Subgroups — C₄³×C₈×C₁₆ #72187111

Order 8192.

315534 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄³×C₈²	C₂	1	1	yes	maximal, characteristic
2	C₄⁴×C₁₆	C₂	1	2	yes	maximal
3	C₂×C₄²×C₈×C₁₆	C₂	1	28	yes	maximal
4	C₄⁴×C₈	C₄	1	2	yes	—
5	C₄⁴×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₈²	C₄	1	14	yes	—
7	C₂×C₄²×C₈²	C₂²	1	14	yes	—
8	C₂×C₄³×C₁₆	C₄	1	32	yes	—
9	C₂×C₄³×C₁₆	C₂²	1	28	yes	—
10	C₂²×C₄×C₈×C₁₆	C₂²	1	112	yes	—
11	C₄²×C₈×C₁₆	C₄	1	448	yes	direct product factor
12	C₄⁵	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄³×C₈	C₈	1	32	yes	—
14	C₂×C₄³×C₈	C₂×C₄	1	7	yes	—
15	C₂×C₄³×C₈	C₂×C₄	1	16	yes	—
16	C₂×C₄³×C₈	C₂×C₄	1	28	yes	—
17	C₂×C₄³×C₈	C₂³	1	7	yes	—
18	C₂²×C₄×C₈²	C₂×C₄	1	84	yes	—
19	C₂²×C₄×C₈²	C₂³	1	28	yes	—
20	C₄²×C₈²	C₈	1	224	yes	—
21	C₄²×C₈²	C₂×C₄	1	224	yes	—
22	C₂²×C₄²×C₁₆	C₂×C₄	1	224	yes	—
23	C₂²×C₄²×C₁₆	C₂³	1	56	yes	—
24	C₂³×C₈×C₁₆	C₂³	1	64	yes	—
25	C₄³×C₁₆	C₈	1	512	yes	direct product factor
26	C₄³×C₁₆	C₂×C₄	1	448	yes	—
27	C₂×C₄×C₈×C₁₆	C₂×C₄	1	2688	yes	—
28	C₂³×C₄×C₁₆	C₂²×C₄	1	224	yes	—
29	C₂³×C₄×C₁₆	C₂⁴	1	16	yes	—
30	C₂²×C₄²×C₈	C₄²	1	56	yes	—
31	C₂²×C₄²×C₈	C₂×C₈	1	224	yes	—
32	C₂²×C₄²×C₈	C₂²×C₄	1	21	yes	—
33	C₂²×C₄²×C₈	C₂²×C₄	1	56	yes	—
34	C₂²×C₄²×C₈	C₂²×C₄	1	56	yes	—
35	C₂²×C₄²×C₈	C₂⁴	1	7	yes	—
36	C₂³×C₈²	C₂²×C₄	1	56	yes	—
37	C₂³×C₈²	C₂⁴	1	8	yes	—
38	C₂×C₄⁴	C₄²	1	8	yes	—
39	C₂×C₄⁴	C₂×C₈	1	16	yes	—
40	C₂×C₄⁴	C₂²×C₄	1	7	yes	—
41	C₂×C₄²×C₁₆	C₄²	1	1792	yes	—
42	C₂×C₄²×C₁₆	C₂×C₈	1	3584	yes	—
43	C₂×C₄²×C₁₆	C₂²×C₄	1	1344	yes	—
44	C₂²×C₈×C₁₆	C₂²×C₄	1	1792	yes	—
45	C₄³×C₈	C₁₆	1	512	yes	direct product factor
46	C₄³×C₈	C₄²	1	224	yes	—
47	C₄³×C₈	C₂×C₈	1	256	yes	—
48	C₄³×C₈	C₂×C₈	1	112	yes	—
49	C₄³×C₈	C₂×C₈	1	224	yes	—
50	C₄³×C₈	C₂²×C₄	1	112	yes	—
51	C₂×C₄×C₈²	C₄²	1	672	yes	—
52	C₂×C₄×C₈²	C₂×C₈	1	1344	yes	—
53	C₂×C₄×C₈²	C₂²×C₄	1	672	yes	—
54	C₄×C₈×C₁₆	C₄²	1	7168	yes	direct product factor
55	C₂⁴×C₁₆	C₂³×C₄	1	32	yes	—
56	C₂³×C₄×C₈	C₂×C₄²	1	84	yes	—
57	C₂³×C₄×C₈	C₂²×C₈	1	224	yes	—
58	C₂³×C₄×C₈	C₂³×C₄	1	7	yes	—
59	C₂³×C₄×C₈	C₂³×C₄	1	16	yes	—
60	C₂³×C₄×C₈	C₂³×C₄	1	28	yes	—
61	C₂³×C₄×C₈	C₂⁵	1	1	yes	characteristic
62	C₂²×C₄³	C₄×C₈	1	64	yes	—
63	C₂²×C₄³	C₂×C₄²	1	28	yes	—
64	C₂²×C₄³	C₂²×C₈	1	56	yes	—
65	C₂²×C₄³	C₂³×C₄	1	7	yes	—
66	C₂²×C₄×C₁₆	C₂×C₄²	1	2688	yes	—
67	C₂²×C₄×C₁₆	C₂²×C₈	1	3584	yes	—
68	C₂²×C₄×C₁₆	C₂³×C₄	1	448	yes	—
69	C₂×C₄²×C₈	C₄×C₈	1	1792	yes	—
70	C₂×C₄²×C₈	C₄×C₈	1	896	yes	—
71	C₂×C₄²×C₈	C₄×C₈	1	448	yes	—
72	C₂×C₄²×C₈	C₂×C₁₆	1	3584	yes	—
73	C₂×C₄²×C₈	C₂×C₄²	1	672	yes	—
74	C₂×C₄²×C₈	C₂×C₄²	1	168	yes	—
75	C₂×C₄²×C₈	C₂×C₄²	1	448	yes	—
76	C₂×C₄²×C₈	C₂²×C₈	1	336	yes	—
77	C₂×C₄²×C₈	C₂²×C₈	1	896	yes	—
78	C₂×C₄²×C₈	C₂²×C₈	1	672	yes	—
79	C₂×C₄²×C₈	C₂³×C₄	1	168	yes	—
80	C₂²×C₈²	C₂×C₄²	1	672	yes	—
81	C₂²×C₈²	C₂²×C₈	1	896	yes	—
82	C₂²×C₈²	C₂³×C₄	1	224	yes	—
83	C₄⁴	C₄×C₈	1	128	yes	—
84	C₄⁴	C₂×C₁₆	1	256	yes	—
85	C₄⁴	C₂×C₄²	1	56	yes	—
86	C₄⁴	C₂²×C₈	1	56	yes	—
87	C₄²×C₁₆	C₄×C₈	1	14336	yes	direct product factor
88	C₄²×C₁₆	C₂×C₄²	1	3584	yes	—
89	C₂×C₈×C₁₆	C₂×C₄²	1	7168	yes	—
90	C₄×C₈²	C₄×C₈	1	5376	yes	—
91	C₄×C₈²	C₂×C₄²	1	1792	yes	—
92	C₂⁴×C₈	C₂²×C₄²	1	14	yes	—
93	C₂⁴×C₈	C₂³×C₈	1	32	yes	—
94	C₂⁴×C₈	C₂⁴×C₄	1	2	yes	—
95	C₂³×C₄²	C₂×C₄×C₈	1	112	yes	—
96	C₂³×C₄²	C₂²×C₄²	1	14	yes	—
97	C₂³×C₄²	C₂³×C₈	1	28	yes	—
98	C₂³×C₄²	C₂⁴×C₄	1	1	yes	characteristic
99	C₂³×C₁₆	C₂²×C₄²	1	448	yes	—
100	C₂³×C₁₆	C₂³×C₈	1	512	yes	—
101	C₂²×C₄×C₈	C₄³	1	336	yes	—
102	C₂²×C₄×C₈	C₂×C₄×C₈	1	1344	yes	—
103	C₂²×C₄×C₈	C₂×C₄×C₈	1	2688	yes	—
104	C₂²×C₄×C₈	C₂×C₄×C₈	1	672	yes	—
105	C₂²×C₄×C₈	C₂²×C₁₆	1	3584	yes	—
106	C₂²×C₄×C₈	C₂²×C₄²	1	336	yes	—
107	C₂²×C₄×C₈	C₂²×C₄²	1	84	yes	—
108	C₂²×C₄×C₈	C₂²×C₄²	1	224	yes	—
109	C₂²×C₄×C₈	C₂³×C₈	1	112	yes	—
110	C₂²×C₄×C₈	C₂³×C₈	1	448	yes	—
111	C₂²×C₄×C₈	C₂³×C₈	1	224	yes	—
112	C₂²×C₄×C₈	C₂⁴×C₄	1	28	yes	—
113	C₂×C₄³	C₈²	1	512	yes	—
114	C₂×C₄³	C₄×C₁₆	1	1024	yes	—
115	C₂×C₄³	C₄³	1	112	yes	—
116	C₂×C₄³	C₂×C₄×C₈	1	112	yes	—
117	C₂×C₄³	C₂×C₄×C₈	1	448	yes	—
118	C₂×C₄³	C₂×C₄×C₈	1	448	yes	—
119	C₂×C₄³	C₂²×C₁₆	1	896	yes	—
120	C₂×C₄³	C₂²×C₄²	1	84	yes	—
121	C₂×C₄³	C₂³×C₈	1	84	yes	—
122	C₂×C₄×C₁₆	C₄³	1	3584	yes	—
123	C₂×C₄×C₁₆	C₂×C₄×C₈	1	21504	yes	—
124	C₂×C₄×C₁₆	C₂²×C₄²	1	1792	yes	—
125	C₄²×C₈	C₈²	1	3584	yes	—
126	C₄²×C₈	C₄×C₁₆	1	14336	yes	direct product factor
127	C₄²×C₈	C₄³	1	896	yes	—
128	C₄²×C₈	C₂×C₄×C₈	1	3584	yes	—
129	C₄²×C₈	C₂×C₄×C₈	1	1344	yes	—
130	C₄²×C₈	C₂×C₄×C₈	1	2688	yes	—
131	C₄²×C₈	C₂²×C₄²	1	448	yes	—
132	C₂×C₈²	C₄³	1	896	yes	—
133	C₂×C₈²	C₂×C₄×C₈	1	5376	yes	—
134	C₂×C₈²	C₂²×C₄²	1	896	yes	—
135	C₈×C₁₆	C₄³	1	4096	yes	direct product factor
136	C₂⁴×C₄	C₂²×C₄×C₈	1	28	yes	—
137	C₂⁴×C₄	C₂³×C₄²	1	1	yes	characteristic
138	C₂⁴×C₄	C₂⁴×C₈	1	2	yes	—
139	C₂³×C₈	C₂×C₄³	1	84	yes	—
140	C₂³×C₈	C₂²×C₄×C₈	1	112	yes	—
141	C₂³×C₈	C₂²×C₄×C₈	1	448	yes	—
142	C₂³×C₈	C₂²×C₄×C₈	1	224	yes	—
143	C₂³×C₈	C₂³×C₁₆	1	512	yes	—
144	C₂³×C₈	C₂³×C₄²	1	28	yes	—
145	C₂³×C₈	C₂⁴×C₈	1	32	yes	—
146	C₂²×C₄²	C₂×C₈²	1	896	yes	—
147	C₂²×C₄²	C₄²×C₈	1	448	yes	—
148	C₂²×C₄²	C₂×C₄×C₁₆	1	1792	yes	—
149	C₂²×C₄²	C₂×C₄³	1	84	yes	—
150	C₂²×C₄²	C₂²×C₄×C₈	1	336	yes	—
151	C₂²×C₄²	C₂²×C₄×C₈	1	84	yes	—
152	C₂²×C₄²	C₂²×C₄×C₈	1	224	yes	—
153	C₂²×C₄²	C₂³×C₁₆	1	448	yes	—
154	C₂²×C₄²	C₂³×C₄²	1	14	yes	—
155	C₂²×C₄²	C₂⁴×C₈	1	14	yes	—
156	C₂²×C₁₆	C₂×C₄³	1	896	yes	—
157	C₂²×C₁₆	C₂²×C₄×C₈	1	3584	yes	—
158	C₂×C₄×C₈	C₂×C₈²	1	5376	yes	—
159	C₂×C₄×C₈	C₄²×C₈	1	3584	yes	—
160	C₂×C₄×C₈	C₄²×C₈	1	1344	yes	—
161	C₂×C₄×C₈	C₄²×C₈	1	2688	yes	—
162	C₂×C₄×C₈	C₂×C₄×C₁₆	1	21504	yes	—
163	C₂×C₄×C₈	C₂×C₄³	1	112	yes	—
164	C₂×C₄×C₈	C₂×C₄³	1	448	yes	—
165	C₂×C₄×C₈	C₂×C₄³	1	448	yes	—
166	C₂×C₄×C₈	C₂²×C₄×C₈	1	1344	yes	—
167	C₂×C₄×C₈	C₂²×C₄×C₈	1	2688	yes	—
168	C₂×C₄×C₈	C₂²×C₄×C₈	1	672	yes	—
169	C₂×C₄×C₈	C₂³×C₄²	1	112	yes	—
170	C₄³	C₈×C₁₆	1	4096	yes	direct product factor
171	C₄³	C₂×C₈²	1	896	yes	—
172	C₄³	C₄²×C₈	1	896	yes	—
173	C₄³	C₂×C₄×C₁₆	1	3584	yes	—
174	C₄³	C₂×C₄³	1	112	yes	—
175	C₄³	C₂²×C₄×C₈	1	336	yes	—
176	C₄×C₁₆	C₄²×C₈	1	14336	yes	direct product factor
177	C₄×C₁₆	C₂×C₄³	1	1024	yes	—
178	C₈²	C₄²×C₈	1	3584	yes	—
179	C₈²	C₂×C₄³	1	512	yes	—
180	C₂⁵	C₂³×C₄×C₈	1	1	yes	characteristic
181	C₂³×C₄	C₂²×C₈²	1	224	yes	—
182	C₂³×C₄	C₂×C₄²×C₈	1	168	yes	—
183	C₂³×C₄	C₂²×C₄×C₁₆	1	448	yes	—
184	C₂³×C₄	C₂²×C₄³	1	7	yes	—
185	C₂³×C₄	C₂³×C₄×C₈	1	7	yes	—
186	C₂³×C₄	C₂³×C₄×C₈	1	16	yes	—
187	C₂³×C₄	C₂³×C₄×C₈	1	28	yes	—
188	C₂³×C₄	C₂⁴×C₁₆	1	32	yes	—
189	C₂²×C₈	C₄⁴	1	56	yes	—
190	C₂²×C₈	C₂²×C₈²	1	896	yes	—
191	C₂²×C₈	C₂×C₄²×C₈	1	336	yes	—
192	C₂²×C₈	C₂×C₄²×C₈	1	896	yes	—
193	C₂²×C₈	C₂×C₄²×C₈	1	672	yes	—
194	C₂²×C₈	C₂²×C₄×C₁₆	1	3584	yes	—
195	C₂²×C₈	C₂²×C₄³	1	56	yes	—
196	C₂²×C₈	C₂³×C₄×C₈	1	224	yes	—
197	C₂×C₄²	C₄×C₈²	1	1792	yes	—
198	C₂×C₄²	C₂×C₈×C₁₆	1	7168	yes	—
199	C₂×C₄²	C₄²×C₁₆	1	3584	yes	—
200	C₂×C₄²	C₄⁴	1	56	yes	—
201	C₂×C₄²	C₂²×C₈²	1	672	yes	—
202	C₂×C₄²	C₂×C₄²×C₈	1	672	yes	—
203	C₂×C₄²	C₂×C₄²×C₈	1	168	yes	—
204	C₂×C₄²	C₂×C₄²×C₈	1	448	yes	—
205	C₂×C₄²	C₂²×C₄×C₁₆	1	2688	yes	—
206	C₂×C₄²	C₂²×C₄³	1	28	yes	—
207	C₂×C₄²	C₂³×C₄×C₈	1	84	yes	—
208	C₂×C₁₆	C₄⁴	1	256	yes	—
209	C₂×C₁₆	C₂×C₄²×C₈	1	3584	yes	—
210	C₄×C₈	C₄×C₈²	1	5376	yes	—
211	C₄×C₈	C₄²×C₁₆	1	14336	yes	direct product factor
212	C₄×C₈	C₄⁴	1	128	yes	—
213	C₄×C₈	C₂×C₄²×C₈	1	1792	yes	—
214	C₄×C₈	C₂×C₄²×C₈	1	896	yes	—
215	C₄×C₈	C₂×C₄²×C₈	1	448	yes	—
216	C₄×C₈	C₂²×C₄³	1	64	yes	—
217	C₂⁴	C₂³×C₈²	1	8	yes	—
218	C₂⁴	C₂²×C₄²×C₈	1	7	yes	—
219	C₂⁴	C₂³×C₄×C₁₆	1	16	yes	—
220	C₂²×C₄	C₂×C₄×C₈²	1	672	yes	—
221	C₂²×C₄	C₄³×C₈	1	112	yes	—
222	C₂²×C₄	C₂²×C₈×C₁₆	1	1792	yes	—
223	C₂²×C₄	C₂×C₄²×C₁₆	1	1344	yes	—
224	C₂²×C₄	C₂×C₄⁴	1	7	yes	—
225	C₂²×C₄	C₂³×C₈²	1	56	yes	—
226	C₂²×C₄	C₂²×C₄²×C₈	1	21	yes	—
227	C₂²×C₄	C₂²×C₄²×C₈	1	56	yes	—
228	C₂²×C₄	C₂²×C₄²×C₈	1	56	yes	—
229	C₂²×C₄	C₂³×C₄×C₁₆	1	224	yes	—
230	C₂×C₈	C₂×C₄×C₈²	1	1344	yes	—
231	C₂×C₈	C₄³×C₈	1	256	yes	—
232	C₂×C₈	C₄³×C₈	1	112	yes	—
233	C₂×C₈	C₄³×C₈	1	224	yes	—
234	C₂×C₈	C₂×C₄²×C₁₆	1	3584	yes	—
235	C₂×C₈	C₂×C₄⁴	1	16	yes	—
236	C₂×C₈	C₂²×C₄²×C₈	1	224	yes	—
237	C₄²	C₄×C₈×C₁₆	1	7168	yes	direct product factor
238	C₄²	C₂×C₄×C₈²	1	672	yes	—
239	C₄²	C₄³×C₈	1	224	yes	—
240	C₄²	C₂×C₄²×C₁₆	1	1792	yes	—
241	C₄²	C₂×C₄⁴	1	8	yes	—
242	C₄²	C₂²×C₄²×C₈	1	56	yes	—
243	C₁₆	C₄³×C₈	1	512	yes	direct product factor
244	C₂³	C₂³×C₈×C₁₆	1	64	yes	—
245	C₂³	C₂²×C₄²×C₁₆	1	56	yes	—
246	C₂³	C₂²×C₄×C₈²	1	28	yes	—
247	C₂³	C₂×C₄³×C₈	1	7	yes	—
248	C₂×C₄	C₂×C₄×C₈×C₁₆	1	2688	yes	—
249	C₂×C₄	C₄³×C₁₆	1	448	yes	—
250	C₂×C₄	C₂²×C₄²×C₁₆	1	224	yes	—
251	C₂×C₄	C₄²×C₈²	1	224	yes	—
252	C₂×C₄	C₂²×C₄×C₈²	1	84	yes	—
253	C₂×C₄	C₂×C₄³×C₈	1	7	yes	—
254	C₂×C₄	C₂×C₄³×C₈	1	16	yes	—
255	C₂×C₄	C₂×C₄³×C₈	1	28	yes	—
256	C₂×C₄	C₄⁵	1	1	yes	characteristic
257	C₈	C₄³×C₁₆	1	512	yes	direct product factor
258	C₈	C₄²×C₈²	1	224	yes	—
259	C₈	C₂×C₄³×C₈	1	32	yes	—
260	C₂²	C₂²×C₄×C₈×C₁₆	1	112	yes	—
261	C₂²	C₂×C₄³×C₁₆	1	28	yes	—
262	C₂²	C₂×C₄²×C₈²	1	14	yes	—
263	C₂²	C₄⁴×C₈	1	1	yes	characteristic
264	C₄	C₄²×C₈×C₁₆	1	448	yes	direct product factor
265	C₄	C₂×C₄³×C₁₆	1	32	yes	—
266	C₄	C₂×C₄²×C₈²	1	14	yes	—
267	C₄	C₄⁴×C₈	1	2	yes	—
268	C₂	C₂×C₄²×C₈×C₁₆	1	28	yes	—
269	C₂	C₄⁴×C₁₆	1	2	yes	—
270	C₂	C₄³×C₈²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³×C₄×C₈	C₂⁵
socle, prime radical	C₂⁵	C₂³×C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₂³×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. C₄²×C₈×C₁₆	C₄	direct (N × H)
25. C₄³×C₁₆	C₈	direct (N × H)
45. C₄³×C₈	C₁₆	direct (N × H)
54. C₄×C₈×C₁₆	C₄²	direct (N × H)
87. C₄²×C₁₆	C₄×C₈	direct (N × H)
126. C₄²×C₈	C₄×C₁₆	direct (N × H)
135. C₈×C₁₆	C₄³	direct (N × H)
170. C₄³	C₈×C₁₆	direct (N × H)
176. C₄×C₁₆	C₄²×C₈	direct (N × H)
211. C₄×C₈	C₄²×C₁₆	direct (N × H)
237. C₄²	C₄×C₈×C₁₆	direct (N × H)
243. C₁₆	C₄³×C₈	direct (N × H)
257. C₈	C₄³×C₁₆	direct (N × H)
264. C₄	C₄²×C₈×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
C₄³×C₈²	2
C₄⁴×C₁₆	2
C₂×C₄²×C₈×C₁₆	2

Subgroups — C43×C8×C16 #72187111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄³×C₈×C₁₆ #72187111