₂×C₃²×C₅	C₂×C₄
prime radical	C₂×C₃×C₅×D₃	C₄
cosocle	C₃×C₄×C₅	C₂²×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₃⋊#168	C₂	semidirect
3. C₃²×#168	C₂	semidirect
5. D₃×#168	C₃	direct (N × H)
12. C₃⋊#168	C₂×C₃	semidirect
15. C₃×#168	D₃	direct (N × H)
16. C₃×#168	C₂×C₃	semidirect
18. C₃×C₅×D₃	C₈	semidirect
32. C₃²×C₅	C₂×C₈	semidirect
35. #168	C₃×D₃	direct (N × H)
43. C₅×D₃	C₃×C₈	semidirect
55. C₃×D₃	#168	direct (N × H)
58. C₃×C₅	D₃×C₈	semidirect
59. C₃×C₅	C₂×C₃×C₈	semidirect
69. C₃²	C₂×#168	semidirect
77. D₃	C₃×#168	direct (N × H)
78. C₅	C₃×D₃×C₈	semidirect
83. C₃	D₃×#168	direct (N × H)
84. C₃	C₂×C₃×#168	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₃	2
C₃×C₃⋊#168	2
C₃²×#168	2
C₂×C₃×#168	3
D₃×#168	3
C₃×D₃×C₈	5

Subgroups — C3×D3×#168 #10623727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×#168 #10623727