Subgroups — C₄×C₅²×C₁₆² #72202808

Order 25600.

7272 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×C₅²×C₈×C₁₆	C₂	1	3	yes	maximal
2	C₂×C₅²×C₁₆²	C₂	1	4	yes	maximal
3	C₄×C₅²×C₈²	C₂²	1	1	yes	characteristic
4	C₄²×C₅²×C₁₆	C₄	1	6	yes	—
5	C₂×C₅²×C₈×C₁₆	C₄	1	6	yes	—
6	C₂×C₅²×C₈×C₁₆	C₂²	1	6	yes	—
7	C₅²×C₁₆²	C₄	1	16	yes	direct product factor
8	C₄×C₅×C₁₆²	C₅	1	6	yes	maximal, direct product factor
9	C₄²×C₅²×C₈	C₂×C₄	1	3	yes	—
10	C₂×C₅²×C₈²	C₂×C₄	1	3	yes	—
11	C₂×C₅²×C₈²	C₂³	1	1	yes	characteristic
12	C₂×C₄×C₅²×C₁₆	C₈	1	24	yes	—
13	C₂×C₄×C₅²×C₁₆	C₂×C₄	1	12	yes	—
14	C₅²×C₈×C₁₆	C₈	1	24	yes	—
15	C₅²×C₈×C₁₆	C₂×C₄	1	24	yes	—
16	C₄×C₅×C₈×C₁₆	C₂×C₅	1	18	yes	—
17	C₂×C₅×C₁₆²	C₂×C₅	1	24	yes	—
18	C₄×C₁₆²	C₅²	1	1	yes	characteristic, direct product factor, Sylow
19	C₂²×C₅²×C₁₆	C₂×C₈	1	24	yes	—
20	C₂×C₄×C₅²×C₈	C₄²	1	3	yes	—
21	C₂×C₄×C₅²×C₈	C₂×C₈	1	12	yes	—
22	C₂×C₄×C₅²×C₈	C₂²×C₄	1	3	yes	—
23	C₄³×C₅²	C₄²	1	1	yes	characteristic
24	C₄×C₅²×C₁₆	C₁₆	1	96	yes	direct product factor
25	C₄×C₅²×C₁₆	C₄²	1	24	yes	—
26	C₄×C₅²×C₁₆	C₂×C₈	1	24	yes	—
27	C₅²×C₈²	C₂×C₈	1	12	yes	—
28	C₅²×C₈²	C₂²×C₄	1	4	yes	—
29	C₄²×C₅×C₁₆	C₄×C₅	1	36	yes	—
30	C₂×C₅×C₈×C₁₆	C₄×C₅	1	36	yes	—
31	C₂×C₅×C₈×C₁₆	C₂²×C₅	1	36	yes	—
32	C₄×C₅×C₈²	C₂²×C₅	1	6	yes	—
33	C₅×C₁₆²	C₄×C₅	1	96	yes	direct product factor
34	C₂²×C₅²×C₈	C₄×C₈	1	6	yes	—
35	C₂²×C₅²×C₈	C₂²×C₈	1	6	yes	—
36	C₂×C₄²×C₅²	C₄×C₈	1	6	yes	—
37	C₂×C₄²×C₅²	C₂×C₄²	1	1	yes	characteristic
38	C₂×C₅²×C₁₆	C₄×C₈	1	48	yes	—
39	C₂×C₅²×C₁₆	C₂×C₁₆	1	96	yes	—
40	C₄×C₅²×C₈	C₄×C₈	1	12	yes	—
41	C₄×C₅²×C₈	C₂×C₁₆	1	48	yes	—
42	C₄×C₅²×C₈	C₂×C₄²	1	6	yes	—
43	C₄×C₅²×C₈	C₂²×C₈	1	6	yes	—
44	C₂×C₄×C₅×C₁₆	C₅×C₈	1	144	yes	—
45	C₂×C₄×C₅×C₁₆	C₂×C₄×C₅	1	72	yes	—
46	C₄²×C₅×C₈	C₂×C₄×C₅	1	18	yes	—
47	C₂×C₅×C₈²	C₂×C₄×C₅	1	18	yes	—
48	C₂×C₅×C₈²	C₂³×C₅	1	6	yes	—
49	C₅×C₈×C₁₆	C₅×C₈	1	144	yes	—
50	C₅×C₈×C₁₆	C₂×C₄×C₅	1	144	yes	—
51	C₄×C₈×C₁₆	C₂×C₅²	1	3	yes	—
52	C₂×C₁₆²	C₂×C₅²	1	4	yes	—
53	C₂²×C₄×C₅²	C₈²	1	4	yes	—
54	C₂²×C₄×C₅²	C₂×C₄×C₈	1	3	yes	—
55	C₂×C₅²×C₈	C₈²	1	12	yes	—
56	C₂×C₅²×C₈	C₄×C₁₆	1	24	yes	—
57	C₂×C₅²×C₈	C₂×C₄×C₈	1	12	yes	—
58	C₂×C₅²×C₈	C₂²×C₁₆	1	24	yes	—
59	C₄²×C₅²	C₄×C₁₆	1	24	yes	—
60	C₄²×C₅²	C₄³	1	1	yes	characteristic
61	C₄²×C₅²	C₂×C₄×C₈	1	3	yes	—
62	C₅²×C₁₆	C₄×C₁₆	1	96	yes	direct product factor
63	C₂²×C₅×C₁₆	C₂×C₅×C₈	1	144	yes	—
64	C₂×C₄×C₅×C₈	C₄²×C₅	1	18	yes	—
65	C₂×C₄×C₅×C₈	C₂×C₅×C₈	1	72	yes	—
66	C₂×C₄×C₅×C₈	C₂²×C₄×C₅	1	18	yes	—
67	C₄³×C₅	C₄²×C₅	1	6	yes	—
68	C₄×C₅×C₁₆	C₅×C₁₆	1	576	yes	direct product factor
69	C₄×C₅×C₁₆	C₄²×C₅	1	144	yes	—
70	C₄×C₅×C₁₆	C₂×C₅×C₈	1	144	yes	—
71	C₅×C₈²	C₂×C₅×C₈	1	72	yes	—
72	C₅×C₈²	C₂²×C₄×C₅	1	24	yes	—
73	C₄²×C₁₆	C₄×C₅²	1	6	yes	—
74	C₂×C₈×C₁₆	C₄×C₅²	1	6	yes	—
75	C₂×C₈×C₁₆	C₂²×C₅²	1	6	yes	—
76	C₄×C₈²	C₂²×C₅²	1	1	yes	characteristic
77	C₁₆²	C₄×C₅²	1	16	yes	direct product factor
78	C₂³×C₅²	C₂×C₈²	1	1	yes	characteristic
79	C₂×C₄×C₅²	C₈×C₁₆	1	24	yes	—
80	C₂×C₄×C₅²	C₂×C₈²	1	3	yes	—
81	C₂×C₄×C₅²	C₄²×C₈	1	3	yes	—
82	C₂×C₄×C₅²	C₂×C₄×C₁₆	1	12	yes	—
83	C₅²×C₈	C₈×C₁₆	1	24	yes	—
84	C₅²×C₈	C₂×C₄×C₁₆	1	24	yes	—
85	C₂²×C₅×C₈	C₄×C₅×C₈	1	36	yes	—
86	C₂²×C₅×C₈	C₂²×C₅×C₈	1	36	yes	—
87	C₂×C₄²×C₅	C₄×C₅×C₈	1	36	yes	—
88	C₂×C₄²×C₅	C₂×C₄²×C₅	1	6	yes	—
89	C₂×C₅×C₁₆	C₄×C₅×C₈	1	288	yes	—
90	C₂×C₅×C₁₆	C₂×C₅×C₁₆	1	576	yes	—
91	C₄×C₅×C₈	C₄×C₅×C₈	1	72	yes	—
92	C₄×C₅×C₈	C₂×C₅×C₁₆	1	288	yes	—
93	C₄×C₅×C₈	C₂×C₄²×C₅	1	36	yes	—
94	C₄×C₅×C₈	C₂²×C₅×C₈	1	36	yes	—
95	C₂×C₄×C₁₆	C₅²×C₈	1	24	yes	—
96	C₂×C₄×C₁₆	C₂×C₄×C₅²	1	12	yes	—
97	C₄²×C₈	C₂×C₄×C₅²	1	3	yes	—
98	C₂×C₈²	C₂×C₄×C₅²	1	3	yes	—
99	C₂×C₈²	C₂³×C₅²	1	1	yes	characteristic
100	C₈×C₁₆	C₅²×C₈	1	24	yes	—
101	C₈×C₁₆	C₂×C₄×C₅²	1	24	yes	—
102	C₂²×C₅²	C₄×C₈²	1	1	yes	characteristic
103	C₂²×C₅²	C₂×C₈×C₁₆	1	6	yes	—
104	C₄×C₅²	C₁₆²	1	16	yes	direct product factor
105	C₄×C₅²	C₂×C₈×C₁₆	1	6	yes	—
106	C₄×C₅²	C₄²×C₁₆	1	6	yes	—
107	C₂²×C₄×C₅	C₅×C₈²	1	24	yes	—
108	C₂²×C₄×C₅	C₂×C₄×C₅×C₈	1	18	yes	—
109	C₂×C₅×C₈	C₅×C₈²	1	72	yes	—
110	C₂×C₅×C₈	C₄×C₅×C₁₆	1	144	yes	—
111	C₂×C₅×C₈	C₂×C₄×C₅×C₈	1	72	yes	—
112	C₂×C₅×C₈	C₂²×C₅×C₁₆	1	144	yes	—
113	C₄²×C₅	C₄×C₅×C₁₆	1	144	yes	—
114	C₄²×C₅	C₄³×C₅	1	6	yes	—
115	C₄²×C₅	C₂×C₄×C₅×C₈	1	18	yes	—
116	C₅×C₁₆	C₄×C₅×C₁₆	1	576	yes	direct product factor
117	C₂²×C₁₆	C₂×C₅²×C₈	1	24	yes	—
118	C₂×C₄×C₈	C₄²×C₅²	1	3	yes	—
119	C₂×C₄×C₈	C₂×C₅²×C₈	1	12	yes	—
120	C₂×C₄×C₈	C₂²×C₄×C₅²	1	3	yes	—
121	C₄³	C₄²×C₅²	1	1	yes	characteristic
122	C₄×C₁₆	C₅²×C₁₆	1	96	yes	direct product factor
123	C₄×C₁₆	C₄²×C₅²	1	24	yes	—
124	C₄×C₁₆	C₂×C₅²×C₈	1	24	yes	—
125	C₈²	C₂×C₅²×C₈	1	12	yes	—
126	C₈²	C₂²×C₄×C₅²	1	4	yes	—
127	C₂×C₅²	C₂×C₁₆²	1	4	yes	—
128	C₂×C₅²	C₄×C₈×C₁₆	1	3	yes	—
129	C₂³×C₅	C₂×C₅×C₈²	1	6	yes	—
130	C₂×C₄×C₅	C₅×C₈×C₁₆	1	144	yes	—
131	C₂×C₄×C₅	C₂×C₅×C₈²	1	18	yes	—
132	C₂×C₄×C₅	C₄²×C₅×C₈	1	18	yes	—
133	C₂×C₄×C₅	C₂×C₄×C₅×C₁₆	1	72	yes	—
134	C₅×C₈	C₅×C₈×C₁₆	1	144	yes	—
135	C₅×C₈	C₂×C₄×C₅×C₁₆	1	144	yes	—
136	C₂²×C₈	C₄×C₅²×C₈	1	6	yes	—
137	C₂²×C₈	C₂²×C₅²×C₈	1	6	yes	—
138	C₂×C₄²	C₄×C₅²×C₈	1	6	yes	—
139	C₂×C₄²	C₂×C₄²×C₅²	1	1	yes	characteristic
140	C₂×C₁₆	C₄×C₅²×C₈	1	48	yes	—
141	C₂×C₁₆	C₂×C₅²×C₁₆	1	96	yes	—
142	C₄×C₈	C₄×C₅²×C₈	1	12	yes	—
143	C₄×C₈	C₂×C₅²×C₁₆	1	48	yes	—
144	C₄×C₈	C₂×C₄²×C₅²	1	6	yes	—
145	C₄×C₈	C₂²×C₅²×C₈	1	6	yes	—
146	C₅²	C₄×C₁₆²	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×C₅	C₄×C₅×C₈²	1	6	yes	—
148	C₂²×C₅	C₂×C₅×C₈×C₁₆	1	36	yes	—
149	C₄×C₅	C₅×C₁₆²	1	96	yes	direct product factor
150	C₄×C₅	C₂×C₅×C₈×C₁₆	1	36	yes	—
151	C₄×C₅	C₄²×C₅×C₁₆	1	36	yes	—
152	C₂²×C₄	C₅²×C₈²	1	4	yes	—
153	C₂²×C₄	C₂×C₄×C₅²×C₈	1	3	yes	—
154	C₂×C₈	C₅²×C₈²	1	12	yes	—
155	C₂×C₈	C₄×C₅²×C₁₆	1	24	yes	—
156	C₂×C₈	C₂×C₄×C₅²×C₈	1	12	yes	—
157	C₂×C₈	C₂²×C₅²×C₁₆	1	24	yes	—
158	C₄²	C₄×C₅²×C₁₆	1	24	yes	—
159	C₄²	C₄³×C₅²	1	1	yes	characteristic
160	C₄²	C₂×C₄×C₅²×C₈	1	3	yes	—
161	C₁₆	C₄×C₅²×C₁₆	1	96	yes	direct product factor
162	C₂×C₅	C₂×C₅×C₁₆²	1	24	yes	—
163	C₂×C₅	C₄×C₅×C₈×C₁₆	1	18	yes	—
164	C₂³	C₂×C₅²×C₈²	1	1	yes	characteristic
165	C₂×C₄	C₅²×C₈×C₁₆	1	24	yes	—
166	C₂×C₄	C₂×C₄×C₅²×C₁₆	1	12	yes	—
167	C₂×C₄	C₂×C₅²×C₈²	1	3	yes	—
168	C₂×C₄	C₄²×C₅²×C₈	1	3	yes	—
169	C₈	C₅²×C₈×C₁₆	1	24	yes	—
170	C₈	C₂×C₄×C₅²×C₁₆	1	24	yes	—
171	C₅	C₄×C₅×C₁₆²	1	6	yes	direct product factor
172	C₂²	C₂×C₅²×C₈×C₁₆	1	6	yes	—
173	C₂²	C₄×C₅²×C₈²	1	1	yes	characteristic
174	C₄	C₅²×C₁₆²	1	16	yes	direct product factor
175	C₄	C₂×C₅²×C₈×C₁₆	1	6	yes	—
176	C₄	C₄²×C₅²×C₁₆	1	6	yes	—
177	C₂	C₂×C₅²×C₁₆²	1	4	yes	—
178	C₂	C₄×C₅²×C₈×C₁₆	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈²	C₂³×C₅²
socle, prime radical	C₂³×C₅²	C₂×C₈²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₅²×C₁₆²	C₄	direct (N × H)
8. C₄×C₅×C₁₆²	C₅	direct (N × H)
18. C₄×C₁₆²	C₅²	direct (N × H)
24. C₄×C₅²×C₁₆	C₁₆	direct (N × H)
33. C₅×C₁₆²	C₄×C₅	direct (N × H)
62. C₅²×C₁₆	C₄×C₁₆	direct (N × H)
68. C₄×C₅×C₁₆	C₅×C₁₆	direct (N × H)
77. C₁₆²	C₄×C₅²	direct (N × H)
104. C₄×C₅²	C₁₆²	direct (N × H)
116. C₅×C₁₆	C₄×C₅×C₁₆	direct (N × H)
122. C₄×C₁₆	C₅²×C₁₆	direct (N × H)
146. C₅²	C₄×C₁₆²	direct (N × H)
149. C₄×C₅	C₅×C₁₆²	direct (N × H)
161. C₁₆	C₄×C₅²×C₁₆	direct (N × H)
171. C₅	C₄×C₅×C₁₆²	direct (N × H)
174. C₄	C₅²×C₁₆²	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×C₅²×C₈×C₁₆	2
C₂×C₅²×C₁₆²	2
C₄×C₅×C₁₆²	5

Subgroups — C4×C52×C162 #72202808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅²×C₁₆² #72202808