Subgroups — C₂²×C₃×#157 #89118

Order 432.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#157	C₂	1	6	yes	maximal, direct product factor
3	C₂²×#157	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂²×C₃³	C₄	1	1	yes	characteristic, complemented
5	C₂×C₃×C₃⋊D₃	C₄	1	3	yes	complemented
6	C₂×C₃×C₃⋊D₃	C₂²	1	3	yes	—
7	C₃×#157	C₂²	1	4	yes	direct product factor
8	C₂²×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
9	C₂²×C₃×D₃	—	6	2	no	—
10	C₂×#157	C₂×C₃	1	6	yes	direct product factor
11	C₂×C₃³	C₂×C₄	1	3	yes	complemented
12	C₃×C₃⋊D₃	C₂×C₄	1	3	yes	complemented
13	C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
14	C₂²×C₃×C₄	—	9	1	no	maximal
15	C₂²×C₃²	—	2	2	no	—
16	C₂²×C₃²	—	4	2	no	—
17	C₂²×C₃²	C₃×C₄	1	1	yes	characteristic, complemented
18	C₂×C₃⋊D₃	C₃×C₄	1	3	yes	complemented
19	C₂×C₃⋊D₃	C₂²×C₃	1	3	yes	—
20	C₂×C₃×D₃	—	6	6	no	—
21	C₂×C₃×D₃	—	6	6	no	—
22	#157	C₂²×C₃	1	4	yes	direct product factor
23	C₃³	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
24	C₂³×C₃	—	9	1	no	—
25	C₂²×D₃	—	6	2	no	—
26	C₂×C₃×C₄	—	9	6	no	—
27	C₂×C₃²	—	2	6	no	—
28	C₂×C₃²	—	4	6	no	—
29	C₂×C₃²	C₂×C₃×C₄	1	3	yes	complemented
30	C₃⋊D₃	C₂×C₃×C₄	1	3	yes	complemented
31	C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
32	C₃×D₃	—	6	6	no	—
33	C₃×D₃	—	6	2	no	—
34	C₂²×C₄	—	9	1	no	Sylow
35	C₂²×C₃	—	2	2	no	—
36	C₂²×C₃	—	4	2	no	—
37	C₂²×C₃	—	9	3	no	—
38	C₂²×C₃	—	9	3	no	—
39	C₂²×C₃	#157	1	1	yes	characteristic, direct product factor
40	C₂×D₃	—	6	6	no	—
41	C₂×D₃	—	6	6	no	—
42	C₃×C₄	—	9	4	no	—
43	C₃²	—	2	2	no	—
44	C₃²	—	4	2	no	—
45	C₃²	C₂²×C₃×C₄	1	1	yes	characteristic, complemented
46	C₂³	—	9	1	no	—
47	C₂×C₄	—	9	6	no	—
48	C₂×C₃	—	2	6	no	—
49	C₂×C₃	—	4	6	no	—
50	C₂×C₃	—	9	1	no	—
51	C₂×C₃	—	9	3	no	—
52	C₂×C₃	C₂×#157	1	3	yes	direct product factor
53	D₃	—	6	6	no	—
54	D₃	—	6	2	no	—
55	C₂²	—	9	3	no	—
56	C₂²	—	9	3	no	—
57	C₂²	C₃×#157	1	1	yes	characteristic, direct product factor
58	C₄	—	9	4	no	—
59	C₃	—	2	2	no	—
60	C₃	—	4	2	no	—
61	C₃	C₂²×#157	1	1	yes	characteristic, direct product factor
62	C₂	—	9	1	no	—
63	C₂	—	9	3	no	—
64	C₂	C₂×C₃×#157	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	#157
commutator, hypocenter	C₃²	C₂²×C₃×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₃³	C₄
prime radical	C₂²×C₃×C₃⋊D₃	C₂
cosocle	C₃⋊D₃	C₂³×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#157	C₂	direct (N × H)
3. C₂²×#157	C₃	direct (N × H)
4. C₂²×C₃³	C₄	semidirect
5. C₂×C₃×C₃⋊D₃	C₄	semidirect
7. C₃×#157	C₂²	direct (N × H)
10. C₂×#157	C₂×C₃	direct (N × H)
11. C₂×C₃³	C₂×C₄	semidirect
12. C₃×C₃⋊D₃	C₂×C₄	semidirect
17. C₂²×C₃²	C₃×C₄	semidirect
18. C₂×C₃⋊D₃	C₃×C₄	semidirect
22. #157	C₂²×C₃	direct (N × H)
23. C₃³	C₂²×C₄	semidirect
29. C₂×C₃²	C₂×C₃×C₄	semidirect
30. C₃⋊D₃	C₂×C₃×C₄	semidirect
39. C₂²×C₃	#157	direct (N × H)
45. C₃²	C₂²×C₃×C₄	semidirect
52. C₂×C₃	C₂×#157	direct (N × H)
57. C₂²	C₃×#157	direct (N × H)
61. C₃	C₂²×#157	direct (N × H)
64. C₂	C₂×C₃×#157	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₃⋊D₃	2
C₂×C₃×#157	2
C₂²×#157	3
C₂²×C₃×C₄	9

Subgroups — C22×C3×#157 #89118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#157 #89118