Subgroups — C₂²×C₃³×C₇×C₂₇ #72197989

Order 20412.

5800 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₇×C₂₇	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃³×C₇×C₉	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₃²×C₇×C₂₇	C₃	1	39	yes	maximal, direct product factor
4	C₂×C₃³×C₇×C₉	C₂×C₃	1	3	yes	—
5	C₂×C₃²×C₇×C₂₇	C₂×C₃	1	117	yes	direct product factor
6	C₂²×C₃³×C₂₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃⁴×C₇	C₉	1	1	yes	characteristic
8	C₂²×C₃²×C₇×C₉	C₉	1	26	yes	—
9	C₂²×C₃²×C₇×C₉	C₃²	1	13	yes	—
10	C₂²×C₃×C₇×C₂₇	C₃²	1	117	yes	direct product factor
11	C₃³×C₇×C₂₇	C₂²	1	1	yes	characteristic, direct product factor, Hall
12	C₃³×C₇×C₉	C₂²×C₃	1	1	yes	characteristic
13	C₃²×C₇×C₂₇	C₂²×C₃	1	39	yes	direct product factor
14	C₂×C₃³×C₂₇	C₂×C₇	1	3	yes	direct product factor
15	C₂×C₃⁴×C₇	C₂×C₉	1	3	yes	—
16	C₂×C₃²×C₇×C₉	C₂×C₉	1	78	yes	—
17	C₂×C₃²×C₇×C₉	C₂×C₃²	1	39	yes	—
18	C₂×C₃×C₇×C₂₇	C₂×C₃²	1	351	yes	direct product factor
19	C₂²×C₃³×C₉	C₃×C₇	1	1	yes	characteristic
20	C₂²×C₃²×C₂₇	C₃×C₇	1	39	yes	direct product factor
21	C₂²×C₃³×C₇	C₂₇	1	27	yes	direct product factor
22	C₂²×C₃³×C₇	C₃×C₉	1	13	yes	—
23	C₂²×C₃×C₇×C₉	C₃×C₉	1	104	yes	—
24	C₂²×C₃×C₇×C₉	C₃³	1	13	yes	—
25	C₂²×C₇×C₂₇	C₃³	1	27	yes	direct product factor
26	C₃³×C₂₇	C₂²×C₇	1	1	yes	characteristic, direct product factor, Sylow
27	C₃⁴×C₇	C₂²×C₉	1	1	yes	characteristic
28	C₃²×C₇×C₉	C₂²×C₉	1	26	yes	—
29	C₃²×C₇×C₉	C₂²×C₃²	1	13	yes	—
30	C₃×C₇×C₂₇	C₂²×C₃²	1	117	yes	direct product factor
31	C₂×C₃³×C₉	C₂×C₃×C₇	1	3	yes	—
32	C₂×C₃²×C₂₇	C₂×C₃×C₇	1	117	yes	direct product factor
33	C₂×C₃³×C₇	C₂×C₂₇	1	81	yes	direct product factor
34	C₂×C₃³×C₇	C₂×C₃×C₉	1	39	yes	—
35	C₂×C₃×C₇×C₉	C₂×C₃×C₉	1	312	yes	—
36	C₂×C₃×C₇×C₉	C₂×C₃³	1	39	yes	—
37	C₂×C₇×C₂₇	C₂×C₃³	1	81	yes	direct product factor
38	C₂²×C₃⁴	C₇×C₉	1	1	yes	characteristic
39	C₂²×C₃²×C₉	C₇×C₉	1	26	yes	—
40	C₂²×C₃²×C₉	C₃²×C₇	1	13	yes	—
41	C₂²×C₃×C₂₇	C₃²×C₇	1	117	yes	direct product factor
42	C₂²×C₃²×C₇	C₃×C₂₇	1	117	yes	direct product factor
43	C₂²×C₃²×C₇	C₃²×C₉	1	13	yes	—
44	C₂²×C₇×C₉	C₃²×C₉	1	26	yes	—
45	C₂²×C₇×C₉	C₃⁴	1	1	yes	characteristic
46	C₃³×C₉	C₂²×C₃×C₇	1	1	yes	characteristic
47	C₃²×C₂₇	C₂²×C₃×C₇	1	39	yes	direct product factor
48	C₃³×C₇	C₂²×C₂₇	1	27	yes	direct product factor
49	C₃³×C₇	C₂²×C₃×C₉	1	13	yes	—
50	C₃×C₇×C₉	C₂²×C₃×C₉	1	104	yes	—
51	C₃×C₇×C₉	C₂²×C₃³	1	13	yes	—
52	C₇×C₂₇	C₂²×C₃³	1	27	yes	direct product factor
53	C₂×C₃⁴	C₂×C₇×C₉	1	3	yes	—
54	C₂×C₃²×C₉	C₂×C₇×C₉	1	78	yes	—
55	C₂×C₃²×C₉	C₂×C₃²×C₇	1	39	yes	—
56	C₂×C₃×C₂₇	C₂×C₃²×C₇	1	351	yes	direct product factor
57	C₂×C₃²×C₇	C₂×C₃×C₂₇	1	351	yes	direct product factor
58	C₂×C₃²×C₇	C₂×C₃²×C₉	1	39	yes	—
59	C₂×C₇×C₉	C₂×C₃²×C₉	1	78	yes	—
60	C₂×C₇×C₉	C₂×C₃⁴	1	3	yes	—
61	C₂²×C₃³	C₇×C₂₇	1	27	yes	direct product factor
62	C₂²×C₃³	C₃×C₇×C₉	1	13	yes	—
63	C₂²×C₃×C₉	C₃×C₇×C₉	1	104	yes	—
64	C₂²×C₃×C₉	C₃³×C₇	1	13	yes	—
65	C₂²×C₂₇	C₃³×C₇	1	27	yes	direct product factor
66	C₂²×C₃×C₇	C₃²×C₂₇	1	39	yes	direct product factor
67	C₂²×C₃×C₇	C₃³×C₉	1	1	yes	characteristic
68	C₃⁴	C₂²×C₇×C₉	1	1	yes	characteristic
69	C₃²×C₉	C₂²×C₇×C₉	1	26	yes	—
70	C₃²×C₉	C₂²×C₃²×C₇	1	13	yes	—
71	C₃×C₂₇	C₂²×C₃²×C₇	1	117	yes	direct product factor
72	C₃²×C₇	C₂²×C₃×C₂₇	1	117	yes	direct product factor
73	C₃²×C₇	C₂²×C₃²×C₉	1	13	yes	—
74	C₇×C₉	C₂²×C₃²×C₉	1	26	yes	—
75	C₇×C₉	C₂²×C₃⁴	1	1	yes	characteristic
76	C₂×C₃³	C₂×C₇×C₂₇	1	81	yes	direct product factor
77	C₂×C₃³	C₂×C₃×C₇×C₉	1	39	yes	—
78	C₂×C₃×C₉	C₂×C₃×C₇×C₉	1	312	yes	—
79	C₂×C₃×C₉	C₂×C₃³×C₇	1	39	yes	—
80	C₂×C₂₇	C₂×C₃³×C₇	1	81	yes	direct product factor
81	C₂×C₃×C₇	C₂×C₃²×C₂₇	1	117	yes	direct product factor
82	C₂×C₃×C₇	C₂×C₃³×C₉	1	3	yes	—
83	C₂²×C₃²	C₃×C₇×C₂₇	1	117	yes	direct product factor
84	C₂²×C₃²	C₃²×C₇×C₉	1	13	yes	—
85	C₂²×C₉	C₃²×C₇×C₉	1	26	yes	—
86	C₂²×C₉	C₃⁴×C₇	1	1	yes	characteristic
87	C₂²×C₇	C₃³×C₂₇	1	1	yes	characteristic, direct product factor, Hall
88	C₃³	C₂²×C₇×C₂₇	1	27	yes	direct product factor
89	C₃³	C₂²×C₃×C₇×C₉	1	13	yes	—
90	C₃×C₉	C₂²×C₃×C₇×C₉	1	104	yes	—
91	C₃×C₉	C₂²×C₃³×C₇	1	13	yes	—
92	C₂₇	C₂²×C₃³×C₇	1	27	yes	direct product factor
93	C₃×C₇	C₂²×C₃²×C₂₇	1	39	yes	direct product factor
94	C₃×C₇	C₂²×C₃³×C₉	1	1	yes	characteristic
95	C₂×C₃²	C₂×C₃×C₇×C₂₇	1	351	yes	direct product factor
96	C₂×C₃²	C₂×C₃²×C₇×C₉	1	39	yes	—
97	C₂×C₉	C₂×C₃²×C₇×C₉	1	78	yes	—
98	C₂×C₉	C₂×C₃⁴×C₇	1	3	yes	—
99	C₂×C₇	C₂×C₃³×C₂₇	1	3	yes	direct product factor
100	C₂²×C₃	C₃²×C₇×C₂₇	1	39	yes	direct product factor
101	C₂²×C₃	C₃³×C₇×C₉	1	1	yes	characteristic
102	C₃²	C₂²×C₃×C₇×C₂₇	1	117	yes	direct product factor
103	C₃²	C₂²×C₃²×C₇×C₉	1	13	yes	—
104	C₉	C₂²×C₃²×C₇×C₉	1	26	yes	—
105	C₉	C₂²×C₃⁴×C₇	1	1	yes	characteristic
106	C₇	C₂²×C₃³×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×C₃	C₂×C₃²×C₇×C₂₇	1	117	yes	direct product factor
108	C₂×C₃	C₂×C₃³×C₇×C₉	1	3	yes	—
109	C₂²	C₃³×C₇×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
110	C₃	C₂²×C₃²×C₇×C₂₇	1	39	yes	direct product factor
111	C₃	C₂²×C₃³×C₇×C₉	1	1	yes	characteristic
112	C₂	C₂×C₃³×C₇×C₂₇	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂²×C₃⁴×C₇
socle, prime radical	C₂²×C₃⁴×C₇	C₉
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×C₇×C₂₇	C₂	direct (N × H)
3. C₂²×C₃²×C₇×C₂₇	C₃	direct (N × H)
5. C₂×C₃²×C₇×C₂₇	C₂×C₃	direct (N × H)
6. C₂²×C₃³×C₂₇	C₇	direct (N × H)
10. C₂²×C₃×C₇×C₂₇	C₃²	direct (N × H)
11. C₃³×C₇×C₂₇	C₂²	direct (N × H)
13. C₃²×C₇×C₂₇	C₂²×C₃	direct (N × H)
14. C₂×C₃³×C₂₇	C₂×C₇	direct (N × H)
18. C₂×C₃×C₇×C₂₇	C₂×C₃²	direct (N × H)
20. C₂²×C₃²×C₂₇	C₃×C₇	direct (N × H)
21. C₂²×C₃³×C₇	C₂₇	direct (N × H)
25. C₂²×C₇×C₂₇	C₃³	direct (N × H)
26. C₃³×C₂₇	C₂²×C₇	direct (N × H)
30. C₃×C₇×C₂₇	C₂²×C₃²	direct (N × H)
32. C₂×C₃²×C₂₇	C₂×C₃×C₇	direct (N × H)
33. C₂×C₃³×C₇	C₂×C₂₇	direct (N × H)
37. C₂×C₇×C₂₇	C₂×C₃³	direct (N × H)
41. C₂²×C₃×C₂₇	C₃²×C₇	direct (N × H)
42. C₂²×C₃²×C₇	C₃×C₂₇	direct (N × H)
47. C₃²×C₂₇	C₂²×C₃×C₇	direct (N × H)
48. C₃³×C₇	C₂²×C₂₇	direct (N × H)
52. C₇×C₂₇	C₂²×C₃³	direct (N × H)
56. C₂×C₃×C₂₇	C₂×C₃²×C₇	direct (N × H)
57. C₂×C₃²×C₇	C₂×C₃×C₂₇	direct (N × H)
61. C₂²×C₃³	C₇×C₂₇	direct (N × H)
65. C₂²×C₂₇	C₃³×C₇	direct (N × H)
66. C₂²×C₃×C₇	C₃²×C₂₇	direct (N × H)
71. C₃×C₂₇	C₂²×C₃²×C₇	direct (N × H)
72. C₃²×C₇	C₂²×C₃×C₂₇	direct (N × H)
76. C₂×C₃³	C₂×C₇×C₂₇	direct (N × H)
80. C₂×C₂₇	C₂×C₃³×C₇	direct (N × H)
81. C₂×C₃×C₇	C₂×C₃²×C₂₇	direct (N × H)
83. C₂²×C₃²	C₃×C₇×C₂₇	direct (N × H)
87. C₂²×C₇	C₃³×C₂₇	direct (N × H)
88. C₃³	C₂²×C₇×C₂₇	direct (N × H)
92. C₂₇	C₂²×C₃³×C₇	direct (N × H)
93. C₃×C₇	C₂²×C₃²×C₂₇	direct (N × H)
95. C₂×C₃²	C₂×C₃×C₇×C₂₇	direct (N × H)
99. C₂×C₇	C₂×C₃³×C₂₇	direct (N × H)
100. C₂²×C₃	C₃²×C₇×C₂₇	direct (N × H)
102. C₃²	C₂²×C₃×C₇×C₂₇	direct (N × H)
106. C₇	C₂²×C₃³×C₂₇	direct (N × H)
107. C₂×C₃	C₂×C₃²×C₇×C₂₇	direct (N × H)
109. C₂²	C₃³×C₇×C₂₇	direct (N × H)
110. C₃	C₂²×C₃²×C₇×C₂₇	direct (N × H)
112. C₂	C₂×C₃³×C₇×C₂₇	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×C₇×C₂₇	2
C₂²×C₃³×C₇×C₉	3
C₂²×C₃²×C₇×C₂₇	3
C₂²×C₃³×C₂₇	7

Subgroups — C22×C33×C7×C27 #72197989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃³×C₇×C₂₇ #72197989