Subgroups — C₆₁×#7562 #29510995

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2295	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1272	C₂	1	2	yes	maximal
5	C₆₁×#1271	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
8	C₂×C₄×C₈×C₆₁	—	2	1	no	—
9	C₂×C₄×C₈×C₆₁	—	2	1	no	—
10	C₆₁×#391	C₄	1	1	yes	characteristic
11	C₆₁×#391	C₂²	1	1	yes	characteristic
12	C₆₁×#389	C₄	1	1	yes	characteristic
13	C₆₁×#389	C₂²	1	1	yes	characteristic
14	C₆₁×#387	—	2	1	no	—
15	C₆₁×#387	—	2	1	no	—
16	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
17	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
18	C₆₁×#376	—	2	1	no	—
19	C₄³×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#336	—	2	1	no	—
21	C₆₁×#336	C₂²	1	2	yes	—
22	C₆₁×#336	C₂²	1	2	yes	—
23	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₄×Q₈×C₆₁	—	2	2	no	—
27	C₄×Q₈×C₆₁	C₈	1	4	yes	complemented
28	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	—	2	1	no	—
35	C₂×C₄²×C₆₁	—	2	1	no	—
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₁	—	2	4	no	—
44	C₄×C₈×C₆₁	—	2	4	no	—
45	C₄×C₈×C₆₁	—	4	2	no	—
46	#95×C₆₁	—	2	1	no	—
47	#95×C₆₁	—	2	1	no	—
48	#95×C₆₁	—	2	1	no	—
49	#95×C₆₁	—	2	1	no	—
50	#95×C₆₁	—	2	1	no	—
51	#95×C₆₁	—	2	1	no	—
52	#95×C₆₁	—	2	1	no	—
53	C₂×Q₈×C₆₁	—	2	1	no	—
54	C₂×Q₈×C₆₁	—	4	1	no	—
55	C₂×Q₈×C₆₁	—	4	1	no	—
56	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₆₁	—	2	1	no	—
58	C₂²×C₄×C₆₁	—	2	1	no	—
59	C₂²×C₄×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	—	2	1	no	—
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₁	—	2	4	no	—
69	C₂×C₈×C₆₁	—	2	4	no	—
70	C₂×C₈×C₆₁	—	4	2	no	—
71	C₂×C₈×C₆₁	—	4	2	no	—
72	C₂×C₈×C₆₁	—	4	2	no	—
73	C₂×C₈×C₆₁	—	4	2	no	—
74	C₄⋊C₄×C₆₁	—	2	4	no	—
75	C₄⋊C₄×C₆₁	—	2	1	no	—
76	C₄⋊C₄×C₆₁	—	4	1	no	—
77	C₄⋊C₄×C₆₁	—	4	1	no	—
78	C₄⋊C₄×C₆₁	—	4	2	no	—
79	C₄⋊C₄×C₆₁	—	4	2	no	—
80	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
81	C₄²×C₆₁	—	2	4	no	—
82	C₄²×C₆₁	—	2	2	no	—
83	C₄²×C₆₁	—	2	2	no	—
84	C₄²×C₆₁	—	2	2	no	—
85	C₄²×C₆₁	—	4	1	no	—
86	C₄²×C₆₁	—	4	1	no	—
87	C₄²×C₆₁	—	4	2	no	—
88	C₄²×C₆₁	C₂×C₈	1	2	yes	—
89	C₄²×C₆₁	M₁₆	1	2	yes	—
90	C₄²×C₆₁	SD₁₆	1	2	yes	—
91	C₄²×C₆₁	Q₁₆	1	2	yes	—
92	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	Q₈×C₆₁	—	4	2	no	—
94	Q₈×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	1	no	—
98	C₂×C₄×C₆₁	—	2	1	no	—
99	C₂×C₄×C₆₁	—	2	1	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	1	no	—
105	C₂×C₄×C₆₁	—	2	1	no	—
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	2	2	no	—
109	C₂×C₄×C₆₁	—	4	1	no	—
110	C₂×C₄×C₆₁	—	4	1	no	—
111	C₂×C₄×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	—	4	1	no	—
113	C₂×C₄×C₆₁	—	4	1	no	—
114	C₂×C₄×C₆₁	—	4	1	no	—
115	C₂×C₄×C₆₁	—	4	1	no	—
116	C₂×C₄×C₆₁	—	4	1	no	—
117	C₂×C₄×C₆₁	—	4	1	no	—
118	C₂×C₄×C₆₁	—	4	1	no	—
119	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₆₁	#103	1	2	yes	—
122	C₂×C₄×C₆₁	#103	1	2	yes	—
123	C₂×C₄×C₆₁	#104	1	2	yes	—
124	C₂×C₄×C₆₁	#104	1	2	yes	—
125	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₁	#135	1	2	yes	—
130	C₈×C₆₁	—	4	4	no	—
131	C₈×C₆₁	—	4	4	no	—
132	#7562	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₆₁	#410	1	1	yes	characteristic
135	C₂²×C₆₁	#413	1	1	yes	characteristic
136	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
137	C₂²×C₆₁	#416	1	1	yes	characteristic
138	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
139	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	—	2	1	no	—
142	C₄×C₆₁	—	2	1	no	—
143	C₄×C₆₁	—	2	2	no	—
144	C₄×C₆₁	—	2	2	no	—
145	C₄×C₆₁	—	2	1	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	—	2	2	no	—
148	C₄×C₆₁	—	4	2	no	—
149	C₄×C₆₁	—	4	2	no	—
150	C₄×C₆₁	—	8	1	no	—
151	C₄×C₆₁	—	8	1	no	—
152	C₄×C₆₁	Q₈⋊C₈	1	4	yes	—
153	#2295	C₂×C₆₁	1	1	yes	characteristic
154	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
155	#1744	C₂×C₆₁	1	1	yes	characteristic
156	#1272	C₂×C₆₁	1	2	yes	—
157	#1271	C₂×C₆₁	1	2	yes	—
158	C₂×C₆₁	#1458	1	1	yes	characteristic
159	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
160	C₂×C₆₁	#1476	1	1	yes	characteristic
161	C₂×C₆₁	#1500	1	1	yes	characteristic
162	C₂×C₆₁	#1503	1	1	yes	characteristic
163	C₂×C₆₁	#1524	1	1	yes	characteristic
164	C₂×C₆₁	#1542	1	1	yes	characteristic
165	C₂×C₄×Q₈	C₄×C₆₁	1	1	yes	characteristic
166	C₂×C₄⋊C₈	—	2	1	no	—
167	C₂×C₄×C₈	—	2	1	no	—
168	C₂×C₄×C₈	—	2	1	no	—
169	#391	C₄×C₆₁	1	1	yes	characteristic
170	#391	C₂²×C₆₁	1	1	yes	characteristic
171	#389	C₄×C₆₁	1	1	yes	characteristic
172	#389	C₂²×C₆₁	1	1	yes	characteristic
173	#387	—	2	1	no	—
174	#387	—	2	1	no	—
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄×C₄⋊C₄	C₄×C₆₁	1	1	yes	characteristic
177	#376	—	2	1	no	—
178	C₄³	C₂²×C₆₁	1	1	yes	characteristic
179	#336	—	2	1	no	—
180	#336	C₂²×C₆₁	1	2	yes	—
181	#336	C₂²×C₆₁	1	2	yes	—
182	C₆₁	#7562	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	C₈×C₆₁	1	4	yes	complemented
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
193	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	1	no	—
196	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
203	C₄⋊C₈	—	2	4	no	—
204	C₄×C₈	—	2	4	no	—
205	C₄×C₈	—	4	2	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	#95	—	2	1	no	—
210	#95	—	2	1	no	—
211	#95	—	2	1	no	—
212	#95	—	2	1	no	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	C₂×C₈×C₆₁	1	2	yes	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	C₂×C₈×C₆₁	1	2	yes	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂×C₈×C₆₁	1	2	yes	—
249	C₄²	M₁₆×C₆₁	1	2	yes	—
250	C₄²	SD₁₆×C₆₁	1	2	yes	—
251	C₄²	Q₁₆×C₆₁	1	2	yes	—
252	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
280	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
281	C₂×C₄	#103×C₆₁	1	2	yes	—
282	C₂×C₄	#103×C₆₁	1	2	yes	—
283	C₂×C₄	#104×C₆₁	1	2	yes	—
284	C₂×C₄	#104×C₆₁	1	2	yes	—
285	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
286	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
287	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
288	C₂×C₄	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
289	C₂×C₄	#135×C₆₁	1	2	yes	—
290	C₈	—	4	4	no	—
291	C₈	—	4	4	no	—
292	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
293	C₂²	C₆₁×#410	1	1	yes	characteristic
294	C₂²	C₆₁×#413	1	1	yes	characteristic
295	C₂²	C₂×#103×C₆₁	1	1	yes	characteristic
296	C₂²	C₆₁×#416	1	1	yes	characteristic
297	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
298	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	1	no	—
310	C₄	—	8	1	no	—
311	C₄	C₆₁×Q₈⋊C₈	1	4	yes	—
312	C₂	C₆₁×#1458	1	1	yes	characteristic
313	C₂	C₂×C₆₁×Q₈⋊C₈	1	1	yes	characteristic
314	C₂	C₆₁×#1476	1	1	yes	characteristic
315	C₂	C₆₁×#1500	1	1	yes	characteristic
316	C₂	C₆₁×#1503	1	1	yes	characteristic
317	C₂	C₆₁×#1524	1	1	yes	characteristic
318	C₂	C₆₁×#1542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₄×Q₈×C₆₁	C₈	semidirect
132. #7562	C₆₁	direct (N × H)
182. C₆₁	#7562	direct (N × H)
187. C₄×Q₈	C₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2295	2
C₄×C₄⋊C₈×C₆₁	2
C₆₁×#1744	2
C₆₁×#1272	2
C₆₁×#1271	2
#7562	61

Subgroups — C61×#7562 #29510995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7562 #29510995