Subgroups — C₂²×C₃²×C₄²×C₅² #72192573

Order 14400.

11952 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₄×C₅²	C₂	1	3	yes	maximal
2	C₂×C₃²×C₄²×C₅²	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₄²×C₅²	C₃	1	4	yes	maximal, direct product factor
4	C₂⁴×C₃²×C₅²	C₂²	1	1	yes	characteristic
5	C₂²×C₃²×C₄×C₅²	C₄	1	24	yes	direct product factor
6	C₂²×C₃²×C₄×C₅²	C₂²	1	18	yes	—
7	C₃²×C₄²×C₅²	C₂²	1	16	yes	direct product factor
8	C₂²×C₃²×C₄²×C₅	C₅	1	6	yes	maximal, direct product factor
9	C₂³×C₃×C₄×C₅²	C₂×C₃	1	12	yes	—
10	C₂×C₃×C₄²×C₅²	C₂×C₃	1	48	yes	direct product factor
11	C₂³×C₃²×C₅²	C₂×C₄	1	12	yes	—
12	C₂³×C₃²×C₅²	C₂³	1	3	yes	—
13	C₂×C₃²×C₄×C₅²	C₂×C₄	1	72	yes	direct product factor
14	C₂×C₃²×C₄×C₅²	C₂³	1	12	yes	—
15	C₂²×C₄²×C₅²	C₃²	1	1	yes	characteristic, direct product factor, Hall
16	C₂³×C₃²×C₄×C₅	C₂×C₅	1	18	yes	—
17	C₂×C₃²×C₄²×C₅	C₂×C₅	1	72	yes	direct product factor
18	C₂⁴×C₃×C₅²	C₂²×C₃	1	4	yes	—
19	C₂²×C₃×C₄×C₅²	C₃×C₄	1	96	yes	direct product factor
20	C₂²×C₃×C₄×C₅²	C₂²×C₃	1	72	yes	—
21	C₃×C₄²×C₅²	C₂²×C₃	1	64	yes	direct product factor
22	C₂²×C₃×C₄²×C₅	C₃×C₅	1	24	yes	direct product factor
23	C₂²×C₃²×C₅²	C₄²	1	16	yes	direct product factor
24	C₂²×C₃²×C₅²	C₂²×C₄	1	18	yes	—
25	C₂²×C₃²×C₅²	C₂⁴	1	1	yes	characteristic
26	C₃²×C₄×C₅²	C₂²×C₄	1	24	yes	direct product factor
27	C₂³×C₄×C₅²	C₂×C₃²	1	3	yes	—
28	C₂×C₄²×C₅²	C₂×C₃²	1	12	yes	direct product factor
29	C₂⁴×C₃²×C₅	C₂²×C₅	1	6	yes	—
30	C₂²×C₃²×C₄×C₅	C₄×C₅	1	144	yes	direct product factor
31	C₂²×C₃²×C₄×C₅	C₂²×C₅	1	108	yes	—
32	C₃²×C₄²×C₅	C₂²×C₅	1	96	yes	direct product factor
33	C₂³×C₃×C₅²	C₂×C₃×C₄	1	48	yes	—
34	C₂³×C₃×C₅²	C₂³×C₃	1	12	yes	—
35	C₂×C₃×C₄×C₅²	C₂×C₃×C₄	1	288	yes	direct product factor
36	C₂×C₃×C₄×C₅²	C₂³×C₃	1	48	yes	—
37	C₂²×C₃²×C₄²	C₅²	1	1	yes	characteristic, direct product factor, Hall
38	C₂³×C₃×C₄×C₅	C₂×C₃×C₅	1	72	yes	—
39	C₂×C₃×C₄²×C₅	C₂×C₃×C₅	1	288	yes	direct product factor
40	C₂×C₃²×C₅²	C₂×C₄²	1	12	yes	direct product factor
41	C₂×C₃²×C₅²	C₂³×C₄	1	3	yes	—
42	C₂⁴×C₅²	C₂²×C₃²	1	1	yes	characteristic
43	C₂²×C₄×C₅²	C₃²×C₄	1	24	yes	direct product factor
44	C₂²×C₄×C₅²	C₂²×C₃²	1	18	yes	—
45	C₄²×C₅²	C₂²×C₃²	1	16	yes	direct product factor
46	C₂³×C₃²×C₅	C₂×C₄×C₅	1	72	yes	—
47	C₂³×C₃²×C₅	C₂³×C₅	1	18	yes	—
48	C₂×C₃²×C₄×C₅	C₂×C₄×C₅	1	432	yes	direct product factor
49	C₂×C₃²×C₄×C₅	C₂³×C₅	1	72	yes	—
50	C₂²×C₄²×C₅	C₃²×C₅	1	6	yes	direct product factor
51	C₂²×C₃×C₅²	C₃×C₄²	1	64	yes	direct product factor
52	C₂²×C₃×C₅²	C₂²×C₃×C₄	1	72	yes	—
53	C₂²×C₃×C₅²	C₂⁴×C₃	1	4	yes	—
54	C₃×C₄×C₅²	C₂²×C₃×C₄	1	96	yes	direct product factor
55	C₂³×C₃²×C₄	C₂×C₅²	1	3	yes	—
56	C₂×C₃²×C₄²	C₂×C₅²	1	12	yes	direct product factor
57	C₂⁴×C₃×C₅	C₂²×C₃×C₅	1	24	yes	—
58	C₂²×C₃×C₄×C₅	C₃×C₄×C₅	1	576	yes	direct product factor
59	C₂²×C₃×C₄×C₅	C₂²×C₃×C₅	1	432	yes	—
60	C₃×C₄²×C₅	C₂²×C₃×C₅	1	384	yes	direct product factor
61	C₃²×C₅²	C₂²×C₄²	1	1	yes	characteristic, direct product factor, Hall
62	C₂³×C₅²	C₂×C₃²×C₄	1	12	yes	—
63	C₂³×C₅²	C₂³×C₃²	1	3	yes	—
64	C₂×C₄×C₅²	C₂×C₃²×C₄	1	72	yes	direct product factor
65	C₂×C₄×C₅²	C₂³×C₃²	1	12	yes	—
66	C₂²×C₃×C₄²	C₃×C₅²	1	4	yes	direct product factor
67	C₂²×C₃²×C₅	C₄²×C₅	1	96	yes	direct product factor
68	C₂²×C₃²×C₅	C₂²×C₄×C₅	1	108	yes	—
69	C₂²×C₃²×C₅	C₂⁴×C₅	1	6	yes	—
70	C₃²×C₄×C₅	C₂²×C₄×C₅	1	144	yes	direct product factor
71	C₂³×C₄×C₅	C₂×C₃²×C₅	1	18	yes	—
72	C₂×C₄²×C₅	C₂×C₃²×C₅	1	72	yes	direct product factor
73	C₂×C₃×C₅²	C₂×C₃×C₄²	1	48	yes	direct product factor
74	C₂×C₃×C₅²	C₂³×C₃×C₄	1	12	yes	—
75	C₂⁴×C₃²	C₂²×C₅²	1	1	yes	characteristic
76	C₂²×C₃²×C₄	C₄×C₅²	1	24	yes	direct product factor
77	C₂²×C₃²×C₄	C₂²×C₅²	1	18	yes	—
78	C₃²×C₄²	C₂²×C₅²	1	16	yes	direct product factor
79	C₂³×C₃×C₅	C₂×C₃×C₄×C₅	1	288	yes	—
80	C₂³×C₃×C₅	C₂³×C₃×C₅	1	72	yes	—
81	C₂×C₃×C₄×C₅	C₂×C₃×C₄×C₅	1	1728	yes	direct product factor
82	C₂×C₃×C₄×C₅	C₂³×C₃×C₅	1	288	yes	—
83	C₂²×C₅²	C₃²×C₄²	1	16	yes	direct product factor
84	C₂²×C₅²	C₂²×C₃²×C₄	1	18	yes	—
85	C₂²×C₅²	C₂⁴×C₃²	1	1	yes	characteristic
86	C₄×C₅²	C₂²×C₃²×C₄	1	24	yes	direct product factor
87	C₂³×C₃×C₄	C₂×C₃×C₅²	1	12	yes	—
88	C₂×C₃×C₄²	C₂×C₃×C₅²	1	48	yes	direct product factor
89	C₂×C₃²×C₅	C₂×C₄²×C₅	1	72	yes	direct product factor
90	C₂×C₃²×C₅	C₂³×C₄×C₅	1	18	yes	—
91	C₂⁴×C₅	C₂²×C₃²×C₅	1	6	yes	—
92	C₂²×C₄×C₅	C₃²×C₄×C₅	1	144	yes	direct product factor
93	C₂²×C₄×C₅	C₂²×C₃²×C₅	1	108	yes	—
94	C₄²×C₅	C₂²×C₃²×C₅	1	96	yes	direct product factor
95	C₃×C₅²	C₂²×C₃×C₄²	1	4	yes	direct product factor
96	C₂³×C₃²	C₂×C₄×C₅²	1	12	yes	—
97	C₂³×C₃²	C₂³×C₅²	1	3	yes	—
98	C₂×C₃²×C₄	C₂×C₄×C₅²	1	72	yes	direct product factor
99	C₂×C₃²×C₄	C₂³×C₅²	1	12	yes	—
100	C₂²×C₄²	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×C₃×C₅	C₃×C₄²×C₅	1	384	yes	direct product factor
102	C₂²×C₃×C₅	C₂²×C₃×C₄×C₅	1	432	yes	—
103	C₂²×C₃×C₅	C₂⁴×C₃×C₅	1	24	yes	—
104	C₃×C₄×C₅	C₂²×C₃×C₄×C₅	1	576	yes	direct product factor
105	C₂×C₅²	C₂×C₃²×C₄²	1	12	yes	direct product factor
106	C₂×C₅²	C₂³×C₃²×C₄	1	3	yes	—
107	C₂⁴×C₃	C₂²×C₃×C₅²	1	4	yes	—
108	C₂²×C₃×C₄	C₃×C₄×C₅²	1	96	yes	direct product factor
109	C₂²×C₃×C₄	C₂²×C₃×C₅²	1	72	yes	—
110	C₃×C₄²	C₂²×C₃×C₅²	1	64	yes	direct product factor
111	C₃²×C₅	C₂²×C₄²×C₅	1	6	yes	direct product factor
112	C₂³×C₅	C₂×C₃²×C₄×C₅	1	72	yes	—
113	C₂³×C₅	C₂³×C₃²×C₅	1	18	yes	—
114	C₂×C₄×C₅	C₂×C₃²×C₄×C₅	1	432	yes	direct product factor
115	C₂×C₄×C₅	C₂³×C₃²×C₅	1	72	yes	—
116	C₂²×C₃²	C₄²×C₅²	1	16	yes	direct product factor
117	C₂²×C₃²	C₂²×C₄×C₅²	1	18	yes	—
118	C₂²×C₃²	C₂⁴×C₅²	1	1	yes	characteristic
119	C₃²×C₄	C₂²×C₄×C₅²	1	24	yes	direct product factor
120	C₂³×C₄	C₂×C₃²×C₅²	1	3	yes	—
121	C₂×C₄²	C₂×C₃²×C₅²	1	12	yes	direct product factor
122	C₂×C₃×C₅	C₂×C₃×C₄²×C₅	1	288	yes	direct product factor
123	C₂×C₃×C₅	C₂³×C₃×C₄×C₅	1	72	yes	—
124	C₅²	C₂²×C₃²×C₄²	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³×C₃	C₂×C₃×C₄×C₅²	1	48	yes	—
126	C₂³×C₃	C₂³×C₃×C₅²	1	12	yes	—
127	C₂×C₃×C₄	C₂×C₃×C₄×C₅²	1	288	yes	direct product factor
128	C₂×C₃×C₄	C₂³×C₃×C₅²	1	48	yes	—
129	C₂²×C₅	C₃²×C₄²×C₅	1	96	yes	direct product factor
130	C₂²×C₅	C₂²×C₃²×C₄×C₅	1	108	yes	—
131	C₂²×C₅	C₂⁴×C₃²×C₅	1	6	yes	—
132	C₄×C₅	C₂²×C₃²×C₄×C₅	1	144	yes	direct product factor
133	C₂×C₃²	C₂×C₄²×C₅²	1	12	yes	direct product factor
134	C₂×C₃²	C₂³×C₄×C₅²	1	3	yes	—
135	C₂⁴	C₂²×C₃²×C₅²	1	1	yes	characteristic
136	C₂²×C₄	C₃²×C₄×C₅²	1	24	yes	direct product factor
137	C₂²×C₄	C₂²×C₃²×C₅²	1	18	yes	—
138	C₄²	C₂²×C₃²×C₅²	1	16	yes	direct product factor
139	C₃×C₅	C₂²×C₃×C₄²×C₅	1	24	yes	direct product factor
140	C₂²×C₃	C₃×C₄²×C₅²	1	64	yes	direct product factor
141	C₂²×C₃	C₂²×C₃×C₄×C₅²	1	72	yes	—
142	C₂²×C₃	C₂⁴×C₃×C₅²	1	4	yes	—
143	C₃×C₄	C₂²×C₃×C₄×C₅²	1	96	yes	direct product factor
144	C₂×C₅	C₂×C₃²×C₄²×C₅	1	72	yes	direct product factor
145	C₂×C₅	C₂³×C₃²×C₄×C₅	1	18	yes	—
146	C₃²	C₂²×C₄²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
147	C₂³	C₂×C₃²×C₄×C₅²	1	12	yes	—
148	C₂³	C₂³×C₃²×C₅²	1	3	yes	—
149	C₂×C₄	C₂×C₃²×C₄×C₅²	1	72	yes	direct product factor
150	C₂×C₄	C₂³×C₃²×C₅²	1	12	yes	—
151	C₂×C₃	C₂×C₃×C₄²×C₅²	1	48	yes	direct product factor
152	C₂×C₃	C₂³×C₃×C₄×C₅²	1	12	yes	—
153	C₅	C₂²×C₃²×C₄²×C₅	1	6	yes	direct product factor
154	C₂²	C₃²×C₄²×C₅²	1	16	yes	direct product factor
155	C₂²	C₂²×C₃²×C₄×C₅²	1	18	yes	—
156	C₂²	C₂⁴×C₃²×C₅²	1	1	yes	characteristic
157	C₄	C₂²×C₃²×C₄×C₅²	1	24	yes	direct product factor
158	C₃	C₂²×C₃×C₄²×C₅²	1	4	yes	direct product factor
159	C₂	C₂×C₃²×C₄²×C₅²	1	12	yes	direct product factor
160	C₂	C₂³×C₃²×C₄×C₅²	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂⁴×C₃²×C₅²
socle, prime radical	C₂⁴×C₃²×C₅²	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×C₃²×C₄²×C₅²	C₂	direct (N × H)
3. C₂²×C₃×C₄²×C₅²	C₃	direct (N × H)
5. C₂²×C₃²×C₄×C₅²	C₄	direct (N × H)
7. C₃²×C₄²×C₅²	C₂²	direct (N × H)
8. C₂²×C₃²×C₄²×C₅	C₅	direct (N × H)
10. C₂×C₃×C₄²×C₅²	C₂×C₃	direct (N × H)
13. C₂×C₃²×C₄×C₅²	C₂×C₄	direct (N × H)
15. C₂²×C₄²×C₅²	C₃²	direct (N × H)
17. C₂×C₃²×C₄²×C₅	C₂×C₅	direct (N × H)
19. C₂²×C₃×C₄×C₅²	C₃×C₄	direct (N × H)
21. C₃×C₄²×C₅²	C₂²×C₃	direct (N × H)
22. C₂²×C₃×C₄²×C₅	C₃×C₅	direct (N × H)
23. C₂²×C₃²×C₅²	C₄²	direct (N × H)
26. C₃²×C₄×C₅²	C₂²×C₄	direct (N × H)
28. C₂×C₄²×C₅²	C₂×C₃²	direct (N × H)
30. C₂²×C₃²×C₄×C₅	C₄×C₅	direct (N × H)
32. C₃²×C₄²×C₅	C₂²×C₅	direct (N × H)
35. C₂×C₃×C₄×C₅²	C₂×C₃×C₄	direct (N × H)
37. C₂²×C₃²×C₄²	C₅²	direct (N × H)
39. C₂×C₃×C₄²×C₅	C₂×C₃×C₅	direct (N × H)
40. C₂×C₃²×C₅²	C₂×C₄²	direct (N × H)
43. C₂²×C₄×C₅²	C₃²×C₄	direct (N × H)
45. C₄²×C₅²	C₂²×C₃²	direct (N × H)
48. C₂×C₃²×C₄×C₅	C₂×C₄×C₅	direct (N × H)
50. C₂²×C₄²×C₅	C₃²×C₅	direct (N × H)
51. C₂²×C₃×C₅²	C₃×C₄²	direct (N × H)
54. C₃×C₄×C₅²	C₂²×C₃×C₄	direct (N × H)
56. C₂×C₃²×C₄²	C₂×C₅²	direct (N × H)
58. C₂²×C₃×C₄×C₅	C₃×C₄×C₅	direct (N × H)
60. C₃×C₄²×C₅	C₂²×C₃×C₅	direct (N × H)
61. C₃²×C₅²	C₂²×C₄²	direct (N × H)
64. C₂×C₄×C₅²	C₂×C₃²×C₄	direct (N × H)
66. C₂²×C₃×C₄²	C₃×C₅²	direct (N × H)
67. C₂²×C₃²×C₅	C₄²×C₅	direct (N × H)
70. C₃²×C₄×C₅	C₂²×C₄×C₅	direct (N × H)
72. C₂×C₄²×C₅	C₂×C₃²×C₅	direct (N × H)
73. C₂×C₃×C₅²	C₂×C₃×C₄²	direct (N × H)
76. C₂²×C₃²×C₄	C₄×C₅²	direct (N × H)
78. C₃²×C₄²	C₂²×C₅²	direct (N × H)
81. C₂×C₃×C₄×C₅	C₂×C₃×C₄×C₅	direct (N × H)
83. C₂²×C₅²	C₃²×C₄²	direct (N × H)
86. C₄×C₅²	C₂²×C₃²×C₄	direct (N × H)
88. C₂×C₃×C₄²	C₂×C₃×C₅²	direct (N × H)
89. C₂×C₃²×C₅	C₂×C₄²×C₅	direct (N × H)
92. C₂²×C₄×C₅	C₃²×C₄×C₅	direct (N × H)
94. C₄²×C₅	C₂²×C₃²×C₅	direct (N × H)
95. C₃×C₅²	C₂²×C₃×C₄²	direct (N × H)
98. C₂×C₃²×C₄	C₂×C₄×C₅²	direct (N × H)
100. C₂²×C₄²	C₃²×C₅²	direct (N × H)
101. C₂²×C₃×C₅	C₃×C₄²×C₅	direct (N × H)
104. C₃×C₄×C₅	C₂²×C₃×C₄×C₅	direct (N × H)
105. C₂×C₅²	C₂×C₃²×C₄²	direct (N × H)
108. C₂²×C₃×C₄	C₃×C₄×C₅²	direct (N × H)
110. C₃×C₄²	C₂²×C₃×C₅²	direct (N × H)
111. C₃²×C₅	C₂²×C₄²×C₅	direct (N × H)
114. C₂×C₄×C₅	C₂×C₃²×C₄×C₅	direct (N × H)
116. C₂²×C₃²	C₄²×C₅²	direct (N × H)
119. C₃²×C₄	C₂²×C₄×C₅²	direct (N × H)
121. C₂×C₄²	C₂×C₃²×C₅²	direct (N × H)
122. C₂×C₃×C₅	C₂×C₃×C₄²×C₅	direct (N × H)
124. C₅²	C₂²×C₃²×C₄²	direct (N × H)
127. C₂×C₃×C₄	C₂×C₃×C₄×C₅²	direct (N × H)
129. C₂²×C₅	C₃²×C₄²×C₅	direct (N × H)
132. C₄×C₅	C₂²×C₃²×C₄×C₅	direct (N × H)
133. C₂×C₃²	C₂×C₄²×C₅²	direct (N × H)
136. C₂²×C₄	C₃²×C₄×C₅²	direct (N × H)
138. C₄²	C₂²×C₃²×C₅²	direct (N × H)
139. C₃×C₅	C₂²×C₃×C₄²×C₅	direct (N × H)
140. C₂²×C₃	C₃×C₄²×C₅²	direct (N × H)
143. C₃×C₄	C₂²×C₃×C₄×C₅²	direct (N × H)
144. C₂×C₅	C₂×C₃²×C₄²×C₅	direct (N × H)
146. C₃²	C₂²×C₄²×C₅²	direct (N × H)
149. C₂×C₄	C₂×C₃²×C₄×C₅²	direct (N × H)
151. C₂×C₃	C₂×C₃×C₄²×C₅²	direct (N × H)
153. C₅	C₂²×C₃²×C₄²×C₅	direct (N × H)
154. C₂²	C₃²×C₄²×C₅²	direct (N × H)
157. C₄	C₂²×C₃²×C₄×C₅²	direct (N × H)
158. C₃	C₂²×C₃×C₄²×C₅²	direct (N × H)
159. C₂	C₂×C₃²×C₄²×C₅²	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×C₃²×C₄×C₅²	2
C₂×C₃²×C₄²×C₅²	2
C₂²×C₃×C₄²×C₅²	3
C₂²×C₃²×C₄²×C₅	5

Subgroups — C22×C32×C42×C52 #72192573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×C₄²×C₅² #72192573