Subgroups — C₅²×C₈×C₉×C₁₆ #72205751

Order 28800.

1248 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₈²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅²×C₉×C₁₆	C₂	1	2	yes	maximal
3	C₃×C₅²×C₈×C₁₆	C₃	1	1	yes	maximal, characteristic
4	C₄×C₅²×C₈×C₉	C₄	1	2	yes	—
5	C₄×C₅²×C₈×C₉	C₂²	1	1	yes	characteristic
6	C₂×C₅²×C₉×C₁₆	C₄	1	4	yes	—
7	C₅×C₈×C₉×C₁₆	C₅	1	6	yes	maximal, direct product factor
8	C₃×C₅²×C₈²	C₂×C₃	1	1	yes	characteristic
9	C₃×C₄×C₅²×C₁₆	C₂×C₃	1	2	yes	—
10	C₄²×C₅²×C₉	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅²×C₈×C₉	C₈	1	4	yes	—
12	C₂×C₅²×C₈×C₉	C₂×C₄	1	2	yes	—
13	C₅²×C₉×C₁₆	C₈	1	8	yes	direct product factor
14	C₅²×C₈×C₁₆	C₉	1	1	yes	characteristic, direct product factor, Hall
15	C₅×C₈²×C₉	C₂×C₅	1	6	yes	—
16	C₄×C₅×C₉×C₁₆	C₂×C₅	1	12	yes	—
17	C₃×C₄×C₅²×C₈	C₃×C₄	1	2	yes	—
18	C₃×C₄×C₅²×C₈	C₂²×C₃	1	1	yes	characteristic
19	C₂×C₃×C₅²×C₁₆	C₃×C₄	1	4	yes	—
20	C₃×C₅×C₈×C₁₆	C₃×C₅	1	6	yes	—
21	C₂×C₄×C₅²×C₉	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₅²×C₉	C₂×C₈	1	2	yes	—
23	C₅²×C₈×C₉	C₁₆	1	8	yes	direct product factor
24	C₅²×C₈×C₉	C₂×C₈	1	4	yes	—
25	C₄×C₅²×C₁₆	C₂×C₉	1	2	yes	—
26	C₅²×C₈²	C₂×C₉	1	1	yes	characteristic
27	C₂×C₅×C₉×C₁₆	C₄×C₅	1	24	yes	—
28	C₄×C₅×C₈×C₉	C₄×C₅	1	12	yes	—
29	C₄×C₅×C₈×C₉	C₂²×C₅	1	6	yes	—
30	C₂×C₃×C₅²×C₈	C₃×C₈	1	4	yes	—
31	C₂×C₃×C₅²×C₈	C₂×C₃×C₄	1	2	yes	—
32	C₃×C₄²×C₅²	C₂×C₃×C₄	1	1	yes	characteristic
33	C₃×C₅²×C₁₆	C₃×C₈	1	8	yes	—
34	C₈×C₉×C₁₆	C₅²	1	1	yes	characteristic, direct product factor, Hall
35	C₃×C₄×C₅×C₁₆	C₂×C₃×C₅	1	12	yes	—
36	C₃×C₅×C₈²	C₂×C₃×C₅	1	6	yes	—
37	C₂²×C₅²×C₉	C₄×C₈	1	1	yes	characteristic
38	C₄×C₅²×C₉	C₄×C₈	1	2	yes	—
39	C₄×C₅²×C₉	C₂×C₁₆	1	4	yes	—
40	C₂×C₅²×C₁₆	C₄×C₉	1	4	yes	—
41	C₄×C₅²×C₈	C₄×C₉	1	2	yes	—
42	C₄×C₅²×C₈	C₂²×C₉	1	1	yes	characteristic
43	C₂×C₅×C₈×C₉	C₅×C₈	1	24	yes	—
44	C₂×C₅×C₈×C₉	C₂×C₄×C₅	1	12	yes	—
45	C₄²×C₅×C₉	C₂×C₄×C₅	1	6	yes	—
46	C₅×C₉×C₁₆	C₅×C₈	1	48	yes	direct product factor
47	C₅×C₈×C₁₆	C₅×C₉	1	6	yes	direct product factor
48	C₂×C₃×C₄×C₅²	C₃×C₄²	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅²	C₂×C₃×C₈	1	2	yes	—
50	C₃×C₅²×C₈	C₃×C₁₆	1	8	yes	—
51	C₃×C₅²×C₈	C₂×C₃×C₈	1	4	yes	—
52	C₄×C₉×C₁₆	C₂×C₅²	1	2	yes	—
53	C₈²×C₉	C₂×C₅²	1	1	yes	characteristic
54	C₂×C₃×C₅×C₁₆	C₃×C₄×C₅	1	24	yes	—
55	C₃×C₄×C₅×C₈	C₃×C₄×C₅	1	12	yes	—
56	C₃×C₄×C₅×C₈	C₂²×C₃×C₅	1	6	yes	—
57	C₂×C₅²×C₉	C₈²	1	1	yes	characteristic
58	C₂×C₅²×C₉	C₄×C₁₆	1	2	yes	—
59	C₂×C₅²×C₈	C₈×C₉	1	4	yes	—
60	C₂×C₅²×C₈	C₂×C₄×C₉	1	2	yes	—
61	C₄²×C₅²	C₂×C₄×C₉	1	1	yes	characteristic
62	C₅²×C₁₆	C₈×C₉	1	8	yes	direct product factor
63	C₃×C₈×C₁₆	C₃×C₅²	1	1	yes	characteristic
64	C₂×C₄×C₅×C₉	C₄²×C₅	1	6	yes	—
65	C₂×C₄×C₅×C₉	C₂×C₅×C₈	1	12	yes	—
66	C₅×C₈×C₉	C₅×C₁₆	1	48	yes	direct product factor
67	C₅×C₈×C₉	C₂×C₅×C₈	1	24	yes	—
68	C₄×C₅×C₁₆	C₂×C₅×C₉	1	12	yes	—
69	C₅×C₈²	C₂×C₅×C₉	1	6	yes	—
70	C₂²×C₃×C₅²	C₃×C₄×C₈	1	1	yes	characteristic
71	C₃×C₄×C₅²	C₃×C₄×C₈	1	2	yes	—
72	C₃×C₄×C₅²	C₂×C₃×C₁₆	1	4	yes	—
73	C₂×C₉×C₁₆	C₄×C₅²	1	4	yes	—
74	C₄×C₈×C₉	C₄×C₅²	1	2	yes	—
75	C₄×C₈×C₉	C₂²×C₅²	1	1	yes	characteristic
76	C₂×C₃×C₅×C₈	C₃×C₅×C₈	1	24	yes	—
77	C₂×C₃×C₅×C₈	C₂×C₃×C₄×C₅	1	12	yes	—
78	C₃×C₄²×C₅	C₂×C₃×C₄×C₅	1	6	yes	—
79	C₃×C₅×C₁₆	C₃×C₅×C₈	1	48	yes	—
80	C₅²×C₉	C₈×C₁₆	1	1	yes	characteristic, direct product factor, Hall
81	C₂×C₄×C₅²	C₄²×C₉	1	1	yes	characteristic
82	C₂×C₄×C₅²	C₂×C₈×C₉	1	2	yes	—
83	C₅²×C₈	C₉×C₁₆	1	8	yes	direct product factor
84	C₅²×C₈	C₂×C₈×C₉	1	4	yes	—
85	C₃×C₄×C₁₆	C₂×C₃×C₅²	1	2	yes	—
86	C₃×C₈²	C₂×C₃×C₅²	1	1	yes	characteristic
87	C₂²×C₅×C₉	C₄×C₅×C₈	1	6	yes	—
88	C₄×C₅×C₉	C₄×C₅×C₈	1	12	yes	—
89	C₄×C₅×C₉	C₂×C₅×C₁₆	1	24	yes	—
90	C₂×C₅×C₁₆	C₄×C₅×C₉	1	24	yes	—
91	C₄×C₅×C₈	C₄×C₅×C₉	1	12	yes	—
92	C₄×C₅×C₈	C₂²×C₅×C₉	1	6	yes	—
93	C₂×C₃×C₅²	C₃×C₈²	1	1	yes	characteristic
94	C₂×C₃×C₅²	C₃×C₄×C₁₆	1	2	yes	—
95	C₂×C₈×C₉	C₅²×C₈	1	4	yes	—
96	C₂×C₈×C₉	C₂×C₄×C₅²	1	2	yes	—
97	C₄²×C₉	C₂×C₄×C₅²	1	1	yes	characteristic
98	C₉×C₁₆	C₅²×C₈	1	8	yes	direct product factor
99	C₈×C₁₆	C₅²×C₉	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₃×C₄×C₅	C₃×C₄²×C₅	1	6	yes	—
101	C₂×C₃×C₄×C₅	C₂×C₃×C₅×C₈	1	12	yes	—
102	C₃×C₅×C₈	C₃×C₅×C₁₆	1	48	yes	—
103	C₃×C₅×C₈	C₂×C₃×C₅×C₈	1	24	yes	—
104	C₂²×C₅²	C₄×C₈×C₉	1	1	yes	characteristic
105	C₄×C₅²	C₄×C₈×C₉	1	2	yes	—
106	C₄×C₅²	C₂×C₉×C₁₆	1	4	yes	—
107	C₂×C₃×C₁₆	C₃×C₄×C₅²	1	4	yes	—
108	C₃×C₄×C₈	C₃×C₄×C₅²	1	2	yes	—
109	C₃×C₄×C₈	C₂²×C₃×C₅²	1	1	yes	characteristic
110	C₂×C₅×C₉	C₅×C₈²	1	6	yes	—
111	C₂×C₅×C₉	C₄×C₅×C₁₆	1	12	yes	—
112	C₂×C₅×C₈	C₅×C₈×C₉	1	24	yes	—
113	C₂×C₅×C₈	C₂×C₄×C₅×C₉	1	12	yes	—
114	C₄²×C₅	C₂×C₄×C₅×C₉	1	6	yes	—
115	C₅×C₁₆	C₅×C₈×C₉	1	48	yes	direct product factor
116	C₃×C₅²	C₃×C₈×C₁₆	1	1	yes	characteristic
117	C₂×C₄×C₉	C₄²×C₅²	1	1	yes	characteristic
118	C₂×C₄×C₉	C₂×C₅²×C₈	1	2	yes	—
119	C₈×C₉	C₅²×C₁₆	1	8	yes	direct product factor
120	C₈×C₉	C₂×C₅²×C₈	1	4	yes	—
121	C₄×C₁₆	C₂×C₅²×C₉	1	2	yes	—
122	C₈²	C₂×C₅²×C₉	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₃×C₄×C₅×C₈	1	6	yes	—
124	C₃×C₄×C₅	C₃×C₄×C₅×C₈	1	12	yes	—
125	C₃×C₄×C₅	C₂×C₃×C₅×C₁₆	1	24	yes	—
126	C₂×C₅²	C₈²×C₉	1	1	yes	characteristic
127	C₂×C₅²	C₄×C₉×C₁₆	1	2	yes	—
128	C₂×C₃×C₈	C₃×C₅²×C₈	1	4	yes	—
129	C₂×C₃×C₈	C₂×C₃×C₄×C₅²	1	2	yes	—
130	C₃×C₄²	C₂×C₃×C₄×C₅²	1	1	yes	characteristic
131	C₃×C₁₆	C₃×C₅²×C₈	1	8	yes	—
132	C₅×C₉	C₅×C₈×C₁₆	1	6	yes	direct product factor
133	C₂×C₄×C₅	C₄²×C₅×C₉	1	6	yes	—
134	C₂×C₄×C₅	C₂×C₅×C₈×C₉	1	12	yes	—
135	C₅×C₈	C₅×C₉×C₁₆	1	48	yes	direct product factor
136	C₅×C₈	C₂×C₅×C₈×C₉	1	24	yes	—
137	C₂²×C₉	C₄×C₅²×C₈	1	1	yes	characteristic
138	C₄×C₉	C₄×C₅²×C₈	1	2	yes	—
139	C₄×C₉	C₂×C₅²×C₁₆	1	4	yes	—
140	C₂×C₁₆	C₄×C₅²×C₉	1	4	yes	—
141	C₄×C₈	C₄×C₅²×C₉	1	2	yes	—
142	C₄×C₈	C₂²×C₅²×C₉	1	1	yes	characteristic
143	C₂×C₃×C₅	C₃×C₅×C₈²	1	6	yes	—
144	C₂×C₃×C₅	C₃×C₄×C₅×C₁₆	1	12	yes	—
145	C₅²	C₈×C₉×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×C₃×C₄	C₃×C₄²×C₅²	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×C₃×C₅²×C₈	1	2	yes	—
148	C₃×C₈	C₃×C₅²×C₁₆	1	8	yes	—
149	C₃×C₈	C₂×C₃×C₅²×C₈	1	4	yes	—
150	C₂²×C₅	C₄×C₅×C₈×C₉	1	6	yes	—
151	C₄×C₅	C₄×C₅×C₈×C₉	1	12	yes	—
152	C₄×C₅	C₂×C₅×C₉×C₁₆	1	24	yes	—
153	C₂×C₉	C₅²×C₈²	1	1	yes	characteristic
154	C₂×C₉	C₄×C₅²×C₁₆	1	2	yes	—
155	C₂×C₈	C₅²×C₈×C₉	1	4	yes	—
156	C₂×C₈	C₂×C₄×C₅²×C₉	1	2	yes	—
157	C₄²	C₂×C₄×C₅²×C₉	1	1	yes	characteristic
158	C₁₆	C₅²×C₈×C₉	1	8	yes	direct product factor
159	C₃×C₅	C₃×C₅×C₈×C₁₆	1	6	yes	—
160	C₂²×C₃	C₃×C₄×C₅²×C₈	1	1	yes	characteristic
161	C₃×C₄	C₂×C₃×C₅²×C₁₆	1	4	yes	—
162	C₃×C₄	C₃×C₄×C₅²×C₈	1	2	yes	—
163	C₂×C₅	C₄×C₅×C₉×C₁₆	1	12	yes	—
164	C₂×C₅	C₅×C₈²×C₉	1	6	yes	—
165	C₉	C₅²×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×C₄	C₂×C₅²×C₈×C₉	1	2	yes	—
167	C₂×C₄	C₄²×C₅²×C₉	1	1	yes	characteristic
168	C₈	C₅²×C₉×C₁₆	1	8	yes	direct product factor
169	C₈	C₂×C₅²×C₈×C₉	1	4	yes	—
170	C₂×C₃	C₃×C₄×C₅²×C₁₆	1	2	yes	—
171	C₂×C₃	C₃×C₅²×C₈²	1	1	yes	characteristic
172	C₅	C₅×C₈×C₉×C₁₆	1	6	yes	direct product factor
173	C₂²	C₄×C₅²×C₈×C₉	1	1	yes	characteristic
174	C₄	C₂×C₅²×C₉×C₁₆	1	4	yes	—
175	C₄	C₄×C₅²×C₈×C₉	1	2	yes	—
176	C₃	C₃×C₅²×C₈×C₁₆	1	1	yes	characteristic
177	C₂	C₄×C₅²×C₉×C₁₆	1	2	yes	—
178	C₂	C₅²×C₈²×C₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄×C₈	C₂²×C₃×C₅²
socle, prime radical	C₂²×C₃×C₅²	C₃×C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₅×C₈×C₉×C₁₆	C₅	direct (N × H)
13. C₅²×C₉×C₁₆	C₈	direct (N × H)
14. C₅²×C₈×C₁₆	C₉	direct (N × H)
23. C₅²×C₈×C₉	C₁₆	direct (N × H)
34. C₈×C₉×C₁₆	C₅²	direct (N × H)
46. C₅×C₉×C₁₆	C₅×C₈	direct (N × H)
47. C₅×C₈×C₁₆	C₅×C₉	direct (N × H)
62. C₅²×C₁₆	C₈×C₉	direct (N × H)
66. C₅×C₈×C₉	C₅×C₁₆	direct (N × H)
80. C₅²×C₉	C₈×C₁₆	direct (N × H)
83. C₅²×C₈	C₉×C₁₆	direct (N × H)
98. C₉×C₁₆	C₅²×C₈	direct (N × H)
99. C₈×C₁₆	C₅²×C₉	direct (N × H)
115. C₅×C₁₆	C₅×C₈×C₉	direct (N × H)
119. C₈×C₉	C₅²×C₁₆	direct (N × H)
132. C₅×C₉	C₅×C₈×C₁₆	direct (N × H)
135. C₅×C₈	C₅×C₉×C₁₆	direct (N × H)
145. C₅²	C₈×C₉×C₁₆	direct (N × H)
158. C₁₆	C₅²×C₈×C₉	direct (N × H)
165. C₉	C₅²×C₈×C₁₆	direct (N × H)
168. C₈	C₅²×C₉×C₁₆	direct (N × H)
172. C₅	C₅×C₈×C₉×C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×C₈²×C₉	2
C₄×C₅²×C₉×C₁₆	2
C₃×C₅²×C₈×C₁₆	3
C₅×C₈×C₉×C₁₆	5

Subgroups — C52×C8×C9×C16 #72205751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₈×C₉×C₁₆ #72205751