Subgroups — C₂×C₃×C₁₆³ #72201873

Order 24576.

47900 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×C₁₆²	C₂	1	7	yes	maximal
2	C₃×C₁₆³	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₁₆³	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂×C₃×C₈²×C₁₆	C₂²	1	7	yes	—
5	C₂×C₃×C₄×C₁₆²	C₄	1	28	yes	—
6	C₃×C₈×C₁₆²	C₄	1	28	yes	—
7	C₃×C₈×C₁₆²	C₂²	1	28	yes	—
8	C₂×C₈×C₁₆²	C₂×C₃	1	7	yes	—
9	C₁₆³	C₂×C₃	1	8	yes	direct product factor
10	C₂×C₃×C₈³	C₂³	1	1	yes	characteristic
11	C₂×C₃×C₄×C₈×C₁₆	C₂×C₄	1	42	yes	—
12	C₃×C₈²×C₁₆	C₂×C₄	1	42	yes	—
13	C₃×C₈²×C₁₆	C₂³	1	14	yes	—
14	C₂²×C₃×C₁₆²	C₈	1	112	yes	—
15	C₃×C₄×C₁₆²	C₈	1	112	yes	—
16	C₃×C₄×C₁₆²	C₂×C₄	1	112	yes	—
17	C₂×C₈²×C₁₆	C₂²×C₃	1	7	yes	—
18	C₂×C₄×C₁₆²	C₃×C₄	1	28	yes	—
19	C₈×C₁₆²	C₃×C₄	1	28	yes	—
20	C₈×C₁₆²	C₂²×C₃	1	28	yes	—
21	C₂×C₃×C₄×C₈²	C₂²×C₄	1	7	yes	—
22	C₃×C₈³	C₂²×C₄	1	7	yes	—
23	C₃×C₈³	C₂⁴	1	1	yes	characteristic
24	C₂×C₃×C₄²×C₁₆	C₄²	1	28	yes	—
25	C₂²×C₃×C₈×C₁₆	C₂×C₈	1	168	yes	—
26	C₃×C₄×C₈×C₁₆	C₄²	1	84	yes	—
27	C₃×C₄×C₈×C₁₆	C₂×C₈	1	168	yes	—
28	C₃×C₄×C₈×C₁₆	C₂²×C₄	1	84	yes	—
29	C₂×C₃×C₁₆²	C₁₆	1	896	yes	direct product factor
30	C₂×C₃×C₁₆²	C₂×C₈	1	448	yes	—
31	C₂×C₈³	C₂³×C₃	1	1	yes	characteristic
32	C₂×C₄×C₈×C₁₆	C₂×C₃×C₄	1	42	yes	—
33	C₈²×C₁₆	C₂×C₃×C₄	1	42	yes	—
34	C₈²×C₁₆	C₂³×C₃	1	14	yes	—
35	C₂²×C₁₆²	C₃×C₈	1	112	yes	—
36	C₄×C₁₆²	C₃×C₈	1	112	yes	—
37	C₄×C₁₆²	C₂×C₃×C₄	1	112	yes	—
38	C₂²×C₃×C₄×C₁₆	C₄×C₈	1	168	yes	—
39	C₂×C₃×C₄²×C₈	C₂×C₄²	1	7	yes	—
40	C₂²×C₃×C₈²	C₂²×C₈	1	28	yes	—
41	C₃×C₄²×C₁₆	C₄×C₈	1	168	yes	—
42	C₃×C₄²×C₁₆	C₂×C₄²	1	56	yes	—
43	C₂×C₃×C₈×C₁₆	C₄×C₈	1	336	yes	—
44	C₂×C₃×C₈×C₁₆	C₂×C₁₆	1	1344	yes	—
45	C₂×C₃×C₈×C₁₆	C₂²×C₈	1	336	yes	—
46	C₃×C₄×C₈²	C₂×C₄²	1	21	yes	—
47	C₃×C₄×C₈²	C₂²×C₈	1	28	yes	—
48	C₃×C₄×C₈²	C₂³×C₄	1	7	yes	—
49	C₃×C₁₆²	C₂×C₁₆	1	1792	yes	direct product factor
50	C₂×C₄²×C₁₆	C₃×C₄²	1	28	yes	—
51	C₂²×C₈×C₁₆	C₂×C₃×C₈	1	168	yes	—
52	C₂×C₄×C₈²	C₂²×C₃×C₄	1	7	yes	—
53	C₄×C₈×C₁₆	C₃×C₄²	1	84	yes	—
54	C₄×C₈×C₁₆	C₂×C₃×C₈	1	168	yes	—
55	C₄×C₈×C₁₆	C₂²×C₃×C₄	1	84	yes	—
56	C₂×C₁₆²	C₃×C₁₆	1	896	yes	direct product factor
57	C₂×C₁₆²	C₂×C₃×C₈	1	448	yes	—
58	C₈³	C₂²×C₃×C₄	1	7	yes	—
59	C₈³	C₂⁴×C₃	1	1	yes	characteristic
60	C₂³×C₃×C₁₆	C₈²	1	112	yes	—
61	C₂²×C₃×C₄×C₈	C₂×C₄×C₈	1	42	yes	—
62	C₂×C₃×C₄³	C₄³	1	1	yes	characteristic
63	C₂×C₃×C₄×C₁₆	C₈²	1	336	yes	—
64	C₂×C₃×C₄×C₁₆	C₄×C₁₆	1	1344	yes	—
65	C₂×C₃×C₄×C₁₆	C₂×C₄×C₈	1	336	yes	—
66	C₃×C₄²×C₈	C₄³	1	7	yes	—
67	C₃×C₄²×C₈	C₂×C₄×C₈	1	42	yes	—
68	C₃×C₄²×C₈	C₂²×C₄²	1	7	yes	—
69	C₂×C₃×C₈²	C₂×C₄×C₈	1	84	yes	—
70	C₂×C₃×C₈²	C₂²×C₁₆	1	224	yes	—
71	C₂×C₃×C₈²	C₂³×C₈	1	28	yes	—
72	C₃×C₈×C₁₆	C₄×C₁₆	1	1344	yes	—
73	C₃×C₈×C₁₆	C₂²×C₁₆	1	1344	yes	—
74	C₂²×C₄×C₁₆	C₃×C₄×C₈	1	168	yes	—
75	C₂×C₄²×C₈	C₂×C₃×C₄²	1	7	yes	—
76	C₂²×C₈²	C₂²×C₃×C₈	1	28	yes	—
77	C₄²×C₁₆	C₃×C₄×C₈	1	168	yes	—
78	C₄²×C₁₆	C₂×C₃×C₄²	1	56	yes	—
79	C₂×C₈×C₁₆	C₃×C₄×C₈	1	336	yes	—
80	C₂×C₈×C₁₆	C₂×C₃×C₁₆	1	1344	yes	—
81	C₂×C₈×C₁₆	C₂²×C₃×C₈	1	336	yes	—
82	C₄×C₈²	C₂×C₃×C₄²	1	21	yes	—
83	C₄×C₈²	C₂²×C₃×C₈	1	28	yes	—
84	C₄×C₈²	C₂³×C₃×C₄	1	7	yes	—
85	C₁₆²	C₂×C₃×C₁₆	1	1792	yes	direct product factor
86	C₂³×C₃×C₈	C₂×C₈²	1	28	yes	—
87	C₂²×C₃×C₄²	C₄²×C₈	1	7	yes	—
88	C₂²×C₃×C₁₆	C₈×C₁₆	1	1344	yes	—
89	C₂²×C₃×C₁₆	C₂×C₈²	1	224	yes	—
90	C₂×C₃×C₄×C₈	C₂×C₈²	1	84	yes	—
91	C₂×C₃×C₄×C₈	C₄²×C₈	1	42	yes	—
92	C₂×C₃×C₄×C₈	C₂×C₄×C₁₆	1	336	yes	—
93	C₂×C₃×C₄×C₈	C₂²×C₄×C₈	1	42	yes	—
94	C₃×C₄³	C₄²×C₈	1	7	yes	—
95	C₃×C₄³	C₂×C₄³	1	1	yes	characteristic
96	C₃×C₄×C₁₆	C₈×C₁₆	1	1344	yes	—
97	C₃×C₄×C₁₆	C₂×C₄×C₁₆	1	1344	yes	—
98	C₃×C₈²	C₂×C₄×C₁₆	1	336	yes	—
99	C₃×C₈²	C₂³×C₁₆	1	112	yes	—
100	C₂³×C₁₆	C₃×C₈²	1	112	yes	—
101	C₂²×C₄×C₈	C₂×C₃×C₄×C₈	1	42	yes	—
102	C₂×C₄³	C₃×C₄³	1	1	yes	characteristic
103	C₂×C₄×C₁₆	C₃×C₈²	1	336	yes	—
104	C₂×C₄×C₁₆	C₃×C₄×C₁₆	1	1344	yes	—
105	C₂×C₄×C₁₆	C₂×C₃×C₄×C₈	1	336	yes	—
106	C₄²×C₈	C₃×C₄³	1	7	yes	—
107	C₄²×C₈	C₂×C₃×C₄×C₈	1	42	yes	—
108	C₄²×C₈	C₂²×C₃×C₄²	1	7	yes	—
109	C₂×C₈²	C₂×C₃×C₄×C₈	1	84	yes	—
110	C₂×C₈²	C₂²×C₃×C₁₆	1	224	yes	—
111	C₂×C₈²	C₂³×C₃×C₈	1	28	yes	—
112	C₈×C₁₆	C₃×C₄×C₁₆	1	1344	yes	—
113	C₈×C₁₆	C₂²×C₃×C₁₆	1	1344	yes	—
114	C₂³×C₃×C₄	C₄×C₈²	1	7	yes	—
115	C₂²×C₃×C₈	C₄×C₈²	1	28	yes	—
116	C₂²×C₃×C₈	C₂×C₈×C₁₆	1	336	yes	—
117	C₂²×C₃×C₈	C₂²×C₈²	1	28	yes	—
118	C₂×C₃×C₄²	C₄×C₈²	1	21	yes	—
119	C₂×C₃×C₄²	C₄²×C₁₆	1	56	yes	—
120	C₂×C₃×C₄²	C₂×C₄²×C₈	1	7	yes	—
121	C₂×C₃×C₁₆	C₁₆²	1	1792	yes	direct product factor
122	C₂×C₃×C₁₆	C₂×C₈×C₁₆	1	1344	yes	—
123	C₃×C₄×C₈	C₂×C₈×C₁₆	1	336	yes	—
124	C₃×C₄×C₈	C₄²×C₁₆	1	168	yes	—
125	C₃×C₄×C₈	C₂²×C₄×C₁₆	1	168	yes	—
126	C₂³×C₈	C₂×C₃×C₈²	1	28	yes	—
127	C₂²×C₄²	C₃×C₄²×C₈	1	7	yes	—
128	C₂²×C₁₆	C₃×C₈×C₁₆	1	1344	yes	—
129	C₂²×C₁₆	C₂×C₃×C₈²	1	224	yes	—
130	C₂×C₄×C₈	C₂×C₃×C₈²	1	84	yes	—
131	C₂×C₄×C₈	C₃×C₄²×C₈	1	42	yes	—
132	C₂×C₄×C₈	C₂×C₃×C₄×C₁₆	1	336	yes	—
133	C₂×C₄×C₈	C₂²×C₃×C₄×C₈	1	42	yes	—
134	C₄³	C₃×C₄²×C₈	1	7	yes	—
135	C₄³	C₂×C₃×C₄³	1	1	yes	characteristic
136	C₄×C₁₆	C₃×C₈×C₁₆	1	1344	yes	—
137	C₄×C₁₆	C₂×C₃×C₄×C₁₆	1	1344	yes	—
138	C₈²	C₂×C₃×C₄×C₁₆	1	336	yes	—
139	C₈²	C₂³×C₃×C₁₆	1	112	yes	—
140	C₂⁴×C₃	C₈³	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₈³	1	7	yes	—
142	C₂²×C₃×C₄	C₄×C₈×C₁₆	1	84	yes	—
143	C₂²×C₃×C₄	C₂×C₄×C₈²	1	7	yes	—
144	C₂×C₃×C₈	C₂×C₁₆²	1	448	yes	—
145	C₂×C₃×C₈	C₄×C₈×C₁₆	1	168	yes	—
146	C₂×C₃×C₈	C₂²×C₈×C₁₆	1	168	yes	—
147	C₃×C₄²	C₄×C₈×C₁₆	1	84	yes	—
148	C₃×C₄²	C₂×C₄²×C₁₆	1	28	yes	—
149	C₃×C₁₆	C₂×C₁₆²	1	896	yes	direct product factor
150	C₂³×C₄	C₃×C₄×C₈²	1	7	yes	—
151	C₂²×C₈	C₃×C₄×C₈²	1	28	yes	—
152	C₂²×C₈	C₂×C₃×C₈×C₁₆	1	336	yes	—
153	C₂²×C₈	C₂²×C₃×C₈²	1	28	yes	—
154	C₂×C₄²	C₃×C₄×C₈²	1	21	yes	—
155	C₂×C₄²	C₃×C₄²×C₁₆	1	56	yes	—
156	C₂×C₄²	C₂×C₃×C₄²×C₈	1	7	yes	—
157	C₂×C₁₆	C₃×C₁₆²	1	1792	yes	direct product factor
158	C₂×C₁₆	C₂×C₃×C₈×C₁₆	1	1344	yes	—
159	C₄×C₈	C₂×C₃×C₈×C₁₆	1	336	yes	—
160	C₄×C₈	C₃×C₄²×C₁₆	1	168	yes	—
161	C₄×C₈	C₂²×C₃×C₄×C₁₆	1	168	yes	—
162	C₂³×C₃	C₈²×C₁₆	1	14	yes	—
163	C₂³×C₃	C₂×C₈³	1	1	yes	characteristic
164	C₂×C₃×C₄	C₄×C₁₆²	1	112	yes	—
165	C₂×C₃×C₄	C₈²×C₁₆	1	42	yes	—
166	C₂×C₃×C₄	C₂×C₄×C₈×C₁₆	1	42	yes	—
167	C₃×C₈	C₄×C₁₆²	1	112	yes	—
168	C₃×C₈	C₂²×C₁₆²	1	112	yes	—
169	C₂⁴	C₃×C₈³	1	1	yes	characteristic
170	C₂²×C₄	C₃×C₄×C₈×C₁₆	1	84	yes	—
171	C₂²×C₄	C₃×C₈³	1	7	yes	—
172	C₂²×C₄	C₂×C₃×C₄×C₈²	1	7	yes	—
173	C₂×C₈	C₂×C₃×C₁₆²	1	448	yes	—
174	C₂×C₈	C₃×C₄×C₈×C₁₆	1	168	yes	—
175	C₂×C₈	C₂²×C₃×C₈×C₁₆	1	168	yes	—
176	C₄²	C₃×C₄×C₈×C₁₆	1	84	yes	—
177	C₄²	C₂×C₃×C₄²×C₁₆	1	28	yes	—
178	C₁₆	C₂×C₃×C₁₆²	1	896	yes	direct product factor
179	C₂²×C₃	C₈×C₁₆²	1	28	yes	—
180	C₂²×C₃	C₂×C₈²×C₁₆	1	7	yes	—
181	C₃×C₄	C₈×C₁₆²	1	28	yes	—
182	C₃×C₄	C₂×C₄×C₁₆²	1	28	yes	—
183	C₂³	C₃×C₈²×C₁₆	1	14	yes	—
184	C₂³	C₂×C₃×C₈³	1	1	yes	characteristic
185	C₂×C₄	C₃×C₄×C₁₆²	1	112	yes	—
186	C₂×C₄	C₃×C₈²×C₁₆	1	42	yes	—
187	C₂×C₄	C₂×C₃×C₄×C₈×C₁₆	1	42	yes	—
188	C₈	C₃×C₄×C₁₆²	1	112	yes	—
189	C₈	C₂²×C₃×C₁₆²	1	112	yes	—
190	C₂×C₃	C₁₆³	1	8	yes	direct product factor
191	C₂×C₃	C₂×C₈×C₁₆²	1	7	yes	—
192	C₂²	C₃×C₈×C₁₆²	1	28	yes	—
193	C₂²	C₂×C₃×C₈²×C₁₆	1	7	yes	—
194	C₄	C₃×C₈×C₁₆²	1	28	yes	—
195	C₄	C₂×C₃×C₄×C₁₆²	1	28	yes	—
196	C₃	C₂×C₁₆³	1	1	yes	characteristic, direct product factor, Sylow
197	C₂	C₃×C₁₆³	1	8	yes	direct product factor
198	C₂	C₂×C₃×C₈×C₁₆²	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈³	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₈³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₁₆³	C₂	direct (N × H)
3. C₂×C₁₆³	C₃	direct (N × H)
9. C₁₆³	C₂×C₃	direct (N × H)
29. C₂×C₃×C₁₆²	C₁₆	direct (N × H)
49. C₃×C₁₆²	C₂×C₁₆	direct (N × H)
56. C₂×C₁₆²	C₃×C₁₆	direct (N × H)
85. C₁₆²	C₂×C₃×C₁₆	direct (N × H)
121. C₂×C₃×C₁₆	C₁₆²	direct (N × H)
149. C₃×C₁₆	C₂×C₁₆²	direct (N × H)
157. C₂×C₁₆	C₃×C₁₆²	direct (N × H)
178. C₁₆	C₂×C₃×C₁₆²	direct (N × H)
190. C₂×C₃	C₁₆³	direct (N × H)
196. C₃	C₂×C₁₆³	direct (N × H)
197. C₂	C₃×C₁₆³	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×C₁₆²	2
C₃×C₁₆³	2
C₂×C₁₆³	3

Subgroups — C2×C3×C163 #72201873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₁₆³ #72201873