Subgroups — C₂×C₃³×C₄×C₉×C₁₃ #72202500

Order 25272.

6336 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₉×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄×C₉×C₁₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⁴×C₄×C₁₃	C₃	1	1	yes	maximal, characteristic
4	C₂×C₃²×C₄×C₉×C₁₃	C₃	1	39	yes	maximal, direct product factor
5	C₂×C₃³×C₉×C₁₃	C₄	1	2	yes	direct product factor
6	C₂×C₃³×C₉×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₃²×C₉×C₁₃	C₂×C₃	1	39	yes	—
8	C₂²×C₃⁴×C₁₃	C₂×C₃	1	1	yes	characteristic
9	C₃⁴×C₄×C₁₃	C₂×C₃	1	2	yes	—
10	C₃²×C₄×C₉×C₁₃	C₂×C₃	1	78	yes	direct product factor
11	C₂×C₃³×C₄×C₁₃	C₉	1	27	yes	direct product factor
12	C₂×C₃³×C₄×C₁₃	C₃²	1	13	yes	—
13	C₂×C₃×C₄×C₉×C₁₃	C₃²	1	117	yes	direct product factor
14	C₂×C₃²×C₉×C₁₃	C₃×C₄	1	78	yes	direct product factor
15	C₂×C₃²×C₉×C₁₃	C₂²×C₃	1	39	yes	—
16	C₂×C₃⁴×C₁₃	C₃×C₄	1	2	yes	—
17	C₂×C₃⁴×C₁₃	C₂²×C₃	1	1	yes	characteristic
18	C₃³×C₉×C₁₃	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃³×C₄×C₉	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₃³×C₁₃	C₂×C₉	1	27	yes	—
21	C₂²×C₃³×C₁₃	C₂×C₃²	1	13	yes	—
22	C₃³×C₄×C₁₃	C₂×C₉	1	54	yes	direct product factor
23	C₃³×C₄×C₁₃	C₂×C₃²	1	26	yes	—
24	C₂²×C₃×C₉×C₁₃	C₂×C₃²	1	117	yes	—
25	C₃×C₄×C₉×C₁₃	C₂×C₃²	1	234	yes	direct product factor
26	C₃⁴×C₁₃	C₂×C₃×C₄	1	1	yes	characteristic
27	C₃²×C₉×C₁₃	C₂×C₃×C₄	1	39	yes	direct product factor
28	C₂²×C₃³×C₉	C₂×C₁₃	1	1	yes	characteristic
29	C₃³×C₄×C₉	C₂×C₁₃	1	2	yes	direct product factor
30	C₂×C₃²×C₄×C₁₃	C₃×C₉	1	117	yes	direct product factor
31	C₂×C₃²×C₄×C₁₃	C₃³	1	13	yes	—
32	C₂×C₄×C₉×C₁₃	C₃³	1	27	yes	direct product factor
33	C₂×C₃³×C₁₃	C₄×C₉	1	54	yes	direct product factor
34	C₂×C₃³×C₁₃	C₂²×C₉	1	27	yes	—
35	C₂×C₃³×C₁₃	C₃²×C₄	1	26	yes	—
36	C₂×C₃³×C₁₃	C₂²×C₃²	1	13	yes	—
37	C₂×C₃×C₉×C₁₃	C₃²×C₄	1	234	yes	direct product factor
38	C₂×C₃×C₉×C₁₃	C₂²×C₃²	1	117	yes	—
39	C₂×C₃⁴×C₄	C₃×C₁₃	1	1	yes	characteristic
40	C₂×C₃²×C₄×C₉	C₃×C₁₃	1	39	yes	direct product factor
41	C₂×C₃³×C₉	C₄×C₁₃	1	2	yes	direct product factor
42	C₂×C₃³×C₉	C₂²×C₁₃	1	1	yes	characteristic
43	C₂²×C₃²×C₁₃	C₂×C₃×C₉	1	117	yes	—
44	C₂²×C₃²×C₁₃	C₂×C₃³	1	13	yes	—
45	C₃²×C₄×C₁₃	C₂×C₃×C₉	1	234	yes	direct product factor
46	C₃²×C₄×C₁₃	C₂×C₃³	1	26	yes	—
47	C₂²×C₉×C₁₃	C₂×C₃³	1	27	yes	—
48	C₄×C₉×C₁₃	C₂×C₃³	1	54	yes	direct product factor
49	C₃³×C₁₃	C₂×C₄×C₉	1	27	yes	direct product factor
50	C₃³×C₁₃	C₂×C₃²×C₄	1	13	yes	—
51	C₃×C₉×C₁₃	C₂×C₃²×C₄	1	117	yes	direct product factor
52	C₂²×C₃⁴	C₂×C₃×C₁₃	1	1	yes	characteristic
53	C₃⁴×C₄	C₂×C₃×C₁₃	1	2	yes	—
54	C₂²×C₃²×C₉	C₂×C₃×C₁₃	1	39	yes	—
55	C₃²×C₄×C₉	C₂×C₃×C₁₃	1	78	yes	direct product factor
56	C₂×C₃×C₄×C₁₃	C₃²×C₉	1	39	yes	direct product factor
57	C₂×C₃×C₄×C₁₃	C₃⁴	1	1	yes	characteristic
58	C₃³×C₉	C₂×C₄×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₃²×C₁₃	C₃×C₄×C₉	1	234	yes	direct product factor
60	C₂×C₃²×C₁₃	C₂²×C₃×C₉	1	117	yes	—
61	C₂×C₃²×C₁₃	C₃³×C₄	1	26	yes	—
62	C₂×C₃²×C₁₃	C₂²×C₃³	1	13	yes	—
63	C₂×C₉×C₁₃	C₃³×C₄	1	54	yes	direct product factor
64	C₂×C₉×C₁₃	C₂²×C₃³	1	27	yes	—
65	C₂×C₃³×C₄	C₉×C₁₃	1	27	yes	direct product factor
66	C₂×C₃³×C₄	C₃²×C₁₃	1	13	yes	—
67	C₂×C₃×C₄×C₉	C₃²×C₁₃	1	117	yes	direct product factor
68	C₂×C₃⁴	C₃×C₄×C₁₃	1	2	yes	—
69	C₂×C₃⁴	C₂²×C₃×C₁₃	1	1	yes	characteristic
70	C₂×C₃²×C₉	C₃×C₄×C₁₃	1	78	yes	direct product factor
71	C₂×C₃²×C₉	C₂²×C₃×C₁₃	1	39	yes	—
72	C₂²×C₃×C₁₃	C₂×C₃²×C₉	1	39	yes	—
73	C₂²×C₃×C₁₃	C₂×C₃⁴	1	1	yes	characteristic
74	C₃×C₄×C₁₃	C₂×C₃²×C₉	1	78	yes	direct product factor
75	C₃×C₄×C₁₃	C₂×C₃⁴	1	2	yes	—
76	C₃²×C₁₃	C₂×C₃×C₄×C₉	1	117	yes	direct product factor
77	C₃²×C₁₃	C₂×C₃³×C₄	1	13	yes	—
78	C₉×C₁₃	C₂×C₃³×C₄	1	27	yes	direct product factor
79	C₂²×C₃³	C₂×C₉×C₁₃	1	27	yes	—
80	C₂²×C₃³	C₂×C₃²×C₁₃	1	13	yes	—
81	C₃³×C₄	C₂×C₉×C₁₃	1	54	yes	direct product factor
82	C₃³×C₄	C₂×C₃²×C₁₃	1	26	yes	—
83	C₂²×C₃×C₉	C₂×C₃²×C₁₃	1	117	yes	—
84	C₃×C₄×C₉	C₂×C₃²×C₁₃	1	234	yes	direct product factor
85	C₂×C₄×C₁₃	C₃³×C₉	1	1	yes	characteristic, direct product factor, Hall
86	C₃⁴	C₂×C₃×C₄×C₁₃	1	1	yes	characteristic
87	C₃²×C₉	C₂×C₃×C₄×C₁₃	1	39	yes	direct product factor
88	C₂×C₃×C₁₃	C₃²×C₄×C₉	1	78	yes	direct product factor
89	C₂×C₃×C₁₃	C₂²×C₃²×C₉	1	39	yes	—
90	C₂×C₃×C₁₃	C₃⁴×C₄	1	2	yes	—
91	C₂×C₃×C₁₃	C₂²×C₃⁴	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₃×C₉×C₁₃	1	117	yes	direct product factor
93	C₂×C₃²×C₄	C₃³×C₁₃	1	13	yes	—
94	C₂×C₄×C₉	C₃³×C₁₃	1	27	yes	direct product factor
95	C₂×C₃³	C₄×C₉×C₁₃	1	54	yes	direct product factor
96	C₂×C₃³	C₂²×C₉×C₁₃	1	27	yes	—
97	C₂×C₃³	C₃²×C₄×C₁₃	1	26	yes	—
98	C₂×C₃³	C₂²×C₃²×C₁₃	1	13	yes	—
99	C₂×C₃×C₉	C₃²×C₄×C₁₃	1	234	yes	direct product factor
100	C₂×C₃×C₉	C₂²×C₃²×C₁₃	1	117	yes	—
101	C₂²×C₁₃	C₂×C₃³×C₉	1	1	yes	characteristic
102	C₄×C₁₃	C₂×C₃³×C₉	1	2	yes	direct product factor
103	C₃×C₁₃	C₂×C₃²×C₄×C₉	1	39	yes	direct product factor
104	C₃×C₁₃	C₂×C₃⁴×C₄	1	1	yes	characteristic
105	C₂²×C₃²	C₂×C₃×C₉×C₁₃	1	117	yes	—
106	C₂²×C₃²	C₂×C₃³×C₁₃	1	13	yes	—
107	C₃²×C₄	C₂×C₃×C₉×C₁₃	1	234	yes	direct product factor
108	C₃²×C₄	C₂×C₃³×C₁₃	1	26	yes	—
109	C₂²×C₉	C₂×C₃³×C₁₃	1	27	yes	—
110	C₄×C₉	C₂×C₃³×C₁₃	1	54	yes	direct product factor
111	C₃³	C₂×C₄×C₉×C₁₃	1	27	yes	direct product factor
112	C₃³	C₂×C₃²×C₄×C₁₃	1	13	yes	—
113	C₃×C₉	C₂×C₃²×C₄×C₁₃	1	117	yes	direct product factor
114	C₂×C₁₃	C₃³×C₄×C₉	1	2	yes	direct product factor
115	C₂×C₁₃	C₂²×C₃³×C₉	1	1	yes	characteristic
116	C₂×C₃×C₄	C₃²×C₉×C₁₃	1	39	yes	direct product factor
117	C₂×C₃×C₄	C₃⁴×C₁₃	1	1	yes	characteristic
118	C₂×C₃²	C₃×C₄×C₉×C₁₃	1	234	yes	direct product factor
119	C₂×C₃²	C₂²×C₃×C₉×C₁₃	1	117	yes	—
120	C₂×C₃²	C₃³×C₄×C₁₃	1	26	yes	—
121	C₂×C₃²	C₂²×C₃³×C₁₃	1	13	yes	—
122	C₂×C₉	C₃³×C₄×C₁₃	1	54	yes	direct product factor
123	C₂×C₉	C₂²×C₃³×C₁₃	1	27	yes	—
124	C₁₃	C₂×C₃³×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₃	C₂×C₃⁴×C₁₃	1	1	yes	characteristic
126	C₂²×C₃	C₂×C₃²×C₉×C₁₃	1	39	yes	—
127	C₃×C₄	C₂×C₃⁴×C₁₃	1	2	yes	—
128	C₃×C₄	C₂×C₃²×C₉×C₁₃	1	78	yes	direct product factor
129	C₃²	C₂×C₃×C₄×C₉×C₁₃	1	117	yes	direct product factor
130	C₃²	C₂×C₃³×C₄×C₁₃	1	13	yes	—
131	C₉	C₂×C₃³×C₄×C₁₃	1	27	yes	direct product factor
132	C₂×C₄	C₃³×C₉×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
133	C₂×C₃	C₃²×C₄×C₉×C₁₃	1	78	yes	direct product factor
134	C₂×C₃	C₃⁴×C₄×C₁₃	1	2	yes	—
135	C₂×C₃	C₂²×C₃⁴×C₁₃	1	1	yes	characteristic
136	C₂×C₃	C₂²×C₃²×C₉×C₁₃	1	39	yes	—
137	C₂²	C₂×C₃³×C₉×C₁₃	1	1	yes	characteristic
138	C₄	C₂×C₃³×C₉×C₁₃	1	2	yes	direct product factor
139	C₃	C₂×C₃²×C₄×C₉×C₁₃	1	39	yes	direct product factor
140	C₃	C₂×C₃⁴×C₄×C₁₃	1	1	yes	characteristic
141	C₂	C₃³×C₄×C₉×C₁₃	1	2	yes	direct product factor
142	C₂	C₂²×C₃³×C₉×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂²×C₃⁴×C₁₃
socle, prime radical	C₂²×C₃⁴×C₁₃	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄×C₉×C₁₃	C₂	direct (N × H)
4. C₂×C₃²×C₄×C₉×C₁₃	C₃	direct (N × H)
5. C₂×C₃³×C₉×C₁₃	C₄	direct (N × H)
10. C₃²×C₄×C₉×C₁₃	C₂×C₃	direct (N × H)
11. C₂×C₃³×C₄×C₁₃	C₉	direct (N × H)
13. C₂×C₃×C₄×C₉×C₁₃	C₃²	direct (N × H)
14. C₂×C₃²×C₉×C₁₃	C₃×C₄	direct (N × H)
18. C₃³×C₉×C₁₃	C₂×C₄	direct (N × H)
19. C₂×C₃³×C₄×C₉	C₁₃	direct (N × H)
22. C₃³×C₄×C₁₃	C₂×C₉	direct (N × H)
25. C₃×C₄×C₉×C₁₃	C₂×C₃²	direct (N × H)
27. C₃²×C₉×C₁₃	C₂×C₃×C₄	direct (N × H)
29. C₃³×C₄×C₉	C₂×C₁₃	direct (N × H)
30. C₂×C₃²×C₄×C₁₃	C₃×C₉	direct (N × H)
32. C₂×C₄×C₉×C₁₃	C₃³	direct (N × H)
33. C₂×C₃³×C₁₃	C₄×C₉	direct (N × H)
37. C₂×C₃×C₉×C₁₃	C₃²×C₄	direct (N × H)
40. C₂×C₃²×C₄×C₉	C₃×C₁₃	direct (N × H)
41. C₂×C₃³×C₉	C₄×C₁₃	direct (N × H)
45. C₃²×C₄×C₁₃	C₂×C₃×C₉	direct (N × H)
48. C₄×C₉×C₁₃	C₂×C₃³	direct (N × H)
49. C₃³×C₁₃	C₂×C₄×C₉	direct (N × H)
51. C₃×C₉×C₁₃	C₂×C₃²×C₄	direct (N × H)
55. C₃²×C₄×C₉	C₂×C₃×C₁₃	direct (N × H)
56. C₂×C₃×C₄×C₁₃	C₃²×C₉	direct (N × H)
58. C₃³×C₉	C₂×C₄×C₁₃	direct (N × H)
59. C₂×C₃²×C₁₃	C₃×C₄×C₉	direct (N × H)
63. C₂×C₉×C₁₃	C₃³×C₄	direct (N × H)
65. C₂×C₃³×C₄	C₉×C₁₃	direct (N × H)
67. C₂×C₃×C₄×C₉	C₃²×C₁₃	direct (N × H)
70. C₂×C₃²×C₉	C₃×C₄×C₁₃	direct (N × H)
74. C₃×C₄×C₁₃	C₂×C₃²×C₉	direct (N × H)
76. C₃²×C₁₃	C₂×C₃×C₄×C₉	direct (N × H)
78. C₉×C₁₃	C₂×C₃³×C₄	direct (N × H)
81. C₃³×C₄	C₂×C₉×C₁₃	direct (N × H)
84. C₃×C₄×C₉	C₂×C₃²×C₁₃	direct (N × H)
85. C₂×C₄×C₁₃	C₃³×C₉	direct (N × H)
87. C₃²×C₉	C₂×C₃×C₄×C₁₃	direct (N × H)
88. C₂×C₃×C₁₃	C₃²×C₄×C₉	direct (N × H)
92. C₂×C₃²×C₄	C₃×C₉×C₁₃	direct (N × H)
94. C₂×C₄×C₉	C₃³×C₁₃	direct (N × H)
95. C₂×C₃³	C₄×C₉×C₁₃	direct (N × H)
99. C₂×C₃×C₉	C₃²×C₄×C₁₃	direct (N × H)
102. C₄×C₁₃	C₂×C₃³×C₉	direct (N × H)
103. C₃×C₁₃	C₂×C₃²×C₄×C₉	direct (N × H)
107. C₃²×C₄	C₂×C₃×C₉×C₁₃	direct (N × H)
110. C₄×C₉	C₂×C₃³×C₁₃	direct (N × H)
111. C₃³	C₂×C₄×C₉×C₁₃	direct (N × H)
113. C₃×C₉	C₂×C₃²×C₄×C₁₃	direct (N × H)
114. C₂×C₁₃	C₃³×C₄×C₉	direct (N × H)
116. C₂×C₃×C₄	C₃²×C₉×C₁₃	direct (N × H)
118. C₂×C₃²	C₃×C₄×C₉×C₁₃	direct (N × H)
122. C₂×C₉	C₃³×C₄×C₁₃	direct (N × H)
124. C₁₃	C₂×C₃³×C₄×C₉	direct (N × H)
128. C₃×C₄	C₂×C₃²×C₉×C₁₃	direct (N × H)
129. C₃²	C₂×C₃×C₄×C₉×C₁₃	direct (N × H)
131. C₉	C₂×C₃³×C₄×C₁₃	direct (N × H)
132. C₂×C₄	C₃³×C₉×C₁₃	direct (N × H)
133. C₂×C₃	C₃²×C₄×C₉×C₁₃	direct (N × H)
138. C₄	C₂×C₃³×C₉×C₁₃	direct (N × H)
139. C₃	C₂×C₃²×C₄×C₉×C₁₃	direct (N × H)
141. C₂	C₃³×C₄×C₉×C₁₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃³×C₉×C₁₃	2
C₃³×C₄×C₉×C₁₃	2
C₂×C₃⁴×C₄×C₁₃	3
C₂×C₃²×C₄×C₉×C₁₃	3
C₂×C₃³×C₄×C₉	13

Subgroups — C2×C33×C4×C9×C13 #72202500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×C₄×C₉×C₁₃ #72202500