Subgroups — C₂²×C₃³×C₃₂ #72183113

Order 3456.

1680 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×C₃₂	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃²×C₃₂	C₃	1	13	yes	maximal, direct product factor
4	C₂²×C₃³×C₈	C₄	1	1	yes	characteristic
5	C₂×C₃³×C₁₆	C₄	1	3	yes	—
6	C₂×C₃³×C₁₆	C₂²	1	3	yes	—
7	C₃³×C₃₂	C₂²	1	4	yes	direct product factor
8	C₂²×C₃²×C₁₆	C₂×C₃	1	13	yes	—
9	C₂×C₃²×C₃₂	C₂×C₃	1	78	yes	direct product factor
10	C₂²×C₃³×C₄	C₈	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₈	1	3	yes	—
12	C₂×C₃³×C₈	C₂×C₄	1	3	yes	—
13	C₃³×C₁₆	C₂×C₄	1	3	yes	—
14	C₃³×C₁₆	C₂³	1	1	yes	characteristic
15	C₂²×C₃×C₃₂	C₃²	1	13	yes	direct product factor
16	C₂²×C₃²×C₈	C₃×C₄	1	13	yes	—
17	C₂×C₃²×C₁₆	C₃×C₄	1	39	yes	—
18	C₂×C₃²×C₁₆	C₂²×C₃	1	39	yes	—
19	C₃²×C₃₂	C₂²×C₃	1	52	yes	direct product factor
20	C₂³×C₃³	C₁₆	1	1	yes	characteristic
21	C₂×C₃³×C₄	C₁₆	1	3	yes	—
22	C₂×C₃³×C₄	C₂×C₈	1	3	yes	—
23	C₃³×C₈	C₂×C₈	1	3	yes	—
24	C₃³×C₈	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₁₆	C₂×C₃²	1	13	yes	—
26	C₂×C₃×C₃₂	C₂×C₃²	1	78	yes	direct product factor
27	C₂²×C₃²×C₄	C₃×C₈	1	13	yes	—
28	C₂×C₃²×C₈	C₃×C₈	1	39	yes	—
29	C₂×C₃²×C₈	C₂×C₃×C₄	1	39	yes	—
30	C₃²×C₁₆	C₂×C₃×C₄	1	39	yes	—
31	C₃²×C₁₆	C₂³×C₃	1	13	yes	—
32	C₂²×C₃₂	C₃³	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²×C₃³	C₃₂	1	4	yes	direct product factor
34	C₂²×C₃³	C₂×C₁₆	1	3	yes	—
35	C₃³×C₄	C₂×C₁₆	1	3	yes	—
36	C₃³×C₄	C₂²×C₈	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₃²×C₄	1	13	yes	—
38	C₂×C₃×C₁₆	C₃²×C₄	1	39	yes	—
39	C₂×C₃×C₁₆	C₂²×C₃²	1	39	yes	—
40	C₃×C₃₂	C₂²×C₃²	1	52	yes	direct product factor
41	C₂³×C₃²	C₃×C₁₆	1	13	yes	—
42	C₂×C₃²×C₄	C₃×C₁₆	1	39	yes	—
43	C₂×C₃²×C₄	C₂×C₃×C₈	1	39	yes	—
44	C₃²×C₈	C₂×C₃×C₈	1	39	yes	—
45	C₃²×C₈	C₂²×C₃×C₄	1	13	yes	—
46	C₂²×C₁₆	C₂×C₃³	1	1	yes	characteristic
47	C₂×C₃₂	C₂×C₃³	1	6	yes	direct product factor
48	C₂×C₃³	C₂×C₃₂	1	6	yes	direct product factor
49	C₂×C₃³	C₂²×C₁₆	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₃²×C₈	1	13	yes	—
51	C₂×C₃×C₈	C₃²×C₈	1	39	yes	—
52	C₂×C₃×C₈	C₂×C₃²×C₄	1	39	yes	—
53	C₃×C₁₆	C₂×C₃²×C₄	1	39	yes	—
54	C₃×C₁₆	C₂³×C₃²	1	13	yes	—
55	C₂²×C₃²	C₃×C₃₂	1	52	yes	direct product factor
56	C₂²×C₃²	C₂×C₃×C₁₆	1	39	yes	—
57	C₃²×C₄	C₂×C₃×C₁₆	1	39	yes	—
58	C₃²×C₄	C₂²×C₃×C₈	1	13	yes	—
59	C₂²×C₈	C₃³×C₄	1	1	yes	characteristic
60	C₂×C₁₆	C₃³×C₄	1	3	yes	—
61	C₂×C₁₆	C₂²×C₃³	1	3	yes	—
62	C₃₂	C₂²×C₃³	1	4	yes	direct product factor
63	C₃³	C₂²×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
64	C₂³×C₃	C₃²×C₁₆	1	13	yes	—
65	C₂×C₃×C₄	C₃²×C₁₆	1	39	yes	—
66	C₂×C₃×C₄	C₂×C₃²×C₈	1	39	yes	—
67	C₃×C₈	C₂×C₃²×C₈	1	39	yes	—
68	C₃×C₈	C₂²×C₃²×C₄	1	13	yes	—
69	C₂×C₃²	C₂×C₃×C₃₂	1	78	yes	direct product factor
70	C₂×C₃²	C₂²×C₃×C₁₆	1	13	yes	—
71	C₂²×C₄	C₃³×C₈	1	1	yes	characteristic
72	C₂×C₈	C₃³×C₈	1	3	yes	—
73	C₂×C₈	C₂×C₃³×C₄	1	3	yes	—
74	C₁₆	C₂×C₃³×C₄	1	3	yes	—
75	C₁₆	C₂³×C₃³	1	1	yes	characteristic
76	C₂²×C₃	C₃²×C₃₂	1	52	yes	direct product factor
77	C₂²×C₃	C₂×C₃²×C₁₆	1	39	yes	—
78	C₃×C₄	C₂×C₃²×C₁₆	1	39	yes	—
79	C₃×C₄	C₂²×C₃²×C₈	1	13	yes	—
80	C₃²	C₂²×C₃×C₃₂	1	13	yes	direct product factor
81	C₂³	C₃³×C₁₆	1	1	yes	characteristic
82	C₂×C₄	C₃³×C₁₆	1	3	yes	—
83	C₂×C₄	C₂×C₃³×C₈	1	3	yes	—
84	C₈	C₂×C₃³×C₈	1	3	yes	—
85	C₈	C₂²×C₃³×C₄	1	1	yes	characteristic
86	C₂×C₃	C₂×C₃²×C₃₂	1	78	yes	direct product factor
87	C₂×C₃	C₂²×C₃²×C₁₆	1	13	yes	—
88	C₂²	C₃³×C₃₂	1	4	yes	direct product factor
89	C₂²	C₂×C₃³×C₁₆	1	3	yes	—
90	C₄	C₂×C₃³×C₁₆	1	3	yes	—
91	C₄	C₂²×C₃³×C₈	1	1	yes	characteristic
92	C₃	C₂²×C₃²×C₃₂	1	13	yes	direct product factor
93	C₂	C₂×C₃³×C₃₂	1	6	yes	direct product factor
94	C₂	C₂²×C₃³×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂³×C₃³
socle, prime radical	C₂³×C₃³	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×C₃₂	C₂	direct (N × H)
3. C₂²×C₃²×C₃₂	C₃	direct (N × H)
7. C₃³×C₃₂	C₂²	direct (N × H)
9. C₂×C₃²×C₃₂	C₂×C₃	direct (N × H)
15. C₂²×C₃×C₃₂	C₃²	direct (N × H)
19. C₃²×C₃₂	C₂²×C₃	direct (N × H)
26. C₂×C₃×C₃₂	C₂×C₃²	direct (N × H)
32. C₂²×C₃₂	C₃³	direct (N × H)
33. C₂²×C₃³	C₃₂	direct (N × H)
40. C₃×C₃₂	C₂²×C₃²	direct (N × H)
47. C₂×C₃₂	C₂×C₃³	direct (N × H)
48. C₂×C₃³	C₂×C₃₂	direct (N × H)
55. C₂²×C₃²	C₃×C₃₂	direct (N × H)
62. C₃₂	C₂²×C₃³	direct (N × H)
63. C₃³	C₂²×C₃₂	direct (N × H)
69. C₂×C₃²	C₂×C₃×C₃₂	direct (N × H)
76. C₂²×C₃	C₃²×C₃₂	direct (N × H)
80. C₃²	C₂²×C₃×C₃₂	direct (N × H)
86. C₂×C₃	C₂×C₃²×C₃₂	direct (N × H)
88. C₂²	C₃³×C₃₂	direct (N × H)
92. C₃	C₂²×C₃²×C₃₂	direct (N × H)
93. C₂	C₂×C₃³×C₃₂	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×C₃³×C₁₆	2
C₂×C₃³×C₃₂	2
C₂²×C₃²×C₃₂	3

Subgroups — C22×C33×C32 #72183113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃³×C₃₂ #72183113