Subgroups — C₆₁×#19689 #29523122

Order 15616.

832 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3207	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3184	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1721	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1721	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1720	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1714	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₆₁	—	2	2	no	—
11	C₆₁×#532	C₂²	1	2	yes	—
12	C₆₁×#532	C₂²	1	2	yes	—
13	C₆₁×#500	C₂²	1	1	yes	characteristic
14	C₆₁×#498	C₂²	1	2	yes	—
15	C₆₁×#495	—	2	2	no	—
16	C₆₁×#494	C₂²	1	1	yes	characteristic
17	C₆₁×#493	C₂²	1	1	yes	characteristic
18	C₆₁×#493	C₂²	1	1	yes	characteristic
19	C₆₁×#493	C₂²	1	2	yes	—
20	C₆₁×#493	C₂²	1	2	yes	—
21	C₆₁×#492	C₂²	1	2	yes	—
22	C₆₁×#492	C₂²	1	2	yes	—
23	C₆₁×#487	C₂²	1	2	yes	—
24	C₆₁×#486	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×#469	—	2	4	no	—
29	C₆₁×#466	—	2	2	no	—
30	C₆₁×#463	—	2	4	no	—
31	C₆₁×#449	—	2	4	no	—
32	C₆₁×#442	—	2	4	no	—
33	C₄×D₈×C₆₁	—	2	2	no	—
34	C₆₁×#431	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#322	C₂²	1	2	yes	—
38	C₆₁×#322	C₂²	1	2	yes	—
39	C₂×#41×C₆₁	—	2	2	no	—
40	C₂×#41×C₆₁	—	2	2	no	—
41	#135×C₆₁	—	4	8	no	—
42	C₂×Q₁₆×C₆₁	—	2	2	no	—
43	C₂×SD₁₆×C₆₁	—	2	4	no	—
44	C₂×SD₁₆×C₆₁	—	2	4	no	—
45	C₂×D₈×C₆₁	—	2	2	no	—
46	C₂×D₈×C₆₁	—	2	2	no	—
47	C₂×D₈×C₆₁	—	2	4	no	—
48	C₂×D₈×C₆₁	—	2	4	no	—
49	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
50	C₂²×C₈×C₆₁	D₄	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	#127×C₆₁	C₂³	1	2	yes	—
55	#124×C₆₁	—	2	2	no	—
56	#124×C₆₁	—	2	2	no	—
57	#121×C₆₁	—	2	2	no	—
58	#121×C₆₁	—	2	2	no	—
59	#121×C₆₁	—	2	2	no	—
60	#121×C₆₁	—	2	2	no	—
61	C₄×D₄×C₆₁	—	2	2	no	—
62	C₄×D₄×C₆₁	—	2	2	no	—
63	C₄×D₄×C₆₁	—	2	2	no	—
64	C₄×D₄×C₆₁	—	2	2	no	—
65	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
66	#107×C₆₁	—	2	2	no	—
67	#106×C₆₁	—	2	4	no	—
68	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
69	#103×C₆₁	—	2	4	no	—
70	#102×C₆₁	—	2	4	no	—
71	#102×C₆₁	—	2	4	no	—
72	#102×C₆₁	—	2	4	no	—
73	#102×C₆₁	—	2	4	no	—
74	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
75	#97×C₆₁	D₄	1	2	yes	—
76	C₄×C₈×C₆₁	D₄	1	2	yes	—
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₆₁	C₂³	1	2	yes	—
80	C₄×C₈×C₆₁	C₂³	1	2	yes	—
81	#41×C₆₁	—	4	8	no	—
82	#41×C₆₁	—	8	4	no	—
83	C₂×Q₈×C₆₁	—	2	2	no	—
84	C₂×Q₈×C₆₁	—	2	2	no	—
85	C₂×D₄×C₆₁	—	2	2	no	—
86	C₂×D₄×C₆₁	—	2	2	no	—
87	C₂×D₄×C₆₁	—	2	2	no	—
88	C₂×D₄×C₆₁	—	2	2	no	—
89	C₂×D₄×C₆₁	—	2	2	no	—
90	C₂×D₄×C₆₁	—	2	2	no	—
91	C₂×D₄×C₆₁	—	4	2	no	—
92	C₂×D₄×C₆₁	—	4	2	no	—
93	C₂²×C₄×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	SD₁₆×C₆₁	—	8	4	no	—
101	D₈×C₆₁	—	4	8	no	—
102	D₈×C₆₁	—	4	4	no	—
103	D₈×C₆₁	—	4	4	no	—
104	D₈×C₆₁	—	8	4	no	—
105	M₁₆×C₆₁	—	4	2	no	—
106	C₂×C₈×C₆₁	—	2	4	no	—
107	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
109	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
110	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
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	C₂²⋊C₄×C₆₁	—	4	2	no	—
116	C₂²⋊C₄×C₆₁	—	4	2	no	—
117	C₂²⋊C₄×C₆₁	—	4	2	no	—
118	C₂²⋊C₄×C₆₁	—	4	2	no	—
119	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
123	C₂³×C₆₁	—	4	2	no	—
124	C₂³×C₆₁	—	4	2	no	—
125	C₂³×C₆₁	#127	1	1	yes	characteristic
126	Q₈×C₆₁	—	4	4	no	—
127	Q₈×C₆₁	—	4	4	no	—
128	D₄×C₆₁	—	4	4	no	—
129	D₄×C₆₁	—	4	4	no	—
130	D₄×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	D₄×C₆₁	—	8	4	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	2	2	no	—
139	C₂×C₄×C₆₁	—	4	2	no	—
140	C₂×C₄×C₆₁	—	4	2	no	—
141	C₂×C₄×C₆₁	—	8	4	no	—
142	C₂×C₄×C₆₁	—	8	4	no	—
143	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
144	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
145	C₂×C₄×C₆₁	#127	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	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₆₁	—	2	2	no	—
150	C₈×C₆₁	—	2	4	no	—
151	C₈×C₆₁	—	2	4	no	—
152	C₈×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
153	#19689	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂²×C₆₁	—	2	1	no	—
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	—	8	2	no	—
161	C₂²×C₆₁	—	8	2	no	—
162	C₂²×C₆₁	—	8	2	no	—
163	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
164	C₄×C₆₁	—	2	1	no	—
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	—	2	1	no	—
167	C₄×C₆₁	—	8	2	no	—
168	C₄×C₆₁	—	8	2	no	—
169	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
170	C₄×C₆₁	#575	1	2	yes	—
171	C₄×C₆₁	#576	1	2	yes	—
172	#3207	C₂×C₆₁	1	2	yes	complemented
173	#3184	C₂×C₆₁	1	2	yes	complemented
174	#3169	C₂×C₆₁	1	1	yes	characteristic, complemented
175	#3147	C₂×C₆₁	1	1	yes	characteristic, complemented
176	#1721	C₂×C₆₁	1	2	yes	complemented
177	#1721	C₂×C₆₁	1	2	yes	complemented
178	#1720	C₂×C₆₁	1	2	yes	complemented
179	#1714	C₂×C₆₁	1	2	yes	complemented
180	#1451	C₂×C₆₁	1	1	yes	characteristic, complemented
181	C₂×C₆₁	—	4	1	no	—
182	C₂×C₆₁	—	16	2	no	—
183	C₂×C₆₁	—	16	2	no	—
184	C₂×C₆₁	#3150	1	1	yes	characteristic
185	C₂×C₆₁	#3151	1	1	yes	characteristic
186	C₂×C₆₁	#3155	1	1	yes	characteristic
187	C₂×#135	—	2	2	no	—
188	#532	C₂²×C₆₁	1	2	yes	—
189	#532	C₂²×C₆₁	1	2	yes	—
190	#500	C₂²×C₆₁	1	1	yes	characteristic
191	#498	C₂²×C₆₁	1	2	yes	—
192	#495	—	2	2	no	—
193	#494	C₂²×C₆₁	1	1	yes	characteristic
194	#493	C₂²×C₆₁	1	1	yes	characteristic
195	#493	C₂²×C₆₁	1	1	yes	characteristic
196	#493	C₂²×C₆₁	1	2	yes	—
197	#493	C₂²×C₆₁	1	2	yes	—
198	#492	C₂²×C₆₁	1	2	yes	—
199	#492	C₂²×C₆₁	1	2	yes	—
200	#487	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	#486	C₂²×C₆₁	1	2	yes	—
205	#469	—	2	4	no	—
206	#466	—	2	2	no	—
207	#463	—	2	4	no	—
208	#449	—	2	4	no	—
209	#442	—	2	4	no	—
210	C₄×D₈	—	2	2	no	—
211	#431	C₂²×C₆₁	1	1	yes	characteristic
212	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
214	#322	C₂²×C₆₁	1	2	yes	—
215	#322	C₂²×C₆₁	1	2	yes	—
216	C₆₁	#19689	1	1	yes	characteristic, direct product factor, Sylow
217	C₂×#41	—	2	2	no	—
218	C₂×#41	—	2	2	no	—
219	#135	—	4	8	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×SD₁₆	—	2	4	no	—
222	C₂×SD₁₆	—	2	4	no	—
223	C₂×D₈	—	2	2	no	—
224	C₂×D₈	—	2	2	no	—
225	C₂×D₈	—	2	4	no	—
226	C₂×D₈	—	2	4	no	—
227	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
228	C₂²×C₈	D₄×C₆₁	1	2	yes	—
229	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
230	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
231	#127	C₂³×C₆₁	1	2	yes	—
232	#127	C₂³×C₆₁	1	2	yes	—
233	#124	—	2	2	no	—
234	#124	—	2	2	no	—
235	#121	—	2	2	no	—
236	#121	—	2	2	no	—
237	#121	—	2	2	no	—
238	#121	—	2	2	no	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	—	2	2	no	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
244	#107	—	2	2	no	—
245	#106	—	2	4	no	—
246	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
247	#103	—	2	4	no	—
248	#102	—	2	4	no	—
249	#102	—	2	4	no	—
250	#102	—	2	4	no	—
251	#102	—	2	4	no	—
252	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
253	#97	D₄×C₆₁	1	2	yes	—
254	C₄×C₈	D₄×C₆₁	1	2	yes	—
255	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₆₁	1	2	yes	—
258	C₄×C₈	C₂³×C₆₁	1	2	yes	—
259	#41	—	4	8	no	—
260	#41	—	8	4	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	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	C₂×D₄	—	4	2	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
276	Q₁₆	—	4	4	no	—
277	SD₁₆	—	4	8	no	—
278	SD₁₆	—	8	4	no	—
279	D₈	—	4	8	no	—
280	D₈	—	4	4	no	—
281	D₈	—	4	4	no	—
282	D₈	—	8	4	no	—
283	M₁₆	—	4	2	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
286	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
287	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
288	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
299	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
300	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	#127×C₆₁	1	1	yes	characteristic
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
322	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
323	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₈	AGL(1,ℤ/8)×C₆₁	1	4	yes	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
347	C₄	C₆₁×#575	1	2	yes	—
348	C₄	C₆₁×#576	1	2	yes	—
349	C₂	—	4	1	no	—
350	C₂	—	16	2	no	—
351	C₂	—	16	2	no	—
352	C₂	C₆₁×#3150	1	1	yes	characteristic
353	C₂	C₆₁×#3151	1	1	yes	characteristic
354	C₂	C₆₁×#3155	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3207	C₂	semidirect
2. C₆₁×#3184	C₂	semidirect
3. C₆₁×#3169	C₂	semidirect
4. C₆₁×#3147	C₂	semidirect
5. C₆₁×#1721	C₂	semidirect
6. C₆₁×#1721	C₂	semidirect
7. C₆₁×#1720	C₂	semidirect
8. C₆₁×#1714	C₂	semidirect
9. C₆₁×#1451	C₂	semidirect
153. #19689	C₆₁	direct (N × H)
172. #3207	C₂×C₆₁	semidirect
173. #3184	C₂×C₆₁	semidirect
174. #3169	C₂×C₆₁	semidirect
175. #3147	C₂×C₆₁	semidirect
176. #1721	C₂×C₆₁	semidirect
177. #1721	C₂×C₆₁	semidirect
178. #1720	C₂×C₆₁	semidirect
179. #1714	C₂×C₆₁	semidirect
180. #1451	C₂×C₆₁	semidirect
216. C₆₁	#19689	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×#3207	2
C₆₁×#3184	2
C₆₁×#3169	2
C₆₁×#3147	2
C₆₁×#1721	2
C₆₁×#1720	2
C₆₁×#1714	2
C₆₁×#1451	2
#19689	61

Subgroups — C61×#19689 #29523122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19689 #29523122