Subgroups — C₂×C₃³×C₇×C₈₁ #72207384

Order 30618.

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×C₇×C₈₁	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃³×C₇×C₂₇	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₇×C₈₁	C₃	1	39	yes	maximal, direct product factor
4	C₂×C₃³×C₈₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃³×C₇×C₉	C₉	1	1	yes	characteristic
6	C₂×C₃²×C₇×C₂₇	C₉	1	26	yes	—
7	C₂×C₃²×C₇×C₂₇	C₃²	1	13	yes	—
8	C₂×C₃×C₇×C₈₁	C₃²	1	117	yes	direct product factor
9	C₃³×C₇×C₂₇	C₂×C₃	1	1	yes	characteristic
10	C₃²×C₇×C₈₁	C₂×C₃	1	39	yes	direct product factor
11	C₃³×C₈₁	C₂×C₇	1	1	yes	characteristic, direct product factor, Sylow
12	C₃³×C₇×C₉	C₂×C₉	1	1	yes	characteristic
13	C₃²×C₇×C₂₇	C₂×C₉	1	26	yes	—
14	C₃²×C₇×C₂₇	C₂×C₃²	1	13	yes	—
15	C₃×C₇×C₈₁	C₂×C₃²	1	117	yes	direct product factor
16	C₂×C₃³×C₂₇	C₃×C₇	1	1	yes	characteristic
17	C₂×C₃²×C₈₁	C₃×C₇	1	39	yes	direct product factor
18	C₂×C₃⁴×C₇	C₂₇	1	1	yes	characteristic
19	C₂×C₃²×C₇×C₉	C₂₇	1	26	yes	—
20	C₂×C₃²×C₇×C₉	C₃×C₉	1	13	yes	—
21	C₂×C₃×C₇×C₂₇	C₃×C₉	1	104	yes	—
22	C₂×C₃×C₇×C₂₇	C₃³	1	13	yes	—
23	C₂×C₇×C₈₁	C₃³	1	27	yes	direct product factor
24	C₃³×C₂₇	C₂×C₃×C₇	1	1	yes	characteristic
25	C₃²×C₈₁	C₂×C₃×C₇	1	39	yes	direct product factor
26	C₃⁴×C₇	C₂×C₂₇	1	1	yes	characteristic
27	C₃²×C₇×C₉	C₂×C₂₇	1	26	yes	—
28	C₃²×C₇×C₉	C₂×C₃×C₉	1	13	yes	—
29	C₃×C₇×C₂₇	C₂×C₃×C₉	1	104	yes	—
30	C₃×C₇×C₂₇	C₂×C₃³	1	13	yes	—
31	C₇×C₈₁	C₂×C₃³	1	27	yes	direct product factor
32	C₂×C₃³×C₉	C₇×C₉	1	1	yes	characteristic
33	C₂×C₃²×C₂₇	C₇×C₉	1	26	yes	—
34	C₂×C₃²×C₂₇	C₃²×C₇	1	13	yes	—
35	C₂×C₃×C₈₁	C₃²×C₇	1	117	yes	direct product factor
36	C₂×C₃³×C₇	C₈₁	1	27	yes	direct product factor
37	C₂×C₃³×C₇	C₃×C₂₇	1	13	yes	—
38	C₂×C₃×C₇×C₉	C₃×C₂₇	1	104	yes	—
39	C₂×C₃×C₇×C₉	C₃²×C₉	1	13	yes	—
40	C₂×C₇×C₂₇	C₃²×C₉	1	26	yes	—
41	C₂×C₇×C₂₇	C₃⁴	1	1	yes	characteristic
42	C₃³×C₉	C₂×C₇×C₉	1	1	yes	characteristic
43	C₃²×C₂₇	C₂×C₇×C₉	1	26	yes	—
44	C₃²×C₂₇	C₂×C₃²×C₇	1	13	yes	—
45	C₃×C₈₁	C₂×C₃²×C₇	1	117	yes	direct product factor
46	C₃³×C₇	C₂×C₈₁	1	27	yes	direct product factor
47	C₃³×C₇	C₂×C₃×C₂₇	1	13	yes	—
48	C₃×C₇×C₉	C₂×C₃×C₂₇	1	104	yes	—
49	C₃×C₇×C₉	C₂×C₃²×C₉	1	13	yes	—
50	C₇×C₂₇	C₂×C₃²×C₉	1	26	yes	—
51	C₇×C₂₇	C₂×C₃⁴	1	1	yes	characteristic
52	C₂×C₃⁴	C₇×C₂₇	1	1	yes	characteristic
53	C₂×C₃²×C₉	C₇×C₂₇	1	26	yes	—
54	C₂×C₃²×C₉	C₃×C₇×C₉	1	13	yes	—
55	C₂×C₃×C₂₇	C₃×C₇×C₉	1	104	yes	—
56	C₂×C₃×C₂₇	C₃³×C₇	1	13	yes	—
57	C₂×C₈₁	C₃³×C₇	1	27	yes	direct product factor
58	C₂×C₃²×C₇	C₃×C₈₁	1	117	yes	direct product factor
59	C₂×C₃²×C₇	C₃²×C₂₇	1	13	yes	—
60	C₂×C₇×C₉	C₃²×C₂₇	1	26	yes	—
61	C₂×C₇×C₉	C₃³×C₉	1	1	yes	characteristic
62	C₃⁴	C₂×C₇×C₂₇	1	1	yes	characteristic
63	C₃²×C₉	C₂×C₇×C₂₇	1	26	yes	—
64	C₃²×C₉	C₂×C₃×C₇×C₉	1	13	yes	—
65	C₃×C₂₇	C₂×C₃×C₇×C₉	1	104	yes	—
66	C₃×C₂₇	C₂×C₃³×C₇	1	13	yes	—
67	C₈₁	C₂×C₃³×C₇	1	27	yes	direct product factor
68	C₃²×C₇	C₂×C₃×C₈₁	1	117	yes	direct product factor
69	C₃²×C₇	C₂×C₃²×C₂₇	1	13	yes	—
70	C₇×C₉	C₂×C₃²×C₂₇	1	26	yes	—
71	C₇×C₉	C₂×C₃³×C₉	1	1	yes	characteristic
72	C₂×C₃³	C₇×C₈₁	1	27	yes	direct product factor
73	C₂×C₃³	C₃×C₇×C₂₇	1	13	yes	—
74	C₂×C₃×C₉	C₃×C₇×C₂₇	1	104	yes	—
75	C₂×C₃×C₉	C₃²×C₇×C₉	1	13	yes	—
76	C₂×C₂₇	C₃²×C₇×C₉	1	26	yes	—
77	C₂×C₂₇	C₃⁴×C₇	1	1	yes	characteristic
78	C₂×C₃×C₇	C₃²×C₈₁	1	39	yes	direct product factor
79	C₂×C₃×C₇	C₃³×C₂₇	1	1	yes	characteristic
80	C₃³	C₂×C₇×C₈₁	1	27	yes	direct product factor
81	C₃³	C₂×C₃×C₇×C₂₇	1	13	yes	—
82	C₃×C₉	C₂×C₃×C₇×C₂₇	1	104	yes	—
83	C₃×C₉	C₂×C₃²×C₇×C₉	1	13	yes	—
84	C₂₇	C₂×C₃²×C₇×C₉	1	26	yes	—
85	C₂₇	C₂×C₃⁴×C₇	1	1	yes	characteristic
86	C₃×C₇	C₂×C₃²×C₈₁	1	39	yes	direct product factor
87	C₃×C₇	C₂×C₃³×C₂₇	1	1	yes	characteristic
88	C₂×C₃²	C₃×C₇×C₈₁	1	117	yes	direct product factor
89	C₂×C₃²	C₃²×C₇×C₂₇	1	13	yes	—
90	C₂×C₉	C₃²×C₇×C₂₇	1	26	yes	—
91	C₂×C₉	C₃³×C₇×C₉	1	1	yes	characteristic
92	C₂×C₇	C₃³×C₈₁	1	1	yes	characteristic, direct product factor, Hall
93	C₃²	C₂×C₃×C₇×C₈₁	1	117	yes	direct product factor
94	C₃²	C₂×C₃²×C₇×C₂₇	1	13	yes	—
95	C₉	C₂×C₃²×C₇×C₂₇	1	26	yes	—
96	C₉	C₂×C₃³×C₇×C₉	1	1	yes	characteristic
97	C₇	C₂×C₃³×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×C₃	C₃²×C₇×C₈₁	1	39	yes	direct product factor
99	C₂×C₃	C₃³×C₇×C₂₇	1	1	yes	characteristic
100	C₃	C₂×C₃²×C₇×C₈₁	1	39	yes	direct product factor
101	C₃	C₂×C₃³×C₇×C₂₇	1	1	yes	characteristic
102	C₂	C₃³×C₇×C₈₁	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₇	C₂×C₃⁴×C₇
socle, prime radical	C₂×C₃⁴×C₇	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×C₇×C₈₁	C₂	direct (N × H)
3. C₂×C₃²×C₇×C₈₁	C₃	direct (N × H)
4. C₂×C₃³×C₈₁	C₇	direct (N × H)
8. C₂×C₃×C₇×C₈₁	C₃²	direct (N × H)
10. C₃²×C₇×C₈₁	C₂×C₃	direct (N × H)
11. C₃³×C₈₁	C₂×C₇	direct (N × H)
15. C₃×C₇×C₈₁	C₂×C₃²	direct (N × H)
17. C₂×C₃²×C₈₁	C₃×C₇	direct (N × H)
23. C₂×C₇×C₈₁	C₃³	direct (N × H)
25. C₃²×C₈₁	C₂×C₃×C₇	direct (N × H)
31. C₇×C₈₁	C₂×C₃³	direct (N × H)
35. C₂×C₃×C₈₁	C₃²×C₇	direct (N × H)
36. C₂×C₃³×C₇	C₈₁	direct (N × H)
45. C₃×C₈₁	C₂×C₃²×C₇	direct (N × H)
46. C₃³×C₇	C₂×C₈₁	direct (N × H)
57. C₂×C₈₁	C₃³×C₇	direct (N × H)
58. C₂×C₃²×C₇	C₃×C₈₁	direct (N × H)
67. C₈₁	C₂×C₃³×C₇	direct (N × H)
68. C₃²×C₇	C₂×C₃×C₈₁	direct (N × H)
72. C₂×C₃³	C₇×C₈₁	direct (N × H)
78. C₂×C₃×C₇	C₃²×C₈₁	direct (N × H)
80. C₃³	C₂×C₇×C₈₁	direct (N × H)
86. C₃×C₇	C₂×C₃²×C₈₁	direct (N × H)
88. C₂×C₃²	C₃×C₇×C₈₁	direct (N × H)
92. C₂×C₇	C₃³×C₈₁	direct (N × H)
93. C₃²	C₂×C₃×C₇×C₈₁	direct (N × H)
97. C₇	C₂×C₃³×C₈₁	direct (N × H)
98. C₂×C₃	C₃²×C₇×C₈₁	direct (N × H)
100. C₃	C₂×C₃²×C₇×C₈₁	direct (N × H)
102. C₂	C₃³×C₇×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×C₇×C₈₁	2
C₂×C₃³×C₇×C₂₇	3
C₂×C₃²×C₇×C₈₁	3
C₂×C₃³×C₈₁	7

Subgroups — C2×C33×C7×C81 #72207384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×C₇×C₈₁ #72207384