Subgroups — C₆₁×#19600 #29523033

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3167	C₂	1	2	yes	maximal, complemented
4	C₆₁×#3131	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1711	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1708	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#533	C₂²	1	2	yes	—
13	C₆₁×#532	C₂²	1	2	yes	—
14	C₆₁×#500	C₂²	1	1	yes	characteristic
15	C₆₁×#500	C₂²	1	1	yes	characteristic
16	C₆₁×#498	C₂²	1	2	yes	—
17	C₆₁×#494	C₂²	1	1	yes	characteristic
18	C₆₁×#493	C₂²	1	1	yes	characteristic
19	C₆₁×#492	C₂²	1	2	yes	—
20	C₆₁×#488	—	2	4	no	—
21	C₆₁×#487	C₂²	1	2	yes	—
22	C₆₁×#487	C₂²	1	2	yes	—
23	C₆₁×#487	C₂²	1	2	yes	—
24	C₆₁×#487	C₂²	1	2	yes	—
25	C₆₁×#486	C₂²	1	2	yes	—
26	C₆₁×#486	C₂²	1	2	yes	—
27	C₆₁×#486	C₂²	1	2	yes	—
28	C₆₁×#486	C₂²	1	2	yes	—
29	C₆₁×#467	—	2	2	no	—
30	C₆₁×#466	—	2	2	no	—
31	C₆₁×#465	—	2	4	no	—
32	C₄×Q₁₆×C₆₁	—	2	2	no	—
33	C₄×SD₁₆×C₆₁	—	2	4	no	—
34	C₄×D₈×C₆₁	—	2	2	no	—
35	C₆₁×#416	—	2	4	no	—
36	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₈²×C₆₁	C₂²	1	1	yes	characteristic
40	C₈²×C₆₁	C₂²	1	1	yes	characteristic
41	C₈²×C₆₁	C₂²	1	1	yes	characteristic
42	C₈²×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×#41×C₆₁	—	2	2	no	—
44	C₂×Q₁₆×C₆₁	—	2	2	no	—
45	C₂×SD₁₆×C₆₁	—	2	4	no	—
46	C₂×D₈×C₆₁	—	2	2	no	—
47	C₂²×C₈×C₆₁	D₄	1	2	yes	—
48	C₂²×C₈×C₆₁	D₄	1	2	yes	—
49	C₂²×C₈×C₆₁	D₄	1	2	yes	—
50	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
53	#127×C₆₁	C₂³	1	2	yes	—
54	#125×C₆₁	—	2	2	no	—
55	#124×C₆₁	—	2	2	no	—
56	C₂²⋊Q₈×C₆₁	—	2	2	no	—
57	C₂²⋊Q₈×C₆₁	—	2	2	no	—
58	#121×C₆₁	—	2	2	no	—
59	#121×C₆₁	—	2	2	no	—
60	C₄×Q₈×C₆₁	—	2	2	no	—
61	C₄×Q₈×C₆₁	—	2	2	no	—
62	C₄×D₄×C₆₁	—	2	2	no	—
63	C₄×D₄×C₆₁	—	2	2	no	—
64	#117×C₆₁	—	2	2	no	—
65	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
66	#107×C₆₁	—	2	2	no	—
67	#107×C₆₁	—	2	2	no	—
68	#106×C₆₁	—	2	4	no	—
69	#103×C₆₁	—	2	4	no	—
70	#103×C₆₁	—	2	4	no	—
71	#103×C₆₁	—	2	4	no	—
72	#103×C₆₁	—	2	4	no	—
73	#102×C₆₁	—	2	4	no	—
74	#102×C₆₁	—	2	4	no	—
75	#102×C₆₁	—	2	4	no	—
76	#102×C₆₁	—	2	4	no	—
77	C₄×C₈×C₆₁	D₄	1	2	yes	—
78	C₄×C₈×C₆₁	D₄	1	2	yes	—
79	C₄×C₈×C₆₁	D₄	1	2	yes	—
80	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
86	#41×C₆₁	—	4	8	no	—
87	C₂×Q₈×C₆₁	—	2	2	no	—
88	C₂×Q₈×C₆₁	—	2	2	no	—
89	C₂×Q₈×C₆₁	—	2	2	no	—
90	C₂×D₄×C₆₁	—	2	2	no	—
91	C₂×D₄×C₆₁	—	2	2	no	—
92	C₂×D₄×C₆₁	—	2	2	no	—
93	C₂×D₄×C₆₁	—	4	2	no	—
94	C₂²×C₄×C₆₁	—	4	2	no	—
95	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₆₁	—	4	4	no	—
99	SD₁₆×C₆₁	—	4	8	no	—
100	D₈×C₆₁	—	4	4	no	—
101	C₂×C₈×C₆₁	—	2	4	no	—
102	C₂×C₈×C₆₁	—	2	4	no	—
103	C₂×C₈×C₆₁	—	2	4	no	—
104	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₆₁	—	2	2	no	—
111	C₄⋊C₄×C₆₁	—	2	2	no	—
112	C₄⋊C₄×C₆₁	—	2	2	no	—
113	C₄⋊C₄×C₆₁	—	2	2	no	—
114	C₄⋊C₄×C₆₁	—	2	2	no	—
115	C₄⋊C₄×C₆₁	—	2	2	no	—
116	C₄⋊C₄×C₆₁	—	4	2	no	—
117	C₄⋊C₄×C₆₁	—	4	2	no	—
118	C₂²⋊C₄×C₆₁	—	4	2	no	—
119	C₂²⋊C₄×C₆₁	—	4	2	no	—
120	C₂²⋊C₄×C₆₁	—	4	2	no	—
121	C₄²×C₆₁	—	4	2	no	—
122	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
126	C₂³×C₆₁	—	4	2	no	—
127	C₂³×C₆₁	#127	1	1	yes	characteristic
128	Q₈×C₆₁	—	4	4	no	—
129	Q₈×C₆₁	—	4	4	no	—
130	Q₈×C₆₁	—	4	4	no	—
131	D₄×C₆₁	—	4	4	no	—
132	D₄×C₆₁	—	4	4	no	—
133	D₄×C₆₁	—	4	4	no	—
134	D₄×C₆₁	—	8	4	no	—
135	C₂×C₄×C₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	4	2	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	—	4	2	no	—
140	C₂×C₄×C₆₁	—	8	4	no	—
141	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
142	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
143	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
144	C₂×C₄×C₆₁	#135	1	4	yes	—
145	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₆₁	—	2	2	no	—
149	C₈×C₆₁	—	2	2	no	—
150	C₈×C₆₁	—	2	2	no	—
151	C₈×C₆₁	—	2	2	no	—
152	C₈×C₆₁	#135	1	4	yes	—
153	C₈×C₆₁	#135	1	4	yes	—
154	#19600	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₆₁	—	2	1	no	—
156	C₂²×C₆₁	—	2	1	no	—
157	C₂²×C₆₁	—	8	2	no	—
158	C₂²×C₆₁	—	8	2	no	—
159	C₂²×C₆₁	—	8	2	no	—
160	C₂²×C₆₁	#495	1	2	yes	—
161	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
162	C₄×C₆₁	—	2	1	no	—
163	C₄×C₆₁	—	2	1	no	—
164	C₄×C₆₁	—	8	2	no	—
165	C₄×C₆₁	—	8	2	no	—
166	C₄×C₆₁	—	8	2	no	—
167	C₄×C₆₁	#495	1	2	yes	—
168	C₄×C₆₁	C₂×#135	1	2	yes	—
169	C₄×C₆₁	C₂×#135	1	2	yes	—
170	C₄×C₆₁	C₂×#135	1	2	yes	—
171	#3170	C₂×C₆₁	1	1	yes	characteristic, complemented
172	#3169	C₂×C₆₁	1	1	yes	characteristic
173	#3167	C₂×C₆₁	1	2	yes	complemented
174	#3131	C₂×C₆₁	1	2	yes	complemented
175	#1711	C₂×C₆₁	1	2	yes	complemented
176	#1710	C₂×C₆₁	1	1	yes	characteristic, complemented
177	#1709	C₂×C₆₁	1	1	yes	characteristic, complemented
178	#1708	C₂×C₆₁	1	2	yes	complemented
179	#1707	C₂×C₆₁	1	1	yes	characteristic, complemented
180	#1706	C₂×C₆₁	1	1	yes	characteristic, complemented
181	C₂×C₈²	C₂×C₆₁	1	1	yes	characteristic, complemented
182	C₂×C₆₁	—	2	2	no	—
183	C₂×C₆₁	—	16	2	no	—
184	C₂×C₆₁	C₂×#495	1	1	yes	characteristic
185	C₂×C₆₁	#3147	1	1	yes	characteristic
186	C₂×C₆₁	#3147	1	1	yes	characteristic
187	#533	C₂²×C₆₁	1	2	yes	—
188	#532	C₂²×C₆₁	1	2	yes	—
189	#500	C₂²×C₆₁	1	1	yes	characteristic
190	#500	C₂²×C₆₁	1	1	yes	characteristic
191	#498	C₂²×C₆₁	1	2	yes	—
192	#494	C₂²×C₆₁	1	1	yes	characteristic
193	#493	C₂²×C₆₁	1	1	yes	characteristic
194	#492	C₂²×C₆₁	1	2	yes	—
195	#488	—	2	4	no	—
196	#487	C₂²×C₆₁	1	2	yes	—
197	#487	C₂²×C₆₁	1	2	yes	—
198	#487	C₂²×C₆₁	1	2	yes	—
199	#487	C₂²×C₆₁	1	2	yes	—
200	#486	C₂²×C₆₁	1	2	yes	—
201	#486	C₂²×C₆₁	1	2	yes	—
202	#486	C₂²×C₆₁	1	2	yes	—
203	#486	C₂²×C₆₁	1	2	yes	—
204	#467	—	2	2	no	—
205	#466	—	2	2	no	—
206	#465	—	2	4	no	—
207	C₄×Q₁₆	—	2	2	no	—
208	C₄×SD₁₆	—	2	4	no	—
209	C₄×D₈	—	2	2	no	—
210	#416	—	2	4	no	—
211	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
212	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
214	C₈²	C₂²×C₆₁	1	1	yes	characteristic
215	C₈²	C₂²×C₆₁	1	1	yes	characteristic
216	C₈²	C₂²×C₆₁	1	1	yes	characteristic
217	C₈²	C₂²×C₆₁	1	1	yes	characteristic
218	C₆₁	#19600	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×#41	—	2	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	2	4	no	—
222	C₂×D₈	—	2	2	no	—
223	C₂²×C₈	D₄×C₆₁	1	2	yes	—
224	C₂²×C₈	D₄×C₆₁	1	2	yes	—
225	C₂²×C₈	D₄×C₆₁	1	2	yes	—
226	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
227	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
228	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
229	#127	C₂³×C₆₁	1	2	yes	—
230	#125	—	2	2	no	—
231	#124	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	#121	—	2	2	no	—
235	#121	—	2	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	2	2	no	—
240	#117	—	2	2	no	—
241	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
242	#107	—	2	2	no	—
243	#107	—	2	2	no	—
244	#106	—	2	4	no	—
245	#103	—	2	4	no	—
246	#103	—	2	4	no	—
247	#103	—	2	4	no	—
248	#103	—	2	4	no	—
249	#102	—	2	4	no	—
250	#102	—	2	4	no	—
251	#102	—	2	4	no	—
252	#102	—	2	4	no	—
253	C₄×C₈	D₄×C₆₁	1	2	yes	—
254	C₄×C₈	D₄×C₆₁	1	2	yes	—
255	C₄×C₈	D₄×C₆₁	1	2	yes	—
256	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
259	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
260	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
261	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
262	#41	—	4	8	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
274	Q₁₆	—	4	4	no	—
275	SD₁₆	—	4	8	no	—
276	D₈	—	4	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
282	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
283	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
284	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
301	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
302	C₂³	—	4	2	no	—
303	C₂³	#127×C₆₁	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
318	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
319	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
320	C₂×C₄	#135×C₆₁	1	4	yes	—
321	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	#135×C₆₁	1	4	yes	—
329	C₈	#135×C₆₁	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	C₆₁×#495	1	2	yes	—
336	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₆₁×#495	1	2	yes	—
343	C₄	C₂×#135×C₆₁	1	2	yes	—
344	C₄	C₂×#135×C₆₁	1	2	yes	—
345	C₄	C₂×#135×C₆₁	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₂×C₆₁×#495	1	1	yes	characteristic
349	C₂	C₆₁×#3147	1	1	yes	characteristic
350	C₂	C₆₁×#3147	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂×#127
prime radical	C₆₁×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3170	C₂	semidirect
3. C₆₁×#3167	C₂	semidirect
4. C₆₁×#3131	C₂	semidirect
5. C₆₁×#1711	C₂	semidirect
6. C₆₁×#1710	C₂	semidirect
7. C₆₁×#1709	C₂	semidirect
8. C₆₁×#1708	C₂	semidirect
9. C₆₁×#1707	C₂	semidirect
10. C₆₁×#1706	C₂	semidirect
11. C₂×C₈²×C₆₁	C₂	semidirect
154. #19600	C₆₁	direct (N × H)
171. #3170	C₂×C₆₁	semidirect
173. #3167	C₂×C₆₁	semidirect
174. #3131	C₂×C₆₁	semidirect
175. #1711	C₂×C₆₁	semidirect
176. #1710	C₂×C₆₁	semidirect
177. #1709	C₂×C₆₁	semidirect
178. #1708	C₂×C₆₁	semidirect
179. #1707	C₂×C₆₁	semidirect
180. #1706	C₂×C₆₁	semidirect
181. C₂×C₈²	C₂×C₆₁	semidirect
218. C₆₁	#19600	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3170	2
C₆₁×#3169	2
C₆₁×#3167	2
C₆₁×#3131	2
C₆₁×#1711	2
C₆₁×#1710	2
C₆₁×#1709	2
C₆₁×#1708	2
C₆₁×#1707	2
C₆₁×#1706	2
C₂×C₈²×C₆₁	2
#19600	61

Subgroups — C61×#19600 #29523033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19600 #29523033