Subgroups — C₂³×C₃²×C₅²×C₈ #72192565

Order 14400.

8112 conjugacy classes of subgroups.

The subgroup lattice has 171 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	1	yes	maximal, characteristic
2	C₂²×C₃²×C₅²×C₈	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₅²×C₈	C₃	1	4	yes	maximal, direct product factor
4	C₂⁴×C₃²×C₅²	C₄	1	1	yes	characteristic
5	C₂²×C₃²×C₄×C₅²	C₄	1	7	yes	—
6	C₂²×C₃²×C₄×C₅²	C₂²	1	7	yes	—
7	C₂×C₃²×C₅²×C₈	C₂²	1	28	yes	direct product factor
8	C₂³×C₃²×C₅×C₈	C₅	1	6	yes	maximal, direct product factor
9	C₂³×C₃×C₄×C₅²	C₂×C₃	1	4	yes	—
10	C₂²×C₃×C₅²×C₈	C₂×C₃	1	56	yes	direct product factor
11	C₂³×C₃²×C₅²	C₈	1	8	yes	direct product factor
12	C₂³×C₃²×C₅²	C₂×C₄	1	7	yes	—
13	C₂×C₃²×C₄×C₅²	C₂×C₄	1	21	yes	—
14	C₂×C₃²×C₄×C₅²	C₂³	1	7	yes	—
15	C₃²×C₅²×C₈	C₂³	1	8	yes	direct product factor
16	C₂³×C₅²×C₈	C₃²	1	1	yes	characteristic, direct product factor, Hall
17	C₂³×C₃²×C₄×C₅	C₂×C₅	1	6	yes	—
18	C₂²×C₃²×C₅×C₈	C₂×C₅	1	84	yes	direct product factor
19	C₂⁴×C₃×C₅²	C₃×C₄	1	4	yes	—
20	C₂²×C₃×C₄×C₅²	C₃×C₄	1	28	yes	—
21	C₂²×C₃×C₄×C₅²	C₂²×C₃	1	28	yes	—
22	C₂×C₃×C₅²×C₈	C₂²×C₃	1	112	yes	direct product factor
23	C₂³×C₃×C₅×C₈	C₃×C₅	1	24	yes	direct product factor
24	C₂²×C₃²×C₅²	C₂×C₈	1	28	yes	direct product factor
25	C₂²×C₃²×C₅²	C₂²×C₄	1	7	yes	—
26	C₃²×C₄×C₅²	C₂²×C₄	1	7	yes	—
27	C₃²×C₄×C₅²	C₂⁴	1	1	yes	characteristic
28	C₂³×C₄×C₅²	C₂×C₃²	1	1	yes	characteristic
29	C₂²×C₅²×C₈	C₂×C₃²	1	14	yes	direct product factor
30	C₂⁴×C₃²×C₅	C₄×C₅	1	6	yes	—
31	C₂²×C₃²×C₄×C₅	C₄×C₅	1	42	yes	—
32	C₂²×C₃²×C₄×C₅	C₂²×C₅	1	42	yes	—
33	C₂×C₃²×C₅×C₈	C₂²×C₅	1	168	yes	direct product factor
34	C₂³×C₃×C₅²	C₃×C₈	1	32	yes	direct product factor
35	C₂³×C₃×C₅²	C₂×C₃×C₄	1	28	yes	—
36	C₂×C₃×C₄×C₅²	C₂×C₃×C₄	1	84	yes	—
37	C₂×C₃×C₄×C₅²	C₂³×C₃	1	28	yes	—
38	C₃×C₅²×C₈	C₂³×C₃	1	32	yes	direct product factor
39	C₂³×C₃²×C₈	C₅²	1	1	yes	characteristic, direct product factor, Hall
40	C₂³×C₃×C₄×C₅	C₂×C₃×C₅	1	24	yes	—
41	C₂²×C₃×C₅×C₈	C₂×C₃×C₅	1	336	yes	direct product factor
42	C₂×C₃²×C₅²	C₂²×C₈	1	14	yes	direct product factor
43	C₂×C₃²×C₅²	C₂³×C₄	1	1	yes	characteristic
44	C₂⁴×C₅²	C₃²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅²	C₃²×C₄	1	7	yes	—
46	C₂²×C₄×C₅²	C₂²×C₃²	1	7	yes	—
47	C₂×C₅²×C₈	C₂²×C₃²	1	28	yes	direct product factor
48	C₂³×C₃²×C₅	C₅×C₈	1	48	yes	direct product factor
49	C₂³×C₃²×C₅	C₂×C₄×C₅	1	42	yes	—
50	C₂×C₃²×C₄×C₅	C₂×C₄×C₅	1	126	yes	—
51	C₂×C₃²×C₄×C₅	C₂³×C₅	1	42	yes	—
52	C₃²×C₅×C₈	C₂³×C₅	1	48	yes	direct product factor
53	C₂³×C₅×C₈	C₃²×C₅	1	6	yes	direct product factor
54	C₂²×C₃×C₅²	C₂×C₃×C₈	1	112	yes	direct product factor
55	C₂²×C₃×C₅²	C₂²×C₃×C₄	1	28	yes	—
56	C₃×C₄×C₅²	C₂²×C₃×C₄	1	28	yes	—
57	C₃×C₄×C₅²	C₂⁴×C₃	1	4	yes	—
58	C₂³×C₃²×C₄	C₂×C₅²	1	1	yes	characteristic
59	C₂²×C₃²×C₈	C₂×C₅²	1	14	yes	direct product factor
60	C₂⁴×C₃×C₅	C₃×C₄×C₅	1	24	yes	—
61	C₂²×C₃×C₄×C₅	C₃×C₄×C₅	1	168	yes	—
62	C₂²×C₃×C₄×C₅	C₂²×C₃×C₅	1	168	yes	—
63	C₂×C₃×C₅×C₈	C₂²×C₃×C₅	1	672	yes	direct product factor
64	C₃²×C₅²	C₂³×C₈	1	1	yes	characteristic, direct product factor, Hall
65	C₂³×C₅²	C₃²×C₈	1	8	yes	direct product factor
66	C₂³×C₅²	C₂×C₃²×C₄	1	7	yes	—
67	C₂×C₄×C₅²	C₂×C₃²×C₄	1	21	yes	—
68	C₂×C₄×C₅²	C₂³×C₃²	1	7	yes	—
69	C₅²×C₈	C₂³×C₃²	1	8	yes	direct product factor
70	C₂³×C₃×C₈	C₃×C₅²	1	4	yes	direct product factor
71	C₂²×C₃²×C₅	C₂×C₅×C₈	1	168	yes	direct product factor
72	C₂²×C₃²×C₅	C₂²×C₄×C₅	1	42	yes	—
73	C₃²×C₄×C₅	C₂²×C₄×C₅	1	42	yes	—
74	C₃²×C₄×C₅	C₂⁴×C₅	1	6	yes	—
75	C₂³×C₄×C₅	C₂×C₃²×C₅	1	6	yes	—
76	C₂²×C₅×C₈	C₂×C₃²×C₅	1	84	yes	direct product factor
77	C₂×C₃×C₅²	C₂²×C₃×C₈	1	56	yes	direct product factor
78	C₂×C₃×C₅²	C₂³×C₃×C₄	1	4	yes	—
79	C₂⁴×C₃²	C₄×C₅²	1	1	yes	characteristic
80	C₂²×C₃²×C₄	C₄×C₅²	1	7	yes	—
81	C₂²×C₃²×C₄	C₂²×C₅²	1	7	yes	—
82	C₂×C₃²×C₈	C₂²×C₅²	1	28	yes	direct product factor
83	C₂³×C₃×C₅	C₃×C₅×C₈	1	192	yes	direct product factor
84	C₂³×C₃×C₅	C₂×C₃×C₄×C₅	1	168	yes	—
85	C₂×C₃×C₄×C₅	C₂×C₃×C₄×C₅	1	504	yes	—
86	C₂×C₃×C₄×C₅	C₂³×C₃×C₅	1	168	yes	—
87	C₃×C₅×C₈	C₂³×C₃×C₅	1	192	yes	direct product factor
88	C₂²×C₅²	C₂×C₃²×C₈	1	28	yes	direct product factor
89	C₂²×C₅²	C₂²×C₃²×C₄	1	7	yes	—
90	C₄×C₅²	C₂²×C₃²×C₄	1	7	yes	—
91	C₄×C₅²	C₂⁴×C₃²	1	1	yes	characteristic
92	C₂³×C₃×C₄	C₂×C₃×C₅²	1	4	yes	—
93	C₂²×C₃×C₈	C₂×C₃×C₅²	1	56	yes	direct product factor
94	C₂×C₃²×C₅	C₂²×C₅×C₈	1	84	yes	direct product factor
95	C₂×C₃²×C₅	C₂³×C₄×C₅	1	6	yes	—
96	C₂⁴×C₅	C₃²×C₄×C₅	1	6	yes	—
97	C₂²×C₄×C₅	C₃²×C₄×C₅	1	42	yes	—
98	C₂²×C₄×C₅	C₂²×C₃²×C₅	1	42	yes	—
99	C₂×C₅×C₈	C₂²×C₃²×C₅	1	168	yes	direct product factor
100	C₃×C₅²	C₂³×C₃×C₈	1	4	yes	direct product factor
101	C₂³×C₃²	C₅²×C₈	1	8	yes	direct product factor
102	C₂³×C₃²	C₂×C₄×C₅²	1	7	yes	—
103	C₂×C₃²×C₄	C₂×C₄×C₅²	1	21	yes	—
104	C₂×C₃²×C₄	C₂³×C₅²	1	7	yes	—
105	C₃²×C₈	C₂³×C₅²	1	8	yes	direct product factor
106	C₂³×C₈	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₃×C₅	C₂×C₃×C₅×C₈	1	672	yes	direct product factor
108	C₂²×C₃×C₅	C₂²×C₃×C₄×C₅	1	168	yes	—
109	C₃×C₄×C₅	C₂²×C₃×C₄×C₅	1	168	yes	—
110	C₃×C₄×C₅	C₂⁴×C₃×C₅	1	24	yes	—
111	C₂×C₅²	C₂²×C₃²×C₈	1	14	yes	direct product factor
112	C₂×C₅²	C₂³×C₃²×C₄	1	1	yes	characteristic
113	C₂⁴×C₃	C₃×C₄×C₅²	1	4	yes	—
114	C₂²×C₃×C₄	C₃×C₄×C₅²	1	28	yes	—
115	C₂²×C₃×C₄	C₂²×C₃×C₅²	1	28	yes	—
116	C₂×C₃×C₈	C₂²×C₃×C₅²	1	112	yes	direct product factor
117	C₃²×C₅	C₂³×C₅×C₈	1	6	yes	direct product factor
118	C₂³×C₅	C₃²×C₅×C₈	1	48	yes	direct product factor
119	C₂³×C₅	C₂×C₃²×C₄×C₅	1	42	yes	—
120	C₂×C₄×C₅	C₂×C₃²×C₄×C₅	1	126	yes	—
121	C₂×C₄×C₅	C₂³×C₃²×C₅	1	42	yes	—
122	C₅×C₈	C₂³×C₃²×C₅	1	48	yes	direct product factor
123	C₂²×C₃²	C₂×C₅²×C₈	1	28	yes	direct product factor
124	C₂²×C₃²	C₂²×C₄×C₅²	1	7	yes	—
125	C₃²×C₄	C₂²×C₄×C₅²	1	7	yes	—
126	C₃²×C₄	C₂⁴×C₅²	1	1	yes	characteristic
127	C₂³×C₄	C₂×C₃²×C₅²	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₃²×C₅²	1	14	yes	direct product factor
129	C₂×C₃×C₅	C₂²×C₃×C₅×C₈	1	336	yes	direct product factor
130	C₂×C₃×C₅	C₂³×C₃×C₄×C₅	1	24	yes	—
131	C₅²	C₂³×C₃²×C₈	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₃	C₃×C₅²×C₈	1	32	yes	direct product factor
133	C₂³×C₃	C₂×C₃×C₄×C₅²	1	28	yes	—
134	C₂×C₃×C₄	C₂×C₃×C₄×C₅²	1	84	yes	—
135	C₂×C₃×C₄	C₂³×C₃×C₅²	1	28	yes	—
136	C₃×C₈	C₂³×C₃×C₅²	1	32	yes	direct product factor
137	C₂²×C₅	C₂×C₃²×C₅×C₈	1	168	yes	direct product factor
138	C₂²×C₅	C₂²×C₃²×C₄×C₅	1	42	yes	—
139	C₄×C₅	C₂²×C₃²×C₄×C₅	1	42	yes	—
140	C₄×C₅	C₂⁴×C₃²×C₅	1	6	yes	—
141	C₂×C₃²	C₂²×C₅²×C₈	1	14	yes	direct product factor
142	C₂×C₃²	C₂³×C₄×C₅²	1	1	yes	characteristic
143	C₂⁴	C₃²×C₄×C₅²	1	1	yes	characteristic
144	C₂²×C₄	C₃²×C₄×C₅²	1	7	yes	—
145	C₂²×C₄	C₂²×C₃²×C₅²	1	7	yes	—
146	C₂×C₈	C₂²×C₃²×C₅²	1	28	yes	direct product factor
147	C₃×C₅	C₂³×C₃×C₅×C₈	1	24	yes	direct product factor
148	C₂²×C₃	C₂×C₃×C₅²×C₈	1	112	yes	direct product factor
149	C₂²×C₃	C₂²×C₃×C₄×C₅²	1	28	yes	—
150	C₃×C₄	C₂²×C₃×C₄×C₅²	1	28	yes	—
151	C₃×C₄	C₂⁴×C₃×C₅²	1	4	yes	—
152	C₂×C₅	C₂²×C₃²×C₅×C₈	1	84	yes	direct product factor
153	C₂×C₅	C₂³×C₃²×C₄×C₅	1	6	yes	—
154	C₃²	C₂³×C₅²×C₈	1	1	yes	characteristic, direct product factor, Sylow
155	C₂³	C₃²×C₅²×C₈	1	8	yes	direct product factor
156	C₂³	C₂×C₃²×C₄×C₅²	1	7	yes	—
157	C₂×C₄	C₂×C₃²×C₄×C₅²	1	21	yes	—
158	C₂×C₄	C₂³×C₃²×C₅²	1	7	yes	—
159	C₈	C₂³×C₃²×C₅²	1	8	yes	direct product factor
160	C₂×C₃	C₂²×C₃×C₅²×C₈	1	56	yes	direct product factor
161	C₂×C₃	C₂³×C₃×C₄×C₅²	1	4	yes	—
162	C₅	C₂³×C₃²×C₅×C₈	1	6	yes	direct product factor
163	C₂²	C₂×C₃²×C₅²×C₈	1	28	yes	direct product factor
164	C₂²	C₂²×C₃²×C₄×C₅²	1	7	yes	—
165	C₄	C₂²×C₃²×C₄×C₅²	1	7	yes	—
166	C₄	C₂⁴×C₃²×C₅²	1	1	yes	characteristic
167	C₃	C₂³×C₃×C₅²×C₈	1	4	yes	direct product factor
168	C₂	C₂²×C₃²×C₅²×C₈	1	14	yes	direct product factor
169	C₂	C₂³×C₃²×C₄×C₅²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂⁴×C₃²×C₅²
socle, prime radical	C₂⁴×C₃²×C₅²	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₅²×C₈	C₂	direct (N × H)
3. C₂³×C₃×C₅²×C₈	C₃	direct (N × H)
7. C₂×C₃²×C₅²×C₈	C₂²	direct (N × H)
8. C₂³×C₃²×C₅×C₈	C₅	direct (N × H)
10. C₂²×C₃×C₅²×C₈	C₂×C₃	direct (N × H)
11. C₂³×C₃²×C₅²	C₈	direct (N × H)
15. C₃²×C₅²×C₈	C₂³	direct (N × H)
16. C₂³×C₅²×C₈	C₃²	direct (N × H)
18. C₂²×C₃²×C₅×C₈	C₂×C₅	direct (N × H)
22. C₂×C₃×C₅²×C₈	C₂²×C₃	direct (N × H)
23. C₂³×C₃×C₅×C₈	C₃×C₅	direct (N × H)
24. C₂²×C₃²×C₅²	C₂×C₈	direct (N × H)
29. C₂²×C₅²×C₈	C₂×C₃²	direct (N × H)
33. C₂×C₃²×C₅×C₈	C₂²×C₅	direct (N × H)
34. C₂³×C₃×C₅²	C₃×C₈	direct (N × H)
38. C₃×C₅²×C₈	C₂³×C₃	direct (N × H)
39. C₂³×C₃²×C₈	C₅²	direct (N × H)
41. C₂²×C₃×C₅×C₈	C₂×C₃×C₅	direct (N × H)
42. C₂×C₃²×C₅²	C₂²×C₈	direct (N × H)
47. C₂×C₅²×C₈	C₂²×C₃²	direct (N × H)
48. C₂³×C₃²×C₅	C₅×C₈	direct (N × H)
52. C₃²×C₅×C₈	C₂³×C₅	direct (N × H)
53. C₂³×C₅×C₈	C₃²×C₅	direct (N × H)
54. C₂²×C₃×C₅²	C₂×C₃×C₈	direct (N × H)
59. C₂²×C₃²×C₈	C₂×C₅²	direct (N × H)
63. C₂×C₃×C₅×C₈	C₂²×C₃×C₅	direct (N × H)
64. C₃²×C₅²	C₂³×C₈	direct (N × H)
65. C₂³×C₅²	C₃²×C₈	direct (N × H)
69. C₅²×C₈	C₂³×C₃²	direct (N × H)
70. C₂³×C₃×C₈	C₃×C₅²	direct (N × H)
71. C₂²×C₃²×C₅	C₂×C₅×C₈	direct (N × H)
76. C₂²×C₅×C₈	C₂×C₃²×C₅	direct (N × H)
77. C₂×C₃×C₅²	C₂²×C₃×C₈	direct (N × H)
82. C₂×C₃²×C₈	C₂²×C₅²	direct (N × H)
83. C₂³×C₃×C₅	C₃×C₅×C₈	direct (N × H)
87. C₃×C₅×C₈	C₂³×C₃×C₅	direct (N × H)
88. C₂²×C₅²	C₂×C₃²×C₈	direct (N × H)
93. C₂²×C₃×C₈	C₂×C₃×C₅²	direct (N × H)
94. C₂×C₃²×C₅	C₂²×C₅×C₈	direct (N × H)
99. C₂×C₅×C₈	C₂²×C₃²×C₅	direct (N × H)
100. C₃×C₅²	C₂³×C₃×C₈	direct (N × H)
101. C₂³×C₃²	C₅²×C₈	direct (N × H)
105. C₃²×C₈	C₂³×C₅²	direct (N × H)
106. C₂³×C₈	C₃²×C₅²	direct (N × H)
107. C₂²×C₃×C₅	C₂×C₃×C₅×C₈	direct (N × H)
111. C₂×C₅²	C₂²×C₃²×C₈	direct (N × H)
116. C₂×C₃×C₈	C₂²×C₃×C₅²	direct (N × H)
117. C₃²×C₅	C₂³×C₅×C₈	direct (N × H)
118. C₂³×C₅	C₃²×C₅×C₈	direct (N × H)
122. C₅×C₈	C₂³×C₃²×C₅	direct (N × H)
123. C₂²×C₃²	C₂×C₅²×C₈	direct (N × H)
128. C₂²×C₈	C₂×C₃²×C₅²	direct (N × H)
129. C₂×C₃×C₅	C₂²×C₃×C₅×C₈	direct (N × H)
131. C₅²	C₂³×C₃²×C₈	direct (N × H)
132. C₂³×C₃	C₃×C₅²×C₈	direct (N × H)
136. C₃×C₈	C₂³×C₃×C₅²	direct (N × H)
137. C₂²×C₅	C₂×C₃²×C₅×C₈	direct (N × H)
141. C₂×C₃²	C₂²×C₅²×C₈	direct (N × H)
146. C₂×C₈	C₂²×C₃²×C₅²	direct (N × H)
147. C₃×C₅	C₂³×C₃×C₅×C₈	direct (N × H)
148. C₂²×C₃	C₂×C₃×C₅²×C₈	direct (N × H)
152. C₂×C₅	C₂²×C₃²×C₅×C₈	direct (N × H)
154. C₃²	C₂³×C₅²×C₈	direct (N × H)
155. C₂³	C₃²×C₅²×C₈	direct (N × H)
159. C₈	C₂³×C₃²×C₅²	direct (N × H)
160. C₂×C₃	C₂²×C₃×C₅²×C₈	direct (N × H)
162. C₅	C₂³×C₃²×C₅×C₈	direct (N × H)
163. C₂²	C₂×C₃²×C₅²×C₈	direct (N × H)
167. C₃	C₂³×C₃×C₅²×C₈	direct (N × H)
168. C₂	C₂²×C₃²×C₅²×C₈	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
C₂²×C₃²×C₅²×C₈	2
C₂³×C₃×C₅²×C₈	3
C₂³×C₃²×C₅×C₈	5

Subgroups — C23×C32×C52×C8 #72192565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃²×C₅²×C₈ #72192565