Subgroups — C₂²×C₃³×C₈₁ #72187607

Order 8748.

3820 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₈₁	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃³×C₂₇	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₃²×C₈₁	C₃	1	39	yes	maximal, direct product factor
4	C₃³×C₈₁	C₂²	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₃³×C₂₇	C₂×C₃	1	3	yes	—
6	C₂×C₃²×C₈₁	C₂×C₃	1	117	yes	direct product factor
7	C₂²×C₃³×C₉	C₉	1	1	yes	characteristic
8	C₂²×C₃²×C₂₇	C₉	1	26	yes	—
9	C₂²×C₃²×C₂₇	C₃²	1	13	yes	—
10	C₂²×C₃×C₈₁	C₃²	1	117	yes	direct product factor
11	C₃³×C₂₇	C₂²×C₃	1	1	yes	characteristic
12	C₃²×C₈₁	C₂²×C₃	1	39	yes	direct product factor
13	C₂×C₃³×C₉	C₂×C₉	1	3	yes	—
14	C₂×C₃²×C₂₇	C₂×C₉	1	78	yes	—
15	C₂×C₃²×C₂₇	C₂×C₃²	1	39	yes	—
16	C₂×C₃×C₈₁	C₂×C₃²	1	351	yes	direct product factor
17	C₂²×C₃⁴	C₂₇	1	1	yes	characteristic
18	C₂²×C₃²×C₉	C₂₇	1	26	yes	—
19	C₂²×C₃²×C₉	C₃×C₉	1	13	yes	—
20	C₂²×C₃×C₂₇	C₃×C₉	1	104	yes	—
21	C₂²×C₃×C₂₇	C₃³	1	13	yes	—
22	C₂²×C₈₁	C₃³	1	27	yes	direct product factor
23	C₃³×C₉	C₂²×C₉	1	1	yes	characteristic
24	C₃²×C₂₇	C₂²×C₉	1	26	yes	—
25	C₃²×C₂₇	C₂²×C₃²	1	13	yes	—
26	C₃×C₈₁	C₂²×C₃²	1	117	yes	direct product factor
27	C₂×C₃⁴	C₂×C₂₇	1	3	yes	—
28	C₂×C₃²×C₉	C₂×C₂₇	1	78	yes	—
29	C₂×C₃²×C₉	C₂×C₃×C₉	1	39	yes	—
30	C₂×C₃×C₂₇	C₂×C₃×C₉	1	312	yes	—
31	C₂×C₃×C₂₇	C₂×C₃³	1	39	yes	—
32	C₂×C₈₁	C₂×C₃³	1	81	yes	direct product factor
33	C₂²×C₃³	C₈₁	1	27	yes	direct product factor
34	C₂²×C₃³	C₃×C₂₇	1	13	yes	—
35	C₂²×C₃×C₉	C₃×C₂₇	1	104	yes	—
36	C₂²×C₃×C₉	C₃²×C₉	1	13	yes	—
37	C₂²×C₂₇	C₃²×C₉	1	26	yes	—
38	C₂²×C₂₇	C₃⁴	1	1	yes	characteristic
39	C₃⁴	C₂²×C₂₇	1	1	yes	characteristic
40	C₃²×C₉	C₂²×C₂₇	1	26	yes	—
41	C₃²×C₉	C₂²×C₃×C₉	1	13	yes	—
42	C₃×C₂₇	C₂²×C₃×C₉	1	104	yes	—
43	C₃×C₂₇	C₂²×C₃³	1	13	yes	—
44	C₈₁	C₂²×C₃³	1	27	yes	direct product factor
45	C₂×C₃³	C₂×C₈₁	1	81	yes	direct product factor
46	C₂×C₃³	C₂×C₃×C₂₇	1	39	yes	—
47	C₂×C₃×C₉	C₂×C₃×C₂₇	1	312	yes	—
48	C₂×C₃×C₉	C₂×C₃²×C₉	1	39	yes	—
49	C₂×C₂₇	C₂×C₃²×C₉	1	78	yes	—
50	C₂×C₂₇	C₂×C₃⁴	1	3	yes	—
51	C₂²×C₃²	C₃×C₈₁	1	117	yes	direct product factor
52	C₂²×C₃²	C₃²×C₂₇	1	13	yes	—
53	C₂²×C₉	C₃²×C₂₇	1	26	yes	—
54	C₂²×C₉	C₃³×C₉	1	1	yes	characteristic
55	C₃³	C₂²×C₈₁	1	27	yes	direct product factor
56	C₃³	C₂²×C₃×C₂₇	1	13	yes	—
57	C₃×C₉	C₂²×C₃×C₂₇	1	104	yes	—
58	C₃×C₉	C₂²×C₃²×C₉	1	13	yes	—
59	C₂₇	C₂²×C₃²×C₉	1	26	yes	—
60	C₂₇	C₂²×C₃⁴	1	1	yes	characteristic
61	C₂×C₃²	C₂×C₃×C₈₁	1	351	yes	direct product factor
62	C₂×C₃²	C₂×C₃²×C₂₇	1	39	yes	—
63	C₂×C₉	C₂×C₃²×C₂₇	1	78	yes	—
64	C₂×C₉	C₂×C₃³×C₉	1	3	yes	—
65	C₂²×C₃	C₃²×C₈₁	1	39	yes	direct product factor
66	C₂²×C₃	C₃³×C₂₇	1	1	yes	characteristic
67	C₃²	C₂²×C₃×C₈₁	1	117	yes	direct product factor
68	C₃²	C₂²×C₃²×C₂₇	1	13	yes	—
69	C₉	C₂²×C₃²×C₂₇	1	26	yes	—
70	C₉	C₂²×C₃³×C₉	1	1	yes	characteristic
71	C₂×C₃	C₂×C₃²×C₈₁	1	117	yes	direct product factor
72	C₂×C₃	C₂×C₃³×C₂₇	1	3	yes	—
73	C₂²	C₃³×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
74	C₃	C₂²×C₃²×C₈₁	1	39	yes	direct product factor
75	C₃	C₂²×C₃³×C₂₇	1	1	yes	characteristic
76	C₂	C₂×C₃³×C₈₁	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₇	C₂²×C₃⁴
socle, prime radical	C₂²×C₃⁴	C₂₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×C₈₁	C₂	direct (N × H)
3. C₂²×C₃²×C₈₁	C₃	direct (N × H)
4. C₃³×C₈₁	C₂²	direct (N × H)
6. C₂×C₃²×C₈₁	C₂×C₃	direct (N × H)
10. C₂²×C₃×C₈₁	C₃²	direct (N × H)
12. C₃²×C₈₁	C₂²×C₃	direct (N × H)
16. C₂×C₃×C₈₁	C₂×C₃²	direct (N × H)
22. C₂²×C₈₁	C₃³	direct (N × H)
26. C₃×C₈₁	C₂²×C₃²	direct (N × H)
32. C₂×C₈₁	C₂×C₃³	direct (N × H)
33. C₂²×C₃³	C₈₁	direct (N × H)
44. C₈₁	C₂²×C₃³	direct (N × H)
45. C₂×C₃³	C₂×C₈₁	direct (N × H)
51. C₂²×C₃²	C₃×C₈₁	direct (N × H)
55. C₃³	C₂²×C₈₁	direct (N × H)
61. C₂×C₃²	C₂×C₃×C₈₁	direct (N × H)
65. C₂²×C₃	C₃²×C₈₁	direct (N × H)
67. C₃²	C₂²×C₃×C₈₁	direct (N × H)
71. C₂×C₃	C₂×C₃²×C₈₁	direct (N × H)
73. C₂²	C₃³×C₈₁	direct (N × H)
74. C₃	C₂²×C₃²×C₈₁	direct (N × H)
76. C₂	C₂×C₃³×C₈₁	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×C₃³×C₈₁	2
C₂²×C₃³×C₂₇	3
C₂²×C₃²×C₈₁	3

Subgroups — C22×C33×C81 #72187607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃³×C₈₁ #72187607