Subgroups — C₄³×C₅×C₉² #72203097

Order 25920.

5934 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₅×C₉²	C₂	1	7	yes	maximal
2	C₃×C₄³×C₅×C₉	C₃	1	4	yes	maximal
3	C₂²×C₄×C₅×C₉²	C₂²	1	7	yes	—
4	C₄²×C₅×C₉²	C₄	1	28	yes	direct product factor
5	C₄³×C₉²	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₄²×C₅×C₉	C₂×C₃	1	28	yes	—
7	C₂³×C₅×C₉²	C₂³	1	1	yes	characteristic
8	C₂×C₄×C₅×C₉²	C₂×C₄	1	42	yes	—
9	C₃²×C₄³×C₅	C₃²	1	1	yes	characteristic
10	C₄³×C₅×C₉	C₉	1	12	yes	direct product factor
11	C₂×C₄²×C₉²	C₂×C₅	1	7	yes	—
12	C₂²×C₃×C₄×C₅×C₉	C₂²×C₃	1	28	yes	—
13	C₃×C₄²×C₅×C₉	C₃×C₄	1	112	yes	—
14	C₃×C₄³×C₉	C₃×C₅	1	4	yes	—
15	C₂²×C₅×C₉²	C₂²×C₄	1	7	yes	—
16	C₄×C₅×C₉²	C₄²	1	28	yes	direct product factor
17	C₂×C₃²×C₄²×C₅	C₂×C₃²	1	7	yes	—
18	C₂×C₄²×C₅×C₉	C₂×C₉	1	84	yes	—
19	C₂²×C₄×C₉²	C₂²×C₅	1	7	yes	—
20	C₄²×C₉²	C₄×C₅	1	28	yes	direct product factor
21	C₂³×C₃×C₅×C₉	C₂³×C₃	1	4	yes	—
22	C₂×C₃×C₄×C₅×C₉	C₂×C₃×C₄	1	168	yes	—
23	C₃×C₄³×C₅	C₃×C₉	1	4	yes	—
24	C₂×C₃×C₄²×C₉	C₂×C₃×C₅	1	28	yes	—
25	C₂×C₅×C₉²	C₂×C₄²	1	7	yes	—
26	C₂²×C₃²×C₄×C₅	C₂²×C₃²	1	7	yes	—
27	C₃²×C₄²×C₅	C₃²×C₄	1	28	yes	—
28	C₂²×C₄×C₅×C₉	C₂²×C₉	1	84	yes	—
29	C₄²×C₅×C₉	C₄×C₉	1	336	yes	direct product factor
30	C₂³×C₉²	C₂³×C₅	1	1	yes	characteristic
31	C₂×C₄×C₉²	C₂×C₄×C₅	1	42	yes	—
32	C₃²×C₄³	C₃²×C₅	1	1	yes	characteristic
33	C₄³×C₉	C₅×C₉	1	12	yes	direct product factor
34	C₂²×C₃×C₅×C₉	C₂²×C₃×C₄	1	28	yes	—
35	C₃×C₄×C₅×C₉	C₃×C₄²	1	112	yes	—
36	C₂×C₃×C₄²×C₅	C₂×C₃×C₉	1	28	yes	—
37	C₂²×C₃×C₄×C₉	C₂²×C₃×C₅	1	28	yes	—
38	C₃×C₄²×C₉	C₃×C₄×C₅	1	112	yes	—
39	C₅×C₉²	C₄³	1	1	yes	characteristic, direct product factor, Hall
40	C₂³×C₃²×C₅	C₂³×C₃²	1	1	yes	characteristic
41	C₂×C₃²×C₄×C₅	C₂×C₃²×C₄	1	42	yes	—
42	C₂³×C₅×C₉	C₂³×C₉	1	12	yes	—
43	C₂×C₄×C₅×C₉	C₂×C₄×C₉	1	504	yes	—
44	C₂²×C₉²	C₂²×C₄×C₅	1	7	yes	—
45	C₄×C₉²	C₄²×C₅	1	28	yes	direct product factor
46	C₄³×C₅	C₉²	1	1	yes	characteristic, direct product factor, Hall
47	C₂×C₃²×C₄²	C₂×C₃²×C₅	1	7	yes	—
48	C₂×C₄²×C₉	C₂×C₅×C₉	1	84	yes	—
49	C₂×C₃×C₅×C₉	C₂×C₃×C₄²	1	28	yes	—
50	C₂²×C₃×C₄×C₅	C₂²×C₃×C₉	1	28	yes	—
51	C₃×C₄²×C₅	C₃×C₄×C₉	1	112	yes	—
52	C₂³×C₃×C₉	C₂³×C₃×C₅	1	4	yes	—
53	C₂×C₃×C₄×C₉	C₂×C₃×C₄×C₅	1	168	yes	—
54	C₃×C₄³	C₃×C₅×C₉	1	4	yes	—
55	C₂²×C₃²×C₅	C₂²×C₃²×C₄	1	7	yes	—
56	C₃²×C₄×C₅	C₃²×C₄²	1	28	yes	—
57	C₂²×C₅×C₉	C₂²×C₄×C₉	1	84	yes	—
58	C₄×C₅×C₉	C₄²×C₉	1	336	yes	direct product factor
59	C₂×C₉²	C₂×C₄²×C₅	1	7	yes	—
60	C₂×C₄²×C₅	C₂×C₉²	1	7	yes	—
61	C₂²×C₃²×C₄	C₂²×C₃²×C₅	1	7	yes	—
62	C₃²×C₄²	C₃²×C₄×C₅	1	28	yes	—
63	C₂²×C₄×C₉	C₂²×C₅×C₉	1	84	yes	—
64	C₄²×C₉	C₄×C₅×C₉	1	336	yes	direct product factor
65	C₃×C₅×C₉	C₃×C₄³	1	4	yes	—
66	C₂³×C₃×C₅	C₂³×C₃×C₉	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₂×C₃×C₄×C₉	1	168	yes	—
68	C₂²×C₃×C₉	C₂²×C₃×C₄×C₅	1	28	yes	—
69	C₃×C₄×C₉	C₃×C₄²×C₅	1	112	yes	—
70	C₂×C₃×C₄²	C₂×C₃×C₅×C₉	1	28	yes	—
71	C₂×C₃²×C₅	C₂×C₃²×C₄²	1	7	yes	—
72	C₂×C₅×C₉	C₂×C₄²×C₉	1	84	yes	—
73	C₉²	C₄³×C₅	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₄×C₅	C₂²×C₉²	1	7	yes	—
75	C₄²×C₅	C₄×C₉²	1	28	yes	direct product factor
76	C₂³×C₃²	C₂³×C₃²×C₅	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×C₃²×C₄×C₅	1	42	yes	—
78	C₂³×C₉	C₂³×C₅×C₉	1	12	yes	—
79	C₂×C₄×C₉	C₂×C₄×C₅×C₉	1	504	yes	—
80	C₄³	C₅×C₉²	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₃×C₅	C₂²×C₃×C₄×C₉	1	28	yes	—
82	C₃×C₄×C₅	C₃×C₄²×C₉	1	112	yes	—
83	C₂×C₃×C₉	C₂×C₃×C₄²×C₅	1	28	yes	—
84	C₂²×C₃×C₄	C₂²×C₃×C₅×C₉	1	28	yes	—
85	C₃×C₄²	C₃×C₄×C₅×C₉	1	112	yes	—
86	C₃²×C₅	C₃²×C₄³	1	1	yes	characteristic
87	C₅×C₉	C₄³×C₉	1	12	yes	direct product factor
88	C₂³×C₅	C₂³×C₉²	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×C₄×C₉²	1	42	yes	—
90	C₂²×C₃²	C₂²×C₃²×C₄×C₅	1	7	yes	—
91	C₃²×C₄	C₃²×C₄²×C₅	1	28	yes	—
92	C₂²×C₉	C₂²×C₄×C₅×C₉	1	84	yes	—
93	C₄×C₉	C₄²×C₅×C₉	1	336	yes	direct product factor
94	C₂×C₄²	C₂×C₅×C₉²	1	7	yes	—
95	C₂×C₃×C₅	C₂×C₃×C₄²×C₉	1	28	yes	—
96	C₃×C₉	C₃×C₄³×C₅	1	4	yes	—
97	C₂³×C₃	C₂³×C₃×C₅×C₉	1	4	yes	—
98	C₂×C₃×C₄	C₂×C₃×C₄×C₅×C₉	1	168	yes	—
99	C₂²×C₅	C₂²×C₄×C₉²	1	7	yes	—
100	C₄×C₅	C₄²×C₉²	1	28	yes	direct product factor
101	C₂×C₃²	C₂×C₃²×C₄²×C₅	1	7	yes	—
102	C₂×C₉	C₂×C₄²×C₅×C₉	1	84	yes	—
103	C₂²×C₄	C₂²×C₅×C₉²	1	7	yes	—
104	C₄²	C₄×C₅×C₉²	1	28	yes	direct product factor
105	C₃×C₅	C₃×C₄³×C₉	1	4	yes	—
106	C₂²×C₃	C₂²×C₃×C₄×C₅×C₉	1	28	yes	—
107	C₃×C₄	C₃×C₄²×C₅×C₉	1	112	yes	—
108	C₂×C₅	C₂×C₄²×C₉²	1	7	yes	—
109	C₃²	C₃²×C₄³×C₅	1	1	yes	characteristic
110	C₉	C₄³×C₅×C₉	1	12	yes	direct product factor
111	C₂³	C₂³×C₅×C₉²	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×C₅×C₉²	1	42	yes	—
113	C₂×C₃	C₂×C₃×C₄²×C₅×C₉	1	28	yes	—
114	C₅	C₄³×C₉²	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²	C₂²×C₄×C₅×C₉²	1	7	yes	—
116	C₄	C₄²×C₅×C₉²	1	28	yes	direct product factor
117	C₃	C₃×C₄³×C₅×C₉	1	4	yes	—
118	C₂	C₂×C₄²×C₅×C₉²	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³×C₃²	C₂³×C₃²×C₅
socle, prime radical	C₂³×C₃²×C₅	C₂³×C₃²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄²×C₅×C₉²	C₄	direct (N × H)
5. C₄³×C₉²	C₅	direct (N × H)
10. C₄³×C₅×C₉	C₉	direct (N × H)
16. C₄×C₅×C₉²	C₄²	direct (N × H)
20. C₄²×C₉²	C₄×C₅	direct (N × H)
29. C₄²×C₅×C₉	C₄×C₉	direct (N × H)
33. C₄³×C₉	C₅×C₉	direct (N × H)
39. C₅×C₉²	C₄³	direct (N × H)
45. C₄×C₉²	C₄²×C₅	direct (N × H)
46. C₄³×C₅	C₉²	direct (N × H)
58. C₄×C₅×C₉	C₄²×C₉	direct (N × H)
64. C₄²×C₉	C₄×C₅×C₉	direct (N × H)
73. C₉²	C₄³×C₅	direct (N × H)
75. C₄²×C₅	C₄×C₉²	direct (N × H)
80. C₄³	C₅×C₉²	direct (N × H)
87. C₅×C₉	C₄³×C₉	direct (N × H)
93. C₄×C₉	C₄²×C₅×C₉	direct (N × H)
100. C₄×C₅	C₄²×C₉²	direct (N × H)
104. C₄²	C₄×C₅×C₉²	direct (N × H)
110. C₉	C₄³×C₅×C₉	direct (N × H)
114. C₅	C₄³×C₉²	direct (N × H)
116. C₄	C₄²×C₅×C₉²	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×C₄²×C₅×C₉²	2
C₃×C₄³×C₅×C₉	3
C₄³×C₉²	5

Subgroups — C43×C5×C92 #72203097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄³×C₅×C₉² #72203097