Subgroups — C₂×C₁₆×#67 #10948335

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×#67	C₂	1	1	yes	maximal, characteristic
3	C₂²×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#4549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×#67	C₂	1	8	yes	maximal, direct product factor
7	C₂×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×C₁₆	—	3	1	no	maximal, Sylow
10	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×#67	C₄	1	1	yes	characteristic
12	C₂²×D₃×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₁₆	—	2	4	no	—
14	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₁₆	C₂²	1	4	yes	complemented
17	C₂×#830	C₄	1	1	yes	characteristic
18	C₈×#67	C₄	1	4	yes	—
19	C₈×#67	C₂²	1	4	yes	—
20	C₂×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
21	C₂×Dic₃⋊C₈	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
23	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₂×D₃×C₁₆	—	2	4	no	—
26	C₂×D₃×C₁₆	C₂²	1	4	yes	complemented
27	#4549	C₂²	1	4	yes	complemented
28	(C₂×D₃)⋊C₁₆	C₂²	1	4	yes	complemented
29	Dic₃⋊C₁₆	C₂²	1	4	yes	complemented
30	Dic₃×C₁₆	C₂²	1	4	yes	complemented
31	C₂³×C₁₆	—	3	1	no	—
32	C₂³×C₁₆	D₃	1	1	yes	characteristic, complemented
33	C₂×C₈×D₄	—	3	1	no	—
34	D₄×C₁₆	—	3	8	no	—
35	C₂×C₄⋊C₁₆	—	3	1	no	—
36	C₂×C₂²⋊C₁₆	—	3	1	no	—
37	C₂×C₂²⋊C₁₆	—	3	1	no	—
38	C₂×C₄×C₁₆	—	3	1	no	—
39	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×#67	C₈	1	1	yes	characteristic
41	C₂²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	—	2	2	no	—
43	C₂²×C₃×C₈	—	2	2	no	—
44	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₃×C₈	C₂³	1	2	yes	—
47	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
48	C₂×#217	C₈	1	1	yes	characteristic
49	C₄×#67	C₈	1	4	yes	—
50	C₄×#67	C₂×C₄	1	4	yes	—
51	C₂×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
52	C₂×Dic₃⋊C₄	C₈	1	1	yes	characteristic
53	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×D₃×C₈	—	2	2	no	—
57	C₂×D₃×C₈	—	2	2	no	—
58	C₂×D₃×C₈	C₂×C₄	1	2	yes	—
59	C₂×D₃×C₈	C₂³	1	2	yes	—
60	C₂×C₃×C₁₆	—	2	2	no	—
61	C₂×C₃×C₁₆	—	2	8	no	—
62	C₂×C₃×C₁₆	D₄	1	4	yes	complemented
63	C₂×C₃×C₁₆	C₂³	1	2	yes	—
64	C₂×C₃×C₁₆	C₂³	1	2	yes	—
65	#830	C₂×C₄	1	4	yes	—
66	(C₂×D₃)⋊C₈	C₂×C₄	1	4	yes	—
67	Dic₃⋊C₈	C₂×C₄	1	4	yes	—
68	C₈×Dic₃	C₂×C₄	1	2	yes	—
69	C₈×Dic₃	C₂³	1	2	yes	—
70	C₂×C₃⋊C₁₆	—	2	2	no	—
71	C₂×C₃⋊C₁₆	—	2	2	no	—
72	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
73	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
74	D₃×C₁₆	—	2	8	no	—
75	C₂³×C₈	—	3	1	no	—
76	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
77	C₂×C₄×D₄	—	3	1	no	—
78	C₂²×C₁₆	—	2	4	no	—
79	C₂²×C₁₆	—	3	1	no	—
80	C₂²×C₁₆	—	3	1	no	—
81	C₂²×C₁₆	—	3	4	no	—
82	C₂²×C₁₆	—	6	4	no	—
83	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
84	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
85	C₂²×C₁₆	C₂×D₃	1	4	yes	complemented
86	C₈×D₄	—	3	4	no	—
87	C₈×D₄	—	3	4	no	—
88	C₂×C₄⋊C₈	—	3	1	no	—
89	C₂×C₂²⋊C₈	—	3	1	no	—
90	C₂×C₂²⋊C₈	—	3	1	no	—
91	C₂×C₄×C₈	—	3	1	no	—
92	C₄⋊C₁₆	—	3	4	no	—
93	C₂²⋊C₁₆	—	3	4	no	—
94	C₂²⋊C₁₆	—	3	4	no	—
95	C₄×C₁₆	—	3	4	no	—
96	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
97	C₂³×D₃	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
102	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
103	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
104	C₂×#67	C₁₆	1	8	yes	direct product factor
105	C₂×#67	C₂×C₈	1	4	yes	—
106	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
107	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
108	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
109	C₂×C₄×D₃	—	2	2	no	—
110	C₂×C₄×D₃	—	2	2	no	—
111	C₂×C₄×D₃	C₂×C₈	1	2	yes	—
112	C₂×C₄×D₃	C₂²×C₄	1	2	yes	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	4	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
120	C₂×C₃×C₈	#41	1	2	yes	—
121	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
122	#217	C₂×C₈	1	4	yes	—
123	(C₂×D₃)⋊C₄	C₂×C₈	1	4	yes	—
124	Dic₃⋊C₄	C₂×C₈	1	4	yes	—
125	C₄×Dic₃	C₂×C₈	1	2	yes	—
126	C₄×Dic₃	C₂²×C₄	1	2	yes	—
127	C₂×C₃⋊C₈	—	2	2	no	—
128	C₂×C₃⋊C₈	—	2	2	no	—
129	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
130	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
131	D₃×C₈	—	2	4	no	—
132	D₃×C₈	—	2	2	no	—
133	D₃×C₈	—	2	2	no	—
134	C₃×C₁₆	—	2	2	no	—
135	C₃×C₁₆	C₂×D₄	1	4	yes	complemented
136	C₃⋊C₁₆	—	2	2	no	—
137	C₃⋊C₁₆	—	2	2	no	—
138	C₂²×D₄	—	3	1	no	—
139	C₂³×C₄	—	3	1	no	—
140	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	3	1	no	—
144	C₂²×C₈	—	3	1	no	—
145	C₂²×C₈	—	3	2	no	—
146	C₂²×C₈	—	3	2	no	—
147	C₂²×C₈	—	6	2	no	—
148	C₂²×C₈	—	6	2	no	—
149	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
150	C₂²×C₈	C₄×D₃	1	2	yes	—
151	C₂²×C₈	C₂²×D₃	1	2	yes	—
152	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
153	C₄×D₄	—	3	4	no	—
154	C₄×D₄	—	3	4	no	—
155	C₂×C₄⋊C₄	—	3	1	no	—
156	C₂×C₂²⋊C₄	—	3	1	no	—
157	C₂×C₂²⋊C₄	—	3	1	no	—
158	C₂×C₄²	—	3	1	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	8	no	—
161	C₂×C₁₆	—	3	2	no	—
162	C₂×C₁₆	—	3	2	no	—
163	C₂×C₁₆	—	6	2	no	—
164	C₂×C₁₆	—	6	2	no	—
165	C₂×C₁₆	—	6	8	no	—
166	C₂×C₁₆	#67	1	4	yes	direct product factor
167	C₂×C₁₆	C₂²×D₃	1	2	yes	—
168	C₂×C₁₆	C₂²×D₃	1	2	yes	—
169	C₄⋊C₈	—	3	4	no	—
170	C₂²⋊C₈	—	3	4	no	—
171	C₂²⋊C₈	—	3	4	no	—
172	C₄×C₈	—	3	2	no	—
173	C₄×C₈	—	3	2	no	—
174	C₂³×C₃	—	2	2	no	—
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	C₂×C₁₆	1	4	yes	complemented
177	C₂³×C₃	C₂²×C₈	1	2	yes	—
178	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
179	C₂²×D₃	—	2	2	no	—
180	C₂²×D₃	—	2	2	no	—
181	C₂²×D₃	C₂×C₁₆	1	4	yes	complemented
182	C₂²×D₃	C₂²×C₈	1	2	yes	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
188	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
189	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
190	C₂×C₃×C₄	#131	1	2	yes	—
191	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
192	#67	C₂×C₁₆	1	16	yes	direct product factor
193	C₂×Dic₃	—	2	2	no	—
194	C₂×Dic₃	—	2	2	no	—
195	C₂×Dic₃	C₂×C₁₆	1	4	yes	complemented
196	C₂×Dic₃	C₂²×C₈	1	2	yes	—
197	C₂×Dic₃	C₂²×C₈	1	2	yes	—
198	C₂×Dic₃	C₂²×C₈	1	2	yes	—
199	C₄×D₃	—	2	2	no	—
200	C₄×D₃	—	2	2	no	—
201	C₄×D₃	—	2	4	no	—
202	C₃×C₈	—	2	2	no	—
203	C₃×C₈	C₄×D₄	1	2	yes	—
204	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
205	C₃×C₈	C₂×#41	1	1	yes	characteristic
206	C₃⋊C₈	—	2	2	no	—
207	C₃⋊C₈	—	2	2	no	—
208	C₂⁴	—	3	1	no	—
209	C₂⁴	D₃×C₈	1	1	yes	characteristic
210	C₂×D₄	—	3	8	no	—
211	C₂×D₄	—	3	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	3	1	no	—
215	C₂²×C₄	—	3	1	no	—
216	C₂²×C₄	—	3	1	no	—
217	C₂²×C₄	—	3	2	no	—
218	C₂²×C₄	—	3	2	no	—
219	C₂²×C₄	—	6	2	no	—
220	C₂²×C₄	—	6	2	no	—
221	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
222	C₂²×C₄	D₃×C₈	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	3	2	no	—
230	C₂×C₈	—	3	2	no	—
231	C₂×C₈	—	6	2	no	—
232	C₂×C₈	—	6	2	no	—
233	C₂×C₈	—	6	4	no	—
234	C₂×C₈	—	6	2	no	—
235	C₂×C₈	—	6	2	no	—
236	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
238	C₂×C₈	#235	1	2	yes	—
239	C₂×C₈	C₂×#67	1	2	yes	—
240	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
241	C₄⋊C₄	—	3	4	no	—
242	C₂²⋊C₄	—	3	4	no	—
243	C₂²⋊C₄	—	3	4	no	—
244	C₄²	—	3	2	no	—
245	C₄²	—	3	2	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	—	6	2	no	—
248	C₁₆	—	6	2	no	—
249	C₁₆	C₂×#67	1	4	yes	direct product factor
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	8	no	—
253	C₂²×C₃	C₈×D₄	1	2	yes	—
254	C₂²×C₃	C₂²×C₁₆	1	2	yes	—
255	C₂²×C₃	C₂²×C₁₆	1	4	yes	complemented
256	C₂²×C₃	#504	1	2	yes	—
257	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
258	C₂×D₃	—	2	2	no	—
259	C₂×D₃	—	2	2	no	—
260	C₂×D₃	—	2	8	no	—
261	C₂×D₃	C₂²×C₁₆	1	4	yes	complemented
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	C₈×D₄	1	2	yes	—
264	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
265	C₃×C₄	C₂×#131	1	1	yes	characteristic
266	Dic₃	—	2	2	no	—
267	Dic₃	—	2	2	no	—
268	Dic₃	C₂²×C₁₆	1	4	yes	complemented
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	3	4	no	—
272	C₂³	—	3	2	no	—
273	C₂³	—	6	2	no	—
274	C₂³	—	6	2	no	—
275	C₂³	D₃×C₁₆	1	4	yes	complemented
276	C₂³	C₂×D₃×C₈	1	2	yes	—
277	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
278	D₄	—	3	16	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	3	4	no	—
284	C₂×C₄	—	3	2	no	—
285	C₂×C₄	—	3	2	no	—
286	C₂×C₄	—	3	2	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	—	6	2	no	—
289	C₂×C₄	—	6	2	no	—
290	C₂×C₄	—	6	2	no	—
291	C₂×C₄	—	6	4	no	—
292	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
293	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
294	C₂×C₄	#901	1	2	yes	—
295	C₂×C₄	C₄×#67	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	6	2	no	—
299	C₈	—	6	2	no	—
300	C₈	C₄×#67	1	2	yes	—
301	C₈	C₂×#235	1	1	yes	characteristic
302	C₈	C₂²×#67	1	1	yes	characteristic
303	C₂×C₃	—	2	4	no	—
304	C₂×C₃	D₄×C₁₆	1	4	yes	complemented
305	C₂×C₃	C₂×C₈×D₄	1	1	yes	characteristic
306	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
307	C₂×C₃	C₂×#504	1	1	yes	characteristic
308	D₃	—	2	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	8	no	—
312	C₂²	—	3	4	no	—
313	C₂²	—	6	2	no	—
314	C₂²	—	6	2	no	—
315	C₂²	—	6	8	no	—
316	C₂²	C₂×D₃×C₁₆	1	2	yes	—
317	C₂²	C₂×D₃×C₁₆	1	4	yes	complemented
318	C₂²	#4901	1	2	yes	—
319	C₂²	C₈×#67	1	2	yes	—
320	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	3	4	no	—
323	C₄	—	6	2	no	—
324	C₄	—	6	2	no	—
325	C₄	C₈×#67	1	2	yes	—
326	C₄	C₂×#901	1	1	yes	characteristic
327	C₄	C₂×C₄×#67	1	1	yes	characteristic
328	C₃	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	4	no	—
330	C₂	—	6	4	no	—
331	C₂	C₁₆×#67	1	4	yes	direct product factor
332	C₂	C₂²×D₃×C₁₆	1	1	yes	characteristic
333	C₂	C₂×#4901	1	1	yes	characteristic
334	C₂	C₂×C₈×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₃
commutator	C₂×C₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂³×C₃×C₁₆	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂²×#67	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂³×C₁₆	D₃
hypocenter	C₃	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₁₆	C₂	semidirect
3. C₂²×D₃×C₁₆	C₂	semidirect
4. C₂×#4549	C₂	semidirect
5. C₂×(C₂×D₃)⋊C₁₆	C₂	semidirect
6. C₁₆×#67	C₂	direct (N × H)
7. C₂×Dic₃⋊C₁₆	C₂	semidirect
8. C₂×Dic₃×C₁₆	C₂	semidirect
16. C₂²×C₃×C₁₆	C₂²	semidirect
26. C₂×D₃×C₁₆	C₂²	semidirect
27. #4549	C₂²	semidirect
28. (C₂×D₃)⋊C₁₆	C₂²	semidirect
29. Dic₃⋊C₁₆	C₂²	semidirect
30. Dic₃×C₁₆	C₂²	semidirect
32. C₂³×C₁₆	D₃	semidirect
62. C₂×C₃×C₁₆	D₄	semidirect
85. C₂²×C₁₆	C₂×D₃	semidirect
104. C₂×#67	C₁₆	direct (N × H)
135. C₃×C₁₆	C₂×D₄	semidirect
166. C₂×C₁₆	#67	direct (N × H)
176. C₂³×C₃	C₂×C₁₆	semidirect
181. C₂²×D₃	C₂×C₁₆	semidirect
192. #67	C₂×C₁₆	direct (N × H)
195. C₂×Dic₃	C₂×C₁₆	semidirect
249. C₁₆	C₂×#67	direct (N × H)
255. C₂²×C₃	C₂²×C₁₆	semidirect
261. C₂×D₃	C₂²×C₁₆	semidirect
268. Dic₃	C₂²×C₁₆	semidirect
275. C₂³	D₃×C₁₆	semidirect
304. C₂×C₃	D₄×C₁₆	semidirect
317. C₂²	C₂×D₃×C₁₆	semidirect
328. C₃	C₂×D₄×C₁₆	semidirect
331. C₂	C₁₆×#67	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₁₆	2
C₂×C₈×#67	2
C₂²×D₃×C₁₆	2
C₂×#4549	2
C₂×(C₂×D₃)⋊C₁₆	2
C₁₆×#67	2
C₂×Dic₃⋊C₁₆	2
C₂×Dic₃×C₁₆	2
C₂×D₄×C₁₆	3

Subgroups — C2×C16×#67 #10948335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×#67 #10948335