Subgroups — C₂²×C₃×C₄²×C₅×C₁₃ #72190827

Order 12480.

1992 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	C₂³×C₃×C₄×C₅×C₁₃	C₂	1	3	yes	maximal
2	C₂×C₃×C₄²×C₅×C₁₃	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₅×C₁₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₄×C₅×C₁₃	C₄	1	24	yes	direct product factor
5	C₂²×C₃×C₄×C₅×C₁₃	C₂²	1	18	yes	—
6	C₂⁴×C₃×C₅×C₁₃	C₂²	1	1	yes	characteristic
7	C₃×C₄²×C₅×C₁₃	C₂²	1	16	yes	direct product factor
8	C₂²×C₃×C₄²×C₁₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₄×C₅×C₁₃	C₂×C₃	1	3	yes	—
10	C₂×C₄²×C₅×C₁₃	C₂×C₃	1	12	yes	direct product factor
11	C₂³×C₃×C₅×C₁₃	C₂×C₄	1	12	yes	—
12	C₂³×C₃×C₅×C₁₃	C₂³	1	3	yes	—
13	C₂×C₃×C₄×C₅×C₁₃	C₂×C₄	1	72	yes	direct product factor
14	C₂×C₃×C₄×C₅×C₁₃	C₂³	1	12	yes	—
15	C₂³×C₃×C₄×C₁₃	C₂×C₅	1	3	yes	—
16	C₂×C₃×C₄²×C₁₃	C₂×C₅	1	12	yes	direct product factor
17	C₂⁴×C₅×C₁₃	C₂²×C₃	1	1	yes	characteristic
18	C₂²×C₄×C₅×C₁₃	C₃×C₄	1	24	yes	direct product factor
19	C₂²×C₄×C₅×C₁₃	C₂²×C₃	1	18	yes	—
20	C₄²×C₅×C₁₃	C₂²×C₃	1	16	yes	direct product factor
21	C₂²×C₃×C₄²×C₅	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂²×C₄²×C₁₃	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
23	C₂²×C₃×C₅×C₁₃	C₄²	1	16	yes	direct product factor
24	C₂²×C₃×C₅×C₁₃	C₂²×C₄	1	18	yes	—
25	C₂²×C₃×C₅×C₁₃	C₂⁴	1	1	yes	characteristic
26	C₃×C₄×C₅×C₁₃	C₂²×C₄	1	24	yes	direct product factor
27	C₂⁴×C₃×C₁₃	C₂²×C₅	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₁₃	C₄×C₅	1	24	yes	direct product factor
29	C₂²×C₃×C₄×C₁₃	C₂²×C₅	1	18	yes	—
30	C₃×C₄²×C₁₃	C₂²×C₅	1	16	yes	direct product factor
31	C₂³×C₅×C₁₃	C₂×C₃×C₄	1	12	yes	—
32	C₂³×C₅×C₁₃	C₂³×C₃	1	3	yes	—
33	C₂×C₄×C₅×C₁₃	C₂×C₃×C₄	1	72	yes	direct product factor
34	C₂×C₄×C₅×C₁₃	C₂³×C₃	1	12	yes	—
35	C₂³×C₃×C₄×C₅	C₂×C₁₃	1	3	yes	—
36	C₂×C₃×C₄²×C₅	C₂×C₁₃	1	12	yes	direct product factor
37	C₂³×C₄×C₁₃	C₂×C₃×C₅	1	3	yes	—
38	C₂×C₄²×C₁₃	C₂×C₃×C₅	1	12	yes	direct product factor
39	C₂×C₃×C₅×C₁₃	C₂×C₄²	1	12	yes	direct product factor
40	C₂×C₃×C₅×C₁₃	C₂³×C₄	1	3	yes	—
41	C₂²×C₄²×C₅	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
42	C₂³×C₃×C₁₃	C₂×C₄×C₅	1	12	yes	—
43	C₂³×C₃×C₁₃	C₂³×C₅	1	3	yes	—
44	C₂×C₃×C₄×C₁₃	C₂×C₄×C₅	1	72	yes	direct product factor
45	C₂×C₃×C₄×C₁₃	C₂³×C₅	1	12	yes	—
46	C₂²×C₅×C₁₃	C₃×C₄²	1	16	yes	direct product factor
47	C₂²×C₅×C₁₃	C₂²×C₃×C₄	1	18	yes	—
48	C₂²×C₅×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
49	C₄×C₅×C₁₃	C₂²×C₃×C₄	1	24	yes	direct product factor
50	C₂⁴×C₃×C₅	C₂²×C₁₃	1	1	yes	characteristic
51	C₂²×C₃×C₄×C₅	C₄×C₁₃	1	24	yes	direct product factor
52	C₂²×C₃×C₄×C₅	C₂²×C₁₃	1	18	yes	—
53	C₃×C₄²×C₅	C₂²×C₁₃	1	16	yes	direct product factor
54	C₂⁴×C₁₃	C₂²×C₃×C₅	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₃×C₄×C₅	1	24	yes	direct product factor
56	C₂²×C₄×C₁₃	C₂²×C₃×C₅	1	18	yes	—
57	C₄²×C₁₃	C₂²×C₃×C₅	1	16	yes	direct product factor
58	C₃×C₅×C₁₃	C₂²×C₄²	1	1	yes	characteristic, direct product factor, Hall
59	C₂²×C₃×C₄²	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
60	C₂³×C₄×C₅	C₂×C₃×C₁₃	1	3	yes	—
61	C₂×C₄²×C₅	C₂×C₃×C₁₃	1	12	yes	direct product factor
62	C₂²×C₃×C₁₃	C₄²×C₅	1	16	yes	direct product factor
63	C₂²×C₃×C₁₃	C₂²×C₄×C₅	1	18	yes	—
64	C₂²×C₃×C₁₃	C₂⁴×C₅	1	1	yes	characteristic
65	C₃×C₄×C₁₃	C₂²×C₄×C₅	1	24	yes	direct product factor
66	C₂×C₅×C₁₃	C₂×C₃×C₄²	1	12	yes	direct product factor
67	C₂×C₅×C₁₃	C₂³×C₃×C₄	1	3	yes	—
68	C₂³×C₃×C₅	C₂×C₄×C₁₃	1	12	yes	—
69	C₂³×C₃×C₅	C₂³×C₁₃	1	3	yes	—
70	C₂×C₃×C₄×C₅	C₂×C₄×C₁₃	1	72	yes	direct product factor
71	C₂×C₃×C₄×C₅	C₂³×C₁₃	1	12	yes	—
72	C₂³×C₁₃	C₂×C₃×C₄×C₅	1	12	yes	—
73	C₂³×C₁₃	C₂³×C₃×C₅	1	3	yes	—
74	C₂×C₄×C₁₃	C₂×C₃×C₄×C₅	1	72	yes	direct product factor
75	C₂×C₄×C₁₃	C₂³×C₃×C₅	1	12	yes	—
76	C₂³×C₃×C₄	C₂×C₅×C₁₃	1	3	yes	—
77	C₂×C₃×C₄²	C₂×C₅×C₁₃	1	12	yes	direct product factor
78	C₂⁴×C₅	C₂²×C₃×C₁₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₃×C₄×C₁₃	1	24	yes	direct product factor
80	C₂²×C₄×C₅	C₂²×C₃×C₁₃	1	18	yes	—
81	C₄²×C₅	C₂²×C₃×C₁₃	1	16	yes	direct product factor
82	C₂×C₃×C₁₃	C₂×C₄²×C₅	1	12	yes	direct product factor
83	C₂×C₃×C₁₃	C₂³×C₄×C₅	1	3	yes	—
84	C₅×C₁₃	C₂²×C₃×C₄²	1	1	yes	characteristic, direct product factor, Hall
85	C₂²×C₄²	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×C₃×C₅	C₄²×C₁₃	1	16	yes	direct product factor
87	C₂²×C₃×C₅	C₂²×C₄×C₁₃	1	18	yes	—
88	C₂²×C₃×C₅	C₂⁴×C₁₃	1	1	yes	characteristic
89	C₃×C₄×C₅	C₂²×C₄×C₁₃	1	24	yes	direct product factor
90	C₂²×C₁₃	C₃×C₄²×C₅	1	16	yes	direct product factor
91	C₂²×C₁₃	C₂²×C₃×C₄×C₅	1	18	yes	—
92	C₂²×C₁₃	C₂⁴×C₃×C₅	1	1	yes	characteristic
93	C₄×C₁₃	C₂²×C₃×C₄×C₅	1	24	yes	direct product factor
94	C₂⁴×C₃	C₂²×C₅×C₁₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₄×C₅×C₁₃	1	24	yes	direct product factor
96	C₂²×C₃×C₄	C₂²×C₅×C₁₃	1	18	yes	—
97	C₃×C₄²	C₂²×C₅×C₁₃	1	16	yes	direct product factor
98	C₂³×C₅	C₂×C₃×C₄×C₁₃	1	12	yes	—
99	C₂³×C₅	C₂³×C₃×C₁₃	1	3	yes	—
100	C₂×C₄×C₅	C₂×C₃×C₄×C₁₃	1	72	yes	direct product factor
101	C₂×C₄×C₅	C₂³×C₃×C₁₃	1	12	yes	—
102	C₃×C₁₃	C₂²×C₄²×C₅	1	1	yes	characteristic, direct product factor, Hall
103	C₂³×C₄	C₂×C₃×C₅×C₁₃	1	3	yes	—
104	C₂×C₄²	C₂×C₃×C₅×C₁₃	1	12	yes	direct product factor
105	C₂×C₃×C₅	C₂×C₄²×C₁₃	1	12	yes	direct product factor
106	C₂×C₃×C₅	C₂³×C₄×C₁₃	1	3	yes	—
107	C₂×C₁₃	C₂×C₃×C₄²×C₅	1	12	yes	direct product factor
108	C₂×C₁₃	C₂³×C₃×C₄×C₅	1	3	yes	—
109	C₂³×C₃	C₂×C₄×C₅×C₁₃	1	12	yes	—
110	C₂³×C₃	C₂³×C₅×C₁₃	1	3	yes	—
111	C₂×C₃×C₄	C₂×C₄×C₅×C₁₃	1	72	yes	direct product factor
112	C₂×C₃×C₄	C₂³×C₅×C₁₃	1	12	yes	—
113	C₂²×C₅	C₃×C₄²×C₁₃	1	16	yes	direct product factor
114	C₂²×C₅	C₂²×C₃×C₄×C₁₃	1	18	yes	—
115	C₂²×C₅	C₂⁴×C₃×C₁₃	1	1	yes	characteristic
116	C₄×C₅	C₂²×C₃×C₄×C₁₃	1	24	yes	direct product factor
117	C₂⁴	C₂²×C₃×C₅×C₁₃	1	1	yes	characteristic
118	C₂²×C₄	C₃×C₄×C₅×C₁₃	1	24	yes	direct product factor
119	C₂²×C₄	C₂²×C₃×C₅×C₁₃	1	18	yes	—
120	C₄²	C₂²×C₃×C₅×C₁₃	1	16	yes	direct product factor
121	C₃×C₅	C₂²×C₄²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
122	C₁₃	C₂²×C₃×C₄²×C₅	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×C₃	C₄²×C₅×C₁₃	1	16	yes	direct product factor
124	C₂²×C₃	C₂²×C₄×C₅×C₁₃	1	18	yes	—
125	C₂²×C₃	C₂⁴×C₅×C₁₃	1	1	yes	characteristic
126	C₃×C₄	C₂²×C₄×C₅×C₁₃	1	24	yes	direct product factor
127	C₂×C₅	C₂×C₃×C₄²×C₁₃	1	12	yes	direct product factor
128	C₂×C₅	C₂³×C₃×C₄×C₁₃	1	3	yes	—
129	C₂³	C₂×C₃×C₄×C₅×C₁₃	1	12	yes	—
130	C₂³	C₂³×C₃×C₅×C₁₃	1	3	yes	—
131	C₂×C₄	C₂×C₃×C₄×C₅×C₁₃	1	72	yes	direct product factor
132	C₂×C₄	C₂³×C₃×C₅×C₁₃	1	12	yes	—
133	C₂×C₃	C₂×C₄²×C₅×C₁₃	1	12	yes	direct product factor
134	C₂×C₃	C₂³×C₄×C₅×C₁₃	1	3	yes	—
135	C₅	C₂²×C₃×C₄²×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²	C₃×C₄²×C₅×C₁₃	1	16	yes	direct product factor
137	C₂²	C₂⁴×C₃×C₅×C₁₃	1	1	yes	characteristic
138	C₂²	C₂²×C₃×C₄×C₅×C₁₃	1	18	yes	—
139	C₄	C₂²×C₃×C₄×C₅×C₁₃	1	24	yes	direct product factor
140	C₃	C₂²×C₄²×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
141	C₂	C₂×C₃×C₄²×C₅×C₁₃	1	12	yes	direct product factor
142	C₂	C₂³×C₃×C₄×C₅×C₁₃	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂⁴×C₃×C₅×C₁₃
socle, prime radical	C₂⁴×C₃×C₅×C₁₃	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₅×C₁₃	C₂	direct (N × H)
3. C₂²×C₄²×C₅×C₁₃	C₃	direct (N × H)
4. C₂²×C₃×C₄×C₅×C₁₃	C₄	direct (N × H)
7. C₃×C₄²×C₅×C₁₃	C₂²	direct (N × H)
8. C₂²×C₃×C₄²×C₁₃	C₅	direct (N × H)
10. C₂×C₄²×C₅×C₁₃	C₂×C₃	direct (N × H)
13. C₂×C₃×C₄×C₅×C₁₃	C₂×C₄	direct (N × H)
16. C₂×C₃×C₄²×C₁₃	C₂×C₅	direct (N × H)
18. C₂²×C₄×C₅×C₁₃	C₃×C₄	direct (N × H)
20. C₄²×C₅×C₁₃	C₂²×C₃	direct (N × H)
21. C₂²×C₃×C₄²×C₅	C₁₃	direct (N × H)
22. C₂²×C₄²×C₁₃	C₃×C₅	direct (N × H)
23. C₂²×C₃×C₅×C₁₃	C₄²	direct (N × H)
26. C₃×C₄×C₅×C₁₃	C₂²×C₄	direct (N × H)
28. C₂²×C₃×C₄×C₁₃	C₄×C₅	direct (N × H)
30. C₃×C₄²×C₁₃	C₂²×C₅	direct (N × H)
33. C₂×C₄×C₅×C₁₃	C₂×C₃×C₄	direct (N × H)
36. C₂×C₃×C₄²×C₅	C₂×C₁₃	direct (N × H)
38. C₂×C₄²×C₁₃	C₂×C₃×C₅	direct (N × H)
39. C₂×C₃×C₅×C₁₃	C₂×C₄²	direct (N × H)
41. C₂²×C₄²×C₅	C₃×C₁₃	direct (N × H)
44. C₂×C₃×C₄×C₁₃	C₂×C₄×C₅	direct (N × H)
46. C₂²×C₅×C₁₃	C₃×C₄²	direct (N × H)
49. C₄×C₅×C₁₃	C₂²×C₃×C₄	direct (N × H)
51. C₂²×C₃×C₄×C₅	C₄×C₁₃	direct (N × H)
53. C₃×C₄²×C₅	C₂²×C₁₃	direct (N × H)
55. C₂²×C₄×C₁₃	C₃×C₄×C₅	direct (N × H)
57. C₄²×C₁₃	C₂²×C₃×C₅	direct (N × H)
58. C₃×C₅×C₁₃	C₂²×C₄²	direct (N × H)
59. C₂²×C₃×C₄²	C₅×C₁₃	direct (N × H)
61. C₂×C₄²×C₅	C₂×C₃×C₁₃	direct (N × H)
62. C₂²×C₃×C₁₃	C₄²×C₅	direct (N × H)
65. C₃×C₄×C₁₃	C₂²×C₄×C₅	direct (N × H)
66. C₂×C₅×C₁₃	C₂×C₃×C₄²	direct (N × H)
70. C₂×C₃×C₄×C₅	C₂×C₄×C₁₃	direct (N × H)
74. C₂×C₄×C₁₃	C₂×C₃×C₄×C₅	direct (N × H)
77. C₂×C₃×C₄²	C₂×C₅×C₁₃	direct (N × H)
79. C₂²×C₄×C₅	C₃×C₄×C₁₃	direct (N × H)
81. C₄²×C₅	C₂²×C₃×C₁₃	direct (N × H)
82. C₂×C₃×C₁₃	C₂×C₄²×C₅	direct (N × H)
84. C₅×C₁₃	C₂²×C₃×C₄²	direct (N × H)
85. C₂²×C₄²	C₃×C₅×C₁₃	direct (N × H)
86. C₂²×C₃×C₅	C₄²×C₁₃	direct (N × H)
89. C₃×C₄×C₅	C₂²×C₄×C₁₃	direct (N × H)
90. C₂²×C₁₃	C₃×C₄²×C₅	direct (N × H)
93. C₄×C₁₃	C₂²×C₃×C₄×C₅	direct (N × H)
95. C₂²×C₃×C₄	C₄×C₅×C₁₃	direct (N × H)
97. C₃×C₄²	C₂²×C₅×C₁₃	direct (N × H)
100. C₂×C₄×C₅	C₂×C₃×C₄×C₁₃	direct (N × H)
102. C₃×C₁₃	C₂²×C₄²×C₅	direct (N × H)
104. C₂×C₄²	C₂×C₃×C₅×C₁₃	direct (N × H)
105. C₂×C₃×C₅	C₂×C₄²×C₁₃	direct (N × H)
107. C₂×C₁₃	C₂×C₃×C₄²×C₅	direct (N × H)
111. C₂×C₃×C₄	C₂×C₄×C₅×C₁₃	direct (N × H)
113. C₂²×C₅	C₃×C₄²×C₁₃	direct (N × H)
116. C₄×C₅	C₂²×C₃×C₄×C₁₃	direct (N × H)
118. C₂²×C₄	C₃×C₄×C₅×C₁₃	direct (N × H)
120. C₄²	C₂²×C₃×C₅×C₁₃	direct (N × H)
121. C₃×C₅	C₂²×C₄²×C₁₃	direct (N × H)
122. C₁₃	C₂²×C₃×C₄²×C₅	direct (N × H)
123. C₂²×C₃	C₄²×C₅×C₁₃	direct (N × H)
126. C₃×C₄	C₂²×C₄×C₅×C₁₃	direct (N × H)
127. C₂×C₅	C₂×C₃×C₄²×C₁₃	direct (N × H)
131. C₂×C₄	C₂×C₃×C₄×C₅×C₁₃	direct (N × H)
133. C₂×C₃	C₂×C₄²×C₅×C₁₃	direct (N × H)
135. C₅	C₂²×C₃×C₄²×C₁₃	direct (N × H)
136. C₂²	C₃×C₄²×C₅×C₁₃	direct (N × H)
139. C₄	C₂²×C₃×C₄×C₅×C₁₃	direct (N × H)
140. C₃	C₂²×C₄²×C₅×C₁₃	direct (N × H)
141. C₂	C₂×C₃×C₄²×C₅×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
C₂³×C₃×C₄×C₅×C₁₃	2
C₂×C₃×C₄²×C₅×C₁₃	2
C₂²×C₄²×C₅×C₁₃	3
C₂²×C₃×C₄²×C₁₃	5
C₂²×C₃×C₄²×C₅	13

Subgroups — C22×C3×C42×C5×C13 #72190827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄²×C₅×C₁₃ #72190827