Subgroups — C₂×C₃³×C₅×C₈₁ #72199334

Order 21870.

3056 conjugacy classes of subgroups.

The subgroup lattice has 104 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₅×C₈₁	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃³×C₅×C₂₇	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₅×C₈₁	C₃	1	39	yes	maximal, direct product factor
4	C₂×C₃³×C₈₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃³×C₅×C₉	C₉	1	1	yes	characteristic
6	C₂×C₃²×C₅×C₂₇	C₉	1	26	yes	—
7	C₂×C₃²×C₅×C₂₇	C₃²	1	13	yes	—
8	C₂×C₃×C₅×C₈₁	C₃²	1	117	yes	direct product factor
9	C₃³×C₅×C₂₇	C₂×C₃	1	1	yes	characteristic
10	C₃²×C₅×C₈₁	C₂×C₃	1	39	yes	direct product factor
11	C₃³×C₈₁	C₂×C₅	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₃³×C₂₇	C₃×C₅	1	1	yes	characteristic
13	C₂×C₃²×C₈₁	C₃×C₅	1	39	yes	direct product factor
14	C₃³×C₅×C₉	C₂×C₉	1	1	yes	characteristic
15	C₃²×C₅×C₂₇	C₂×C₉	1	26	yes	—
16	C₃²×C₅×C₂₇	C₂×C₃²	1	13	yes	—
17	C₃×C₅×C₈₁	C₂×C₃²	1	117	yes	direct product factor
18	C₂×C₃⁴×C₅	C₂₇	1	1	yes	characteristic
19	C₂×C₃²×C₅×C₉	C₂₇	1	26	yes	—
20	C₂×C₃²×C₅×C₉	C₃×C₉	1	13	yes	—
21	C₂×C₃×C₅×C₂₇	C₃×C₉	1	104	yes	—
22	C₂×C₃×C₅×C₂₇	C₃³	1	13	yes	—
23	C₂×C₅×C₈₁	C₃³	1	27	yes	direct product factor
24	C₃³×C₂₇	C₂×C₃×C₅	1	1	yes	characteristic
25	C₃²×C₈₁	C₂×C₃×C₅	1	39	yes	direct product factor
26	C₂×C₃³×C₉	C₅×C₉	1	1	yes	characteristic
27	C₂×C₃²×C₂₇	C₅×C₉	1	26	yes	—
28	C₂×C₃²×C₂₇	C₃²×C₅	1	13	yes	—
29	C₂×C₃×C₈₁	C₃²×C₅	1	117	yes	direct product factor
30	C₃⁴×C₅	C₂×C₂₇	1	1	yes	characteristic
31	C₃²×C₅×C₉	C₂×C₂₇	1	26	yes	—
32	C₃²×C₅×C₉	C₂×C₃×C₉	1	13	yes	—
33	C₃×C₅×C₂₇	C₂×C₃×C₉	1	104	yes	—
34	C₃×C₅×C₂₇	C₂×C₃³	1	13	yes	—
35	C₅×C₈₁	C₂×C₃³	1	27	yes	direct product factor
36	C₂×C₃³×C₅	C₈₁	1	27	yes	direct product factor
37	C₂×C₃³×C₅	C₃×C₂₇	1	13	yes	—
38	C₂×C₃×C₅×C₉	C₃×C₂₇	1	104	yes	—
39	C₂×C₃×C₅×C₉	C₃²×C₉	1	13	yes	—
40	C₂×C₅×C₂₇	C₃²×C₉	1	26	yes	—
41	C₂×C₅×C₂₇	C₃⁴	1	1	yes	characteristic
42	C₃³×C₉	C₂×C₅×C₉	1	1	yes	characteristic
43	C₃²×C₂₇	C₂×C₅×C₉	1	26	yes	—
44	C₃²×C₂₇	C₂×C₃²×C₅	1	13	yes	—
45	C₃×C₈₁	C₂×C₃²×C₅	1	117	yes	direct product factor
46	C₂×C₃⁴	C₅×C₂₇	1	1	yes	characteristic
47	C₂×C₃²×C₉	C₅×C₂₇	1	26	yes	—
48	C₂×C₃²×C₉	C₃×C₅×C₉	1	13	yes	—
49	C₂×C₃×C₂₇	C₃×C₅×C₉	1	104	yes	—
50	C₂×C₃×C₂₇	C₃³×C₅	1	13	yes	—
51	C₂×C₈₁	C₃³×C₅	1	27	yes	direct product factor
52	C₃³×C₅	C₂×C₈₁	1	27	yes	direct product factor
53	C₃³×C₅	C₂×C₃×C₂₇	1	13	yes	—
54	C₃×C₅×C₉	C₂×C₃×C₂₇	1	104	yes	—
55	C₃×C₅×C₉	C₂×C₃²×C₉	1	13	yes	—
56	C₅×C₂₇	C₂×C₃²×C₉	1	26	yes	—
57	C₅×C₂₇	C₂×C₃⁴	1	1	yes	characteristic
58	C₂×C₃²×C₅	C₃×C₈₁	1	117	yes	direct product factor
59	C₂×C₃²×C₅	C₃²×C₂₇	1	13	yes	—
60	C₂×C₅×C₉	C₃²×C₂₇	1	26	yes	—
61	C₂×C₅×C₉	C₃³×C₉	1	1	yes	characteristic
62	C₃⁴	C₂×C₅×C₂₇	1	1	yes	characteristic
63	C₃²×C₉	C₂×C₅×C₂₇	1	26	yes	—
64	C₃²×C₉	C₂×C₃×C₅×C₉	1	13	yes	—
65	C₃×C₂₇	C₂×C₃×C₅×C₉	1	104	yes	—
66	C₃×C₂₇	C₂×C₃³×C₅	1	13	yes	—
67	C₈₁	C₂×C₃³×C₅	1	27	yes	direct product factor
68	C₂×C₃³	C₅×C₈₁	1	27	yes	direct product factor
69	C₂×C₃³	C₃×C₅×C₂₇	1	13	yes	—
70	C₂×C₃×C₉	C₃×C₅×C₂₇	1	104	yes	—
71	C₂×C₃×C₉	C₃²×C₅×C₉	1	13	yes	—
72	C₂×C₂₇	C₃²×C₅×C₉	1	26	yes	—
73	C₂×C₂₇	C₃⁴×C₅	1	1	yes	characteristic
74	C₃²×C₅	C₂×C₃×C₈₁	1	117	yes	direct product factor
75	C₃²×C₅	C₂×C₃²×C₂₇	1	13	yes	—
76	C₅×C₉	C₂×C₃²×C₂₇	1	26	yes	—
77	C₅×C₉	C₂×C₃³×C₉	1	1	yes	characteristic
78	C₂×C₃×C₅	C₃²×C₈₁	1	39	yes	direct product factor
79	C₂×C₃×C₅	C₃³×C₂₇	1	1	yes	characteristic
80	C₃³	C₂×C₅×C₈₁	1	27	yes	direct product factor
81	C₃³	C₂×C₃×C₅×C₂₇	1	13	yes	—
82	C₃×C₉	C₂×C₃×C₅×C₂₇	1	104	yes	—
83	C₃×C₉	C₂×C₃²×C₅×C₉	1	13	yes	—
84	C₂₇	C₂×C₃²×C₅×C₉	1	26	yes	—
85	C₂₇	C₂×C₃⁴×C₅	1	1	yes	characteristic
86	C₂×C₃²	C₃×C₅×C₈₁	1	117	yes	direct product factor
87	C₂×C₃²	C₃²×C₅×C₂₇	1	13	yes	—
88	C₂×C₉	C₃²×C₅×C₂₇	1	26	yes	—
89	C₂×C₉	C₃³×C₅×C₉	1	1	yes	characteristic
90	C₃×C₅	C₂×C₃²×C₈₁	1	39	yes	direct product factor
91	C₃×C₅	C₂×C₃³×C₂₇	1	1	yes	characteristic
92	C₂×C₅	C₃³×C₈₁	1	1	yes	characteristic, direct product factor, Hall
93	C₃²	C₂×C₃×C₅×C₈₁	1	117	yes	direct product factor
94	C₃²	C₂×C₃²×C₅×C₂₇	1	13	yes	—
95	C₉	C₂×C₃²×C₅×C₂₇	1	26	yes	—
96	C₉	C₂×C₃³×C₅×C₉	1	1	yes	characteristic
97	C₂×C₃	C₃²×C₅×C₈₁	1	39	yes	direct product factor
98	C₂×C₃	C₃³×C₅×C₂₇	1	1	yes	characteristic
99	C₅	C₂×C₃³×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
100	C₃	C₂×C₃²×C₅×C₈₁	1	39	yes	direct product factor
101	C₃	C₂×C₃³×C₅×C₂₇	1	1	yes	characteristic
102	C₂	C₃³×C₅×C₈₁	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₇	C₂×C₃⁴×C₅
socle, prime radical	C₂×C₃⁴×C₅	C₂₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃³×C₅×C₈₁	C₂	direct (N × H)
3. C₂×C₃²×C₅×C₈₁	C₃	direct (N × H)
4. C₂×C₃³×C₈₁	C₅	direct (N × H)
8. C₂×C₃×C₅×C₈₁	C₃²	direct (N × H)
10. C₃²×C₅×C₈₁	C₂×C₃	direct (N × H)
11. C₃³×C₈₁	C₂×C₅	direct (N × H)
13. C₂×C₃²×C₈₁	C₃×C₅	direct (N × H)
17. C₃×C₅×C₈₁	C₂×C₃²	direct (N × H)
23. C₂×C₅×C₈₁	C₃³	direct (N × H)
25. C₃²×C₈₁	C₂×C₃×C₅	direct (N × H)
29. C₂×C₃×C₈₁	C₃²×C₅	direct (N × H)
35. C₅×C₈₁	C₂×C₃³	direct (N × H)
36. C₂×C₃³×C₅	C₈₁	direct (N × H)
45. C₃×C₈₁	C₂×C₃²×C₅	direct (N × H)
51. C₂×C₈₁	C₃³×C₅	direct (N × H)
52. C₃³×C₅	C₂×C₈₁	direct (N × H)
58. C₂×C₃²×C₅	C₃×C₈₁	direct (N × H)
67. C₈₁	C₂×C₃³×C₅	direct (N × H)
68. C₂×C₃³	C₅×C₈₁	direct (N × H)
74. C₃²×C₅	C₂×C₃×C₈₁	direct (N × H)
78. C₂×C₃×C₅	C₃²×C₈₁	direct (N × H)
80. C₃³	C₂×C₅×C₈₁	direct (N × H)
86. C₂×C₃²	C₃×C₅×C₈₁	direct (N × H)
90. C₃×C₅	C₂×C₃²×C₈₁	direct (N × H)
92. C₂×C₅	C₃³×C₈₁	direct (N × H)
93. C₃²	C₂×C₃×C₅×C₈₁	direct (N × H)
97. C₂×C₃	C₃²×C₅×C₈₁	direct (N × H)
99. C₅	C₂×C₃³×C₈₁	direct (N × H)
100. C₃	C₂×C₃²×C₅×C₈₁	direct (N × H)
102. C₂	C₃³×C₅×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
C₃³×C₅×C₈₁	2
C₂×C₃³×C₅×C₂₇	3
C₂×C₃²×C₅×C₈₁	3
C₂×C₃³×C₈₁	5

Subgroups — C2×C33×C5×C81 #72199334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×C₅×C₈₁ #72199334