Subgroups — C₂²×C₃×C₅²×C₈₁ #72201581

Order 24300.

720 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²×C₈₁	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₂₇	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₅²×C₈₁	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅²×C₈₁	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₈₁	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₃×C₅²×C₂₇	C₂×C₃	1	3	yes	—
7	C₂×C₅²×C₈₁	C₂×C₃	1	9	yes	direct product factor
8	C₂²×C₃×C₅²×C₉	C₉	1	1	yes	characteristic
9	C₂²×C₅²×C₂₇	C₉	1	2	yes	—
10	C₂²×C₅²×C₂₇	C₃²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₈₁	C₂×C₅	1	18	yes	direct product factor
12	C₃×C₅²×C₂₇	C₂²×C₃	1	1	yes	characteristic
13	C₅²×C₈₁	C₂²×C₃	1	3	yes	direct product factor
14	C₂²×C₃×C₅×C₂₇	C₃×C₅	1	6	yes	—
15	C₂²×C₅×C₈₁	C₃×C₅	1	18	yes	direct product factor
16	C₂×C₃×C₅²×C₉	C₂×C₉	1	3	yes	—
17	C₂×C₅²×C₂₇	C₂×C₉	1	6	yes	—
18	C₂×C₅²×C₂₇	C₂×C₃²	1	3	yes	—
19	C₃×C₅×C₈₁	C₂²×C₅	1	6	yes	direct product factor
20	C₂²×C₃×C₈₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₂²×C₃²×C₅²	C₂₇	1	1	yes	characteristic
22	C₂²×C₅²×C₉	C₂₇	1	2	yes	—
23	C₂²×C₅²×C₉	C₃×C₉	1	1	yes	characteristic
24	C₂×C₃×C₅×C₂₇	C₂×C₃×C₅	1	18	yes	—
25	C₂×C₅×C₈₁	C₂×C₃×C₅	1	54	yes	direct product factor
26	C₃×C₅²×C₉	C₂²×C₉	1	1	yes	characteristic
27	C₅²×C₂₇	C₂²×C₉	1	2	yes	—
28	C₅²×C₂₇	C₂²×C₃²	1	1	yes	characteristic
29	C₂²×C₃×C₅×C₉	C₅×C₉	1	6	yes	—
30	C₂²×C₅×C₂₇	C₅×C₉	1	12	yes	—
31	C₂²×C₅×C₂₇	C₃²×C₅	1	6	yes	—
32	C₂×C₃×C₈₁	C₂×C₅²	1	3	yes	direct product factor
33	C₂×C₃²×C₅²	C₂×C₂₇	1	3	yes	—
34	C₂×C₅²×C₉	C₂×C₂₇	1	6	yes	—
35	C₂×C₅²×C₉	C₂×C₃×C₉	1	3	yes	—
36	C₃×C₅×C₂₇	C₂²×C₃×C₅	1	6	yes	—
37	C₅×C₈₁	C₂²×C₃×C₅	1	18	yes	direct product factor
38	C₂²×C₃×C₂₇	C₃×C₅²	1	1	yes	characteristic
39	C₂²×C₈₁	C₃×C₅²	1	3	yes	direct product factor
40	C₂²×C₃×C₅²	C₈₁	1	3	yes	direct product factor
41	C₂²×C₃×C₅²	C₃×C₂₇	1	1	yes	characteristic
42	C₂×C₃×C₅×C₉	C₂×C₅×C₉	1	18	yes	—
43	C₂×C₅×C₂₇	C₂×C₅×C₉	1	36	yes	—
44	C₂×C₅×C₂₇	C₂×C₃²×C₅	1	18	yes	—
45	C₃×C₈₁	C₂²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
46	C₃²×C₅²	C₂²×C₂₇	1	1	yes	characteristic
47	C₅²×C₉	C₂²×C₂₇	1	2	yes	—
48	C₅²×C₉	C₂²×C₃×C₉	1	1	yes	characteristic
49	C₂²×C₃²×C₅	C₅×C₂₇	1	6	yes	—
50	C₂²×C₅×C₉	C₅×C₂₇	1	12	yes	—
51	C₂²×C₅×C₉	C₃×C₅×C₉	1	6	yes	—
52	C₂×C₃×C₂₇	C₂×C₃×C₅²	1	3	yes	—
53	C₂×C₈₁	C₂×C₃×C₅²	1	9	yes	direct product factor
54	C₂×C₃×C₅²	C₂×C₈₁	1	9	yes	direct product factor
55	C₂×C₃×C₅²	C₂×C₃×C₂₇	1	3	yes	—
56	C₃×C₅×C₉	C₂²×C₅×C₉	1	6	yes	—
57	C₅×C₂₇	C₂²×C₅×C₉	1	12	yes	—
58	C₅×C₂₇	C₂²×C₃²×C₅	1	6	yes	—
59	C₂²×C₃×C₉	C₅²×C₉	1	1	yes	characteristic
60	C₂²×C₂₇	C₅²×C₉	1	2	yes	—
61	C₂²×C₂₇	C₃²×C₅²	1	1	yes	characteristic
62	C₂²×C₅²	C₃×C₈₁	1	1	yes	characteristic, direct product factor, Hall
63	C₂×C₃²×C₅	C₂×C₅×C₂₇	1	18	yes	—
64	C₂×C₅×C₉	C₂×C₅×C₂₇	1	36	yes	—
65	C₂×C₅×C₉	C₂×C₃×C₅×C₉	1	18	yes	—
66	C₃×C₂₇	C₂²×C₃×C₅²	1	1	yes	characteristic
67	C₈₁	C₂²×C₃×C₅²	1	3	yes	direct product factor
68	C₃×C₅²	C₂²×C₈₁	1	3	yes	direct product factor
69	C₃×C₅²	C₂²×C₃×C₂₇	1	1	yes	characteristic
70	C₂²×C₃×C₅	C₅×C₈₁	1	18	yes	direct product factor
71	C₂²×C₃×C₅	C₃×C₅×C₂₇	1	6	yes	—
72	C₂×C₃×C₉	C₂×C₅²×C₉	1	3	yes	—
73	C₂×C₂₇	C₂×C₅²×C₉	1	6	yes	—
74	C₂×C₂₇	C₂×C₃²×C₅²	1	3	yes	—
75	C₂×C₅²	C₂×C₃×C₈₁	1	3	yes	direct product factor
76	C₃²×C₅	C₂²×C₅×C₂₇	1	6	yes	—
77	C₅×C₉	C₂²×C₅×C₂₇	1	12	yes	—
78	C₅×C₉	C₂²×C₃×C₅×C₉	1	6	yes	—
79	C₂²×C₃²	C₅²×C₂₇	1	1	yes	characteristic
80	C₂²×C₉	C₅²×C₂₇	1	2	yes	—
81	C₂²×C₉	C₃×C₅²×C₉	1	1	yes	characteristic
82	C₂×C₃×C₅	C₂×C₅×C₈₁	1	54	yes	direct product factor
83	C₂×C₃×C₅	C₂×C₃×C₅×C₂₇	1	18	yes	—
84	C₃×C₉	C₂²×C₅²×C₉	1	1	yes	characteristic
85	C₂₇	C₂²×C₅²×C₉	1	2	yes	—
86	C₂₇	C₂²×C₃²×C₅²	1	1	yes	characteristic
87	C₅²	C₂²×C₃×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₅	C₃×C₅×C₈₁	1	6	yes	direct product factor
89	C₂×C₃²	C₂×C₅²×C₂₇	1	3	yes	—
90	C₂×C₉	C₂×C₅²×C₂₇	1	6	yes	—
91	C₂×C₉	C₂×C₃×C₅²×C₉	1	3	yes	—
92	C₃×C₅	C₂²×C₅×C₈₁	1	18	yes	direct product factor
93	C₃×C₅	C₂²×C₃×C₅×C₂₇	1	6	yes	—
94	C₂²×C₃	C₅²×C₈₁	1	3	yes	direct product factor
95	C₂²×C₃	C₃×C₅²×C₂₇	1	1	yes	characteristic
96	C₂×C₅	C₂×C₃×C₅×C₈₁	1	18	yes	direct product factor
97	C₃²	C₂²×C₅²×C₂₇	1	1	yes	characteristic
98	C₉	C₂²×C₅²×C₂₇	1	2	yes	—
99	C₉	C₂²×C₃×C₅²×C₉	1	1	yes	characteristic
100	C₂×C₃	C₂×C₅²×C₈₁	1	9	yes	direct product factor
101	C₂×C₃	C₂×C₃×C₅²×C₂₇	1	3	yes	—
102	C₅	C₂²×C₃×C₅×C₈₁	1	6	yes	direct product factor
103	C₂²	C₃×C₅²×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
104	C₃	C₂²×C₅²×C₈₁	1	3	yes	direct product factor
105	C₃	C₂²×C₃×C₅²×C₂₇	1	1	yes	characteristic
106	C₂	C₂×C₃×C₅²×C₈₁	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₇	C₂²×C₃²×C₅²
socle, prime radical	C₂²×C₃²×C₅²	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×C₃×C₅²×C₈₁	C₂	direct (N × H)
3. C₂²×C₅²×C₈₁	C₃	direct (N × H)
4. C₃×C₅²×C₈₁	C₂²	direct (N × H)
5. C₂²×C₃×C₅×C₈₁	C₅	direct (N × H)
7. C₂×C₅²×C₈₁	C₂×C₃	direct (N × H)
11. C₂×C₃×C₅×C₈₁	C₂×C₅	direct (N × H)
13. C₅²×C₈₁	C₂²×C₃	direct (N × H)
15. C₂²×C₅×C₈₁	C₃×C₅	direct (N × H)
19. C₃×C₅×C₈₁	C₂²×C₅	direct (N × H)
20. C₂²×C₃×C₈₁	C₅²	direct (N × H)
25. C₂×C₅×C₈₁	C₂×C₃×C₅	direct (N × H)
32. C₂×C₃×C₈₁	C₂×C₅²	direct (N × H)
37. C₅×C₈₁	C₂²×C₃×C₅	direct (N × H)
39. C₂²×C₈₁	C₃×C₅²	direct (N × H)
40. C₂²×C₃×C₅²	C₈₁	direct (N × H)
45. C₃×C₈₁	C₂²×C₅²	direct (N × H)
53. C₂×C₈₁	C₂×C₃×C₅²	direct (N × H)
54. C₂×C₃×C₅²	C₂×C₈₁	direct (N × H)
62. C₂²×C₅²	C₃×C₈₁	direct (N × H)
67. C₈₁	C₂²×C₃×C₅²	direct (N × H)
68. C₃×C₅²	C₂²×C₈₁	direct (N × H)
70. C₂²×C₃×C₅	C₅×C₈₁	direct (N × H)
75. C₂×C₅²	C₂×C₃×C₈₁	direct (N × H)
82. C₂×C₃×C₅	C₂×C₅×C₈₁	direct (N × H)
87. C₅²	C₂²×C₃×C₈₁	direct (N × H)
88. C₂²×C₅	C₃×C₅×C₈₁	direct (N × H)
92. C₃×C₅	C₂²×C₅×C₈₁	direct (N × H)
94. C₂²×C₃	C₅²×C₈₁	direct (N × H)
96. C₂×C₅	C₂×C₃×C₅×C₈₁	direct (N × H)
100. C₂×C₃	C₂×C₅²×C₈₁	direct (N × H)
102. C₅	C₂²×C₃×C₅×C₈₁	direct (N × H)
103. C₂²	C₃×C₅²×C₈₁	direct (N × H)
104. C₃	C₂²×C₅²×C₈₁	direct (N × H)
106. C₂	C₂×C₃×C₅²×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×C₃×C₅²×C₈₁	2
C₂²×C₃×C₅²×C₂₇	3
C₂²×C₅²×C₈₁	3
C₂²×C₃×C₅×C₈₁	5

Subgroups — C22×C3×C52×C81 #72201581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅²×C₈₁ #72201581