Subgroups — C₄×C₅×C₁₆×C₈₁ #72203071

Order 25920.

290 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	C₄×C₅×C₈×C₈₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₁₆×C₈₁	C₂	1	2	yes	maximal
3	C₄×C₅×C₁₆×C₂₇	C₃	1	1	yes	maximal, characteristic
4	C₄²×C₅×C₈₁	C₄	1	1	yes	characteristic
5	C₂×C₅×C₈×C₈₁	C₄	1	1	yes	characteristic
6	C₂×C₅×C₈×C₈₁	C₂²	1	1	yes	characteristic
7	C₅×C₁₆×C₈₁	C₄	1	4	yes	direct product factor
8	C₄×C₁₆×C₈₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₄×C₅×C₈×C₂₇	C₂×C₃	1	1	yes	characteristic
10	C₂×C₅×C₁₆×C₂₇	C₂×C₃	1	2	yes	—
11	C₂×C₄×C₅×C₈₁	C₈	1	2	yes	—
12	C₂×C₄×C₅×C₈₁	C₂×C₄	1	1	yes	characteristic
13	C₅×C₈×C₈₁	C₈	1	2	yes	—
14	C₅×C₈×C₈₁	C₂×C₄	1	2	yes	—
15	C₄×C₅×C₉×C₁₆	C₉	1	1	yes	characteristic
16	C₄×C₈×C₈₁	C₂×C₅	1	1	yes	characteristic
17	C₂×C₁₆×C₈₁	C₂×C₅	1	2	yes	—
18	C₄²×C₅×C₂₇	C₃×C₄	1	1	yes	characteristic
19	C₂×C₅×C₈×C₂₇	C₃×C₄	1	1	yes	characteristic
20	C₂×C₅×C₈×C₂₇	C₂²×C₃	1	1	yes	characteristic
21	C₅×C₁₆×C₂₇	C₃×C₄	1	4	yes	—
22	C₄×C₁₆×C₂₇	C₃×C₅	1	1	yes	characteristic
23	C₂²×C₅×C₈₁	C₂×C₈	1	1	yes	characteristic
24	C₄×C₅×C₈₁	C₁₆	1	4	yes	direct product factor
25	C₄×C₅×C₈₁	C₄²	1	1	yes	characteristic
26	C₄×C₅×C₈₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₅×C₉×C₁₆	C₂×C₉	1	2	yes	—
28	C₄×C₅×C₈×C₉	C₂×C₉	1	1	yes	characteristic
29	C₂×C₈×C₈₁	C₄×C₅	1	1	yes	characteristic
30	C₂×C₈×C₈₁	C₂²×C₅	1	1	yes	characteristic
31	C₄²×C₈₁	C₄×C₅	1	1	yes	characteristic
32	C₁₆×C₈₁	C₄×C₅	1	4	yes	direct product factor
33	C₂×C₄×C₅×C₂₇	C₃×C₈	1	2	yes	—
34	C₂×C₄×C₅×C₂₇	C₂×C₃×C₄	1	1	yes	characteristic
35	C₅×C₈×C₂₇	C₃×C₈	1	2	yes	—
36	C₅×C₈×C₂₇	C₂×C₃×C₄	1	2	yes	—
37	C₃×C₄×C₅×C₁₆	C₂₇	1	1	yes	characteristic
38	C₂×C₁₆×C₂₇	C₂×C₃×C₅	1	2	yes	—
39	C₄×C₈×C₂₇	C₂×C₃×C₅	1	1	yes	characteristic
40	C₂×C₅×C₈₁	C₄×C₈	1	1	yes	characteristic
41	C₂×C₅×C₈₁	C₂×C₁₆	1	2	yes	—
42	C₂×C₅×C₈×C₉	C₄×C₉	1	1	yes	characteristic
43	C₂×C₅×C₈×C₉	C₂²×C₉	1	1	yes	characteristic
44	C₄²×C₅×C₉	C₄×C₉	1	1	yes	characteristic
45	C₅×C₉×C₁₆	C₄×C₉	1	4	yes	—
46	C₂×C₄×C₈₁	C₅×C₈	1	2	yes	—
47	C₂×C₄×C₈₁	C₂×C₄×C₅	1	1	yes	characteristic
48	C₈×C₈₁	C₅×C₈	1	2	yes	—
49	C₈×C₈₁	C₂×C₄×C₅	1	2	yes	—
50	C₄×C₉×C₁₆	C₅×C₉	1	1	yes	characteristic
51	C₂²×C₅×C₂₇	C₂×C₃×C₈	1	1	yes	characteristic
52	C₄×C₅×C₂₇	C₃×C₁₆	1	4	yes	—
53	C₄×C₅×C₂₇	C₃×C₄²	1	1	yes	characteristic
54	C₄×C₅×C₂₇	C₂×C₃×C₈	1	1	yes	characteristic
55	C₂×C₃×C₅×C₁₆	C₂×C₂₇	1	2	yes	—
56	C₃×C₄×C₅×C₈	C₂×C₂₇	1	1	yes	characteristic
57	C₂×C₈×C₂₇	C₃×C₄×C₅	1	1	yes	characteristic
58	C₂×C₈×C₂₇	C₂²×C₃×C₅	1	1	yes	characteristic
59	C₄²×C₂₇	C₃×C₄×C₅	1	1	yes	characteristic
60	C₁₆×C₂₇	C₃×C₄×C₅	1	4	yes	—
61	C₅×C₈₁	C₄×C₁₆	1	1	yes	characteristic, direct product factor, Hall
62	C₂×C₄×C₅×C₉	C₈×C₉	1	2	yes	—
63	C₂×C₄×C₅×C₉	C₂×C₄×C₉	1	1	yes	characteristic
64	C₅×C₈×C₉	C₈×C₉	1	2	yes	—
65	C₅×C₈×C₉	C₂×C₄×C₉	1	2	yes	—
66	C₂²×C₈₁	C₂×C₅×C₈	1	1	yes	characteristic
67	C₄×C₈₁	C₅×C₁₆	1	4	yes	direct product factor
68	C₄×C₈₁	C₄²×C₅	1	1	yes	characteristic
69	C₄×C₈₁	C₂×C₅×C₈	1	1	yes	characteristic
70	C₄×C₅×C₁₆	C₈₁	1	1	yes	characteristic, direct product factor, Hall
71	C₂×C₉×C₁₆	C₂×C₅×C₉	1	2	yes	—
72	C₄×C₈×C₉	C₂×C₅×C₉	1	1	yes	characteristic
73	C₂×C₅×C₂₇	C₃×C₄×C₈	1	1	yes	characteristic
74	C₂×C₅×C₂₇	C₂×C₃×C₁₆	1	2	yes	—
75	C₂×C₃×C₅×C₈	C₄×C₂₇	1	1	yes	characteristic
76	C₂×C₃×C₅×C₈	C₂²×C₂₇	1	1	yes	characteristic
77	C₃×C₄²×C₅	C₄×C₂₇	1	1	yes	characteristic
78	C₃×C₅×C₁₆	C₄×C₂₇	1	4	yes	—
79	C₂×C₄×C₂₇	C₃×C₅×C₈	1	2	yes	—
80	C₂×C₄×C₂₇	C₂×C₃×C₄×C₅	1	1	yes	characteristic
81	C₈×C₂₇	C₃×C₅×C₈	1	2	yes	—
82	C₈×C₂₇	C₂×C₃×C₄×C₅	1	2	yes	—
83	C₃×C₄×C₁₆	C₅×C₂₇	1	1	yes	characteristic
84	C₂²×C₅×C₉	C₂×C₈×C₉	1	1	yes	characteristic
85	C₄×C₅×C₉	C₉×C₁₆	1	4	yes	—
86	C₄×C₅×C₉	C₄²×C₉	1	1	yes	characteristic
87	C₄×C₅×C₉	C₂×C₈×C₉	1	1	yes	characteristic
88	C₂×C₈₁	C₄×C₅×C₈	1	1	yes	characteristic
89	C₂×C₈₁	C₂×C₅×C₁₆	1	2	yes	—
90	C₂×C₅×C₁₆	C₂×C₈₁	1	2	yes	—
91	C₄×C₅×C₈	C₂×C₈₁	1	1	yes	characteristic
92	C₂×C₈×C₉	C₄×C₅×C₉	1	1	yes	characteristic
93	C₂×C₈×C₉	C₂²×C₅×C₉	1	1	yes	characteristic
94	C₄²×C₉	C₄×C₅×C₉	1	1	yes	characteristic
95	C₉×C₁₆	C₄×C₅×C₉	1	4	yes	—
96	C₅×C₂₇	C₃×C₄×C₁₆	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	C₈×C₂₇	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂×C₄×C₂₇	1	1	yes	characteristic
99	C₃×C₅×C₈	C₈×C₂₇	1	2	yes	—
100	C₃×C₅×C₈	C₂×C₄×C₂₇	1	2	yes	—
101	C₂²×C₂₇	C₂×C₃×C₅×C₈	1	1	yes	characteristic
102	C₄×C₂₇	C₃×C₅×C₁₆	1	4	yes	—
103	C₄×C₂₇	C₃×C₄²×C₅	1	1	yes	characteristic
104	C₄×C₂₇	C₂×C₃×C₅×C₈	1	1	yes	characteristic
105	C₂×C₃×C₁₆	C₂×C₅×C₂₇	1	2	yes	—
106	C₃×C₄×C₈	C₂×C₅×C₂₇	1	1	yes	characteristic
107	C₂×C₅×C₉	C₄×C₈×C₉	1	1	yes	characteristic
108	C₂×C₅×C₉	C₂×C₉×C₁₆	1	2	yes	—
109	C₈₁	C₄×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
110	C₂×C₅×C₈	C₄×C₈₁	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂²×C₈₁	1	1	yes	characteristic
112	C₄²×C₅	C₄×C₈₁	1	1	yes	characteristic
113	C₅×C₁₆	C₄×C₈₁	1	4	yes	direct product factor
114	C₂×C₄×C₉	C₅×C₈×C₉	1	2	yes	—
115	C₂×C₄×C₉	C₂×C₄×C₅×C₉	1	1	yes	characteristic
116	C₈×C₉	C₅×C₈×C₉	1	2	yes	—
117	C₈×C₉	C₂×C₄×C₅×C₉	1	2	yes	—
118	C₄×C₁₆	C₅×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×C₃×C₅	C₂×C₈×C₂₇	1	1	yes	characteristic
120	C₃×C₄×C₅	C₁₆×C₂₇	1	4	yes	—
121	C₃×C₄×C₅	C₄²×C₂₇	1	1	yes	characteristic
122	C₃×C₄×C₅	C₂×C₈×C₂₇	1	1	yes	characteristic
123	C₂×C₂₇	C₃×C₄×C₅×C₈	1	1	yes	characteristic
124	C₂×C₂₇	C₂×C₃×C₅×C₁₆	1	2	yes	—
125	C₂×C₃×C₈	C₄×C₅×C₂₇	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂²×C₅×C₂₇	1	1	yes	characteristic
127	C₃×C₄²	C₄×C₅×C₂₇	1	1	yes	characteristic
128	C₃×C₁₆	C₄×C₅×C₂₇	1	4	yes	—
129	C₅×C₉	C₄×C₉×C₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅	C₈×C₈₁	1	2	yes	—
131	C₂×C₄×C₅	C₂×C₄×C₈₁	1	1	yes	characteristic
132	C₅×C₈	C₈×C₈₁	1	2	yes	—
133	C₅×C₈	C₂×C₄×C₈₁	1	2	yes	—
134	C₂²×C₉	C₂×C₅×C₈×C₉	1	1	yes	characteristic
135	C₄×C₉	C₅×C₉×C₁₆	1	4	yes	—
136	C₄×C₉	C₄²×C₅×C₉	1	1	yes	characteristic
137	C₄×C₉	C₂×C₅×C₈×C₉	1	1	yes	characteristic
138	C₂×C₁₆	C₂×C₅×C₈₁	1	2	yes	—
139	C₄×C₈	C₂×C₅×C₈₁	1	1	yes	characteristic
140	C₂×C₃×C₅	C₄×C₈×C₂₇	1	1	yes	characteristic
141	C₂×C₃×C₅	C₂×C₁₆×C₂₇	1	2	yes	—
142	C₂₇	C₃×C₄×C₅×C₁₆	1	1	yes	characteristic
143	C₂×C₃×C₄	C₅×C₈×C₂₇	1	2	yes	—
144	C₂×C₃×C₄	C₂×C₄×C₅×C₂₇	1	1	yes	characteristic
145	C₃×C₈	C₅×C₈×C₂₇	1	2	yes	—
146	C₃×C₈	C₂×C₄×C₅×C₂₇	1	2	yes	—
147	C₂²×C₅	C₂×C₈×C₈₁	1	1	yes	characteristic
148	C₄×C₅	C₁₆×C₈₁	1	4	yes	direct product factor
149	C₄×C₅	C₄²×C₈₁	1	1	yes	characteristic
150	C₄×C₅	C₂×C₈×C₈₁	1	1	yes	characteristic
151	C₂×C₉	C₄×C₅×C₈×C₉	1	1	yes	characteristic
152	C₂×C₉	C₂×C₅×C₉×C₁₆	1	2	yes	—
153	C₂×C₈	C₄×C₅×C₈₁	1	1	yes	characteristic
154	C₂×C₈	C₂²×C₅×C₈₁	1	1	yes	characteristic
155	C₄²	C₄×C₅×C₈₁	1	1	yes	characteristic
156	C₁₆	C₄×C₅×C₈₁	1	4	yes	direct product factor
157	C₃×C₅	C₄×C₁₆×C₂₇	1	1	yes	characteristic
158	C₂²×C₃	C₂×C₅×C₈×C₂₇	1	1	yes	characteristic
159	C₃×C₄	C₅×C₁₆×C₂₇	1	4	yes	—
160	C₃×C₄	C₂×C₅×C₈×C₂₇	1	1	yes	characteristic
161	C₃×C₄	C₄²×C₅×C₂₇	1	1	yes	characteristic
162	C₂×C₅	C₂×C₁₆×C₈₁	1	2	yes	—
163	C₂×C₅	C₄×C₈×C₈₁	1	1	yes	characteristic
164	C₉	C₄×C₅×C₉×C₁₆	1	1	yes	characteristic
165	C₂×C₄	C₅×C₈×C₈₁	1	2	yes	—
166	C₂×C₄	C₂×C₄×C₅×C₈₁	1	1	yes	characteristic
167	C₈	C₅×C₈×C₈₁	1	2	yes	—
168	C₈	C₂×C₄×C₅×C₈₁	1	2	yes	—
169	C₂×C₃	C₂×C₅×C₁₆×C₂₇	1	2	yes	—
170	C₂×C₃	C₄×C₅×C₈×C₂₇	1	1	yes	characteristic
171	C₅	C₄×C₁₆×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²	C₂×C₅×C₈×C₈₁	1	1	yes	characteristic
173	C₄	C₅×C₁₆×C₈₁	1	4	yes	direct product factor
174	C₄	C₂×C₅×C₈×C₈₁	1	1	yes	characteristic
175	C₄	C₄²×C₅×C₈₁	1	1	yes	characteristic
176	C₃	C₄×C₅×C₁₆×C₂₇	1	1	yes	characteristic
177	C₂	C₂×C₅×C₁₆×C₈₁	1	2	yes	—
178	C₂	C₄×C₅×C₈×C₈₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈×C₂₇	C₂²×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₂×C₈×C₂₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₅×C₁₆×C₈₁	C₄	direct (N × H)
8. C₄×C₁₆×C₈₁	C₅	direct (N × H)
24. C₄×C₅×C₈₁	C₁₆	direct (N × H)
32. C₁₆×C₈₁	C₄×C₅	direct (N × H)
61. C₅×C₈₁	C₄×C₁₆	direct (N × H)
67. C₄×C₈₁	C₅×C₁₆	direct (N × H)
70. C₄×C₅×C₁₆	C₈₁	direct (N × H)
109. C₈₁	C₄×C₅×C₁₆	direct (N × H)
113. C₅×C₁₆	C₄×C₈₁	direct (N × H)
118. C₄×C₁₆	C₅×C₈₁	direct (N × H)
148. C₄×C₅	C₁₆×C₈₁	direct (N × H)
156. C₁₆	C₄×C₅×C₈₁	direct (N × H)
171. C₅	C₄×C₁₆×C₈₁	direct (N × H)
173. C₄	C₅×C₁₆×C₈₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×C₈×C₈₁	2
C₂×C₅×C₁₆×C₈₁	2
C₄×C₅×C₁₆×C₂₇	3
C₄×C₁₆×C₈₁	5

Subgroups — C4×C5×C16×C81 #72203071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₁₆×C₈₁ #72203071