Subgroups — C₆₁×#8166 #29511599

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₄×#100×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1460	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₆₁	C₄	1	2	yes	—
9	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#436	—	2	1	no	—
11	C₆₁×#436	—	2	1	no	—
12	C₆₁×#436	—	2	1	no	—
13	C₆₁×#436	—	2	1	no	—
14	C₆₁×#431	C₂²	1	1	yes	characteristic
15	C₆₁×#431	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#100×C₆₁	C₄	1	2	yes	—
18	C₂×C₂²⋊C₈×C₆₁	C₄	1	2	yes	—
19	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#343	C₄	1	2	yes	—
23	C₆₁×#323	C₄	1	4	yes	—
24	C₆₁×#322	—	2	1	no	—
25	C₆₁×#322	—	2	1	no	—
26	C₆₁×#322	—	2	1	no	—
27	C₈²×C₆₁	—	2	1	no	—
28	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₆₁	—	2	2	no	—
31	C₂×M₁₆×C₆₁	—	2	2	no	—
32	C₂×M₁₆×C₆₁	—	4	4	no	—
33	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₆₁	—	2	1	no	—
39	C₄×D₄×C₆₁	—	2	1	no	—
40	C₄×D₄×C₆₁	—	2	1	no	—
41	C₄×D₄×C₆₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₆₁	—	2	1	no	—
46	C₄⋊C₈×C₆₁	—	2	1	no	—
47	C₄⋊C₈×C₆₁	—	2	1	no	—
48	C₄⋊C₈×C₆₁	—	2	1	no	—
49	#100×C₆₁	—	2	4	no	—
50	C₂²⋊C₈×C₆₁	—	2	2	no	—
51	C₂²⋊C₈×C₆₁	—	2	2	no	—
52	C₂²⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
53	C₂²⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
54	#97×C₆₁	—	2	1	no	—
55	#97×C₆₁	—	2	1	no	—
56	C₄×C₈×C₆₁	—	2	1	no	—
57	C₄×C₈×C₆₁	—	2	1	no	—
58	C₄×C₈×C₆₁	—	2	1	no	—
59	C₄×C₈×C₆₁	—	2	1	no	—
60	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
64	C₂⁴×C₆₁	C₄²	1	2	yes	—
65	C₂×D₄×C₆₁	—	2	1	no	—
66	C₂×D₄×C₆₁	—	2	1	no	—
67	C₂×D₄×C₆₁	—	2	2	no	—
68	C₂×D₄×C₆₁	—	2	2	no	—
69	C₂²×C₄×C₆₁	—	2	2	no	—
70	C₂²×C₄×C₆₁	—	2	2	no	—
71	C₂²×C₄×C₆₁	—	4	2	no	—
72	C₂²×C₄×C₆₁	—	4	2	no	—
73	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
74	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	M₁₆×C₆₁	—	2	2	no	—
78	M₁₆×C₆₁	—	2	2	no	—
79	M₁₆×C₆₁	—	2	2	no	—
80	M₁₆×C₆₁	—	4	4	no	—
81	M₁₆×C₆₁	—	4	4	no	—
82	M₁₆×C₆₁	M₁₆	1	4	yes	complemented
83	C₂×C₈×C₆₁	—	2	2	no	—
84	C₂×C₈×C₆₁	—	2	2	no	—
85	C₂×C₈×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	—	2	2	no	—
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	—	4	1	no	—
90	C₂×C₈×C₆₁	—	4	2	no	—
91	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₆₁	—	2	1	no	—
96	C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₂²⋊C₄×C₆₁	—	2	2	no	—
98	C₂²⋊C₄×C₆₁	—	2	2	no	—
99	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₁	#41	1	1	yes	characteristic
102	C₄²×C₆₁	#41	1	1	yes	characteristic
103	C₂³×C₆₁	—	2	2	no	—
104	C₂³×C₆₁	—	2	2	no	—
105	C₂³×C₆₁	—	2	2	no	—
106	C₂³×C₆₁	—	4	2	no	—
107	C₂³×C₆₁	—	4	2	no	—
108	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
109	D₄×C₆₁	—	4	2	no	—
110	D₄×C₆₁	—	4	2	no	—
111	C₂×C₄×C₆₁	—	2	1	no	—
112	C₂×C₄×C₆₁	—	2	1	no	—
113	C₂×C₄×C₆₁	—	2	2	no	—
114	C₂×C₄×C₆₁	—	4	1	no	—
115	C₂×C₄×C₆₁	—	4	4	no	—
116	C₂×C₄×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	—	4	2	no	—
118	C₂×C₄×C₆₁	—	4	2	no	—
119	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₆₁	#131	1	2	yes	—
127	C₈×C₆₁	—	2	1	no	—
128	C₈×C₆₁	—	2	1	no	—
129	C₈×C₆₁	—	2	1	no	—
130	C₈×C₆₁	—	2	4	no	—
131	C₈×C₆₁	—	4	2	no	—
132	C₈×C₆₁	—	4	2	no	—
133	C₈×C₆₁	—	4	2	no	—
134	C₈×C₆₁	—	4	2	no	—
135	C₈×C₆₁	#100	1	2	yes	complemented
136	#8166	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	—	2	1	no	—
139	C₂²×C₆₁	—	4	2	no	—
140	C₂²×C₆₁	—	4	2	no	—
141	C₂²×C₆₁	—	4	2	no	—
142	C₂²×C₆₁	—	8	2	no	—
143	C₂²×C₆₁	—	8	2	no	—
144	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
146	C₂²×C₆₁	#405	1	1	yes	characteristic
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	1	no	—
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	4	2	no	—
151	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
152	C₄×C₆₁	#413	1	1	yes	characteristic
153	C₄×C₆₁	#435	1	2	yes	—
154	C₄×C₆₁	#436	1	2	yes	—
155	C₂×#436	C₂×C₆₁	1	1	yes	characteristic
156	C₄×#100	C₂×C₆₁	1	1	yes	characteristic
157	#1460	C₂×C₆₁	1	1	yes	characteristic
158	#1451	C₂×C₆₁	1	1	yes	characteristic, complemented
159	C₈×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
160	#1283	C₂×C₆₁	1	1	yes	characteristic, complemented
161	#1283	C₂×C₆₁	1	1	yes	characteristic, complemented
162	C₂×C₆₁	—	2	1	no	—
163	C₂×C₆₁	—	2	1	no	—
164	C₂×C₆₁	—	8	2	no	—
165	C₂×C₆₁	#1753	1	1	yes	characteristic
166	C₂×C₆₁	C₄×#100	1	1	yes	characteristic
167	C₂×C₆₁	#1757	1	1	yes	characteristic
168	C₂²×M₁₆	C₄×C₆₁	1	2	yes	—
169	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
170	#436	—	2	1	no	—
171	#436	—	2	1	no	—
172	#436	—	2	1	no	—
173	#436	—	2	1	no	—
174	#431	C₂²×C₆₁	1	1	yes	characteristic
175	#431	C₂²×C₆₁	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
177	C₂×#100	C₄×C₆₁	1	2	yes	—
178	C₂×C₂²⋊C₈	C₄×C₆₁	1	2	yes	—
179	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
180	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
182	#343	C₄×C₆₁	1	2	yes	—
183	#323	C₄×C₆₁	1	4	yes	—
184	#322	—	2	1	no	—
185	#322	—	2	1	no	—
186	#322	—	2	1	no	—
187	C₈²	—	2	1	no	—
188	C₆₁	#8166	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
190	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	4	4	no	—
194	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
195	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
196	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
197	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
198	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	—	2	1	no	—
203	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
205	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	#100	—	2	4	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	2	no	—
213	C₂²⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
214	C₂²⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
215	#97	—	2	1	no	—
216	#97	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
222	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
223	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
224	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
225	C₂⁴	C₄²×C₆₁	1	2	yes	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
237	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	4	4	no	—
242	M₁₆	—	4	4	no	—
243	M₁₆	M₁₆×C₆₁	1	4	yes	complemented
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
262	C₄²	#41×C₆₁	1	1	yes	characteristic
263	C₄²	#41×C₆₁	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
281	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
284	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
286	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
287	C₂×C₄	#131×C₆₁	1	2	yes	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	4	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	4	2	no	—
296	C₈	#100×C₆₁	1	2	yes	complemented
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
305	C₂²	C₄×M₁₆×C₆₁	1	1	yes	characteristic
306	C₂²	C₆₁×#405	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	2	no	—
311	C₄	C₂×#100×C₆₁	1	1	yes	characteristic
312	C₄	C₆₁×#413	1	1	yes	characteristic
313	C₄	C₆₁×#435	1	2	yes	—
314	C₄	C₆₁×#436	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	8	2	no	—
318	C₂	C₆₁×#1753	1	1	yes	characteristic
319	C₂	C₄×#100×C₆₁	1	1	yes	characteristic
320	C₂	C₆₁×#1757	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₆₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1451	C₂	semidirect
5. C₈×M₁₆×C₆₁	C₂	semidirect
6. C₆₁×#1283	C₂	semidirect
7. C₆₁×#1283	C₂	semidirect
82. M₁₆×C₆₁	M₁₆	semidirect
135. C₈×C₆₁	#100	semidirect
136. #8166	C₆₁	direct (N × H)
158. #1451	C₂×C₆₁	semidirect
159. C₈×M₁₆	C₂×C₆₁	semidirect
160. #1283	C₂×C₆₁	semidirect
161. #1283	C₂×C₆₁	semidirect
188. C₆₁	#8166	direct (N × H)
243. M₁₆	M₁₆×C₆₁	semidirect
296. C₈	#100×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#436	2
C₄×#100×C₆₁	2
C₆₁×#1460	2
C₆₁×#1451	2
C₈×M₁₆×C₆₁	2
C₆₁×#1283	2
#8166	61

Subgroups — C61×#8166 #29511599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8166 #29511599