Subgroups — C₆₁×#25691 #29529124

Order 15616.

920 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3509	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3166	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3080	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2642	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2038	C₂	1	4	yes	maximal
6	C₆₁×#1995	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1857	C₂	1	4	yes	maximal, complemented
9	C₆₁×#1759	C₂	1	1	yes	maximal, characteristic
10	C₆₁×#557	—	2	2	no	—
11	C₆₁×#556	—	2	2	no	—
12	C₆₁×#552	—	2	4	no	—
13	C₆₁×#541	—	2	2	no	—
14	C₆₁×#536	C₂²	1	2	yes	—
15	C₆₁×#533	—	2	2	no	—
16	C₆₁×#533	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#519	C₂²	1	2	yes	—
20	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
21	C₆₁×#501	—	2	2	no	—
22	C₆₁×#499	—	2	2	no	—
23	C₆₁×#479	—	2	2	no	—
24	C₆₁×#478	—	2	2	no	—
25	C₆₁×#428	C₂²	1	4	yes	—
26	C₆₁×#428	C₂²	1	2	yes	—
27	C₆₁×#423	C₂²	1	1	yes	characteristic
28	C₂×#104×C₆₁	C₂²	1	2	yes	—
29	C₆₁×#417	C₂²	1	2	yes	—
30	C₂×#101×C₆₁	C₂²	1	4	yes	—
31	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#400	—	2	4	no	—
33	C₆₁×#391	C₂²	1	4	yes	—
34	C₆₁×#376	C₂²	1	1	yes	characteristic
35	C₆₁×#339	C₂²	1	2	yes	—
36	C₆₁×#339	C₂²	1	4	yes	—
37	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
38	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
39	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₆₁	—	2	2	no	—
42	C₄⋊Q₈×C₆₁	—	2	1	no	—
43	C₄⋊Q₈×C₆₁	—	2	1	no	—
44	C₄⋊Q₈×C₆₁	—	2	2	no	—
45	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
46	#126×C₆₁	—	2	8	no	—
47	#125×C₆₁	—	2	4	no	—
48	#125×C₆₁	—	2	8	no	—
49	#125×C₆₁	—	2	2	no	—
50	#125×C₆₁	—	2	1	no	—
51	#125×C₆₁	—	2	1	no	—
52	#125×C₆₁	D₄	1	2	yes	—
53	#124×C₆₁	—	2	2	no	—
54	#123×C₆₁	—	2	4	no	—
55	C₂²⋊Q₈×C₆₁	—	2	4	no	—
56	C₂²⋊Q₈×C₆₁	—	2	2	no	—
57	C₂²⋊Q₈×C₆₁	—	2	4	no	—
58	C₄×Q₈×C₆₁	—	2	4	no	—
59	C₄×Q₈×C₆₁	—	2	4	no	—
60	C₄×Q₈×C₆₁	—	2	4	no	—
61	C₄×Q₈×C₆₁	—	2	2	no	—
62	C₄×Q₈×C₆₁	D₄	1	2	yes	—
63	C₄×D₄×C₆₁	—	2	2	no	—
64	C₄×D₄×C₆₁	D₄	1	2	yes	—
65	#117×C₆₁	—	2	4	no	—
66	#117×C₆₁	—	2	4	no	—
67	#117×C₆₁	—	2	1	no	—
68	#117×C₆₁	C₂³	1	2	yes	—
69	#117×C₆₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₁	D₄	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
73	C₂×C₄²×C₆₁	—	2	1	no	—
74	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
76	#107×C₆₁	—	2	2	no	—
77	#107×C₆₁	—	2	4	no	—
78	#106×C₆₁	—	2	2	no	—
79	#106×C₆₁	—	2	4	no	—
80	C₄⋊C₈×C₆₁	—	2	2	no	—
81	#104×C₆₁	—	2	4	no	—
82	#103×C₆₁	—	2	2	no	—
83	#102×C₆₁	—	2	2	no	—
84	#101×C₆₁	—	2	8	no	—
85	#97×C₆₁	—	2	1	no	—
86	C₄×C₈×C₆₁	—	2	1	no	—
87	#95×C₆₁	—	2	2	no	—
88	#95×C₆₁	—	2	8	no	—
89	#41×C₆₁	—	2	2	no	—
90	#41×C₆₁	—	2	2	no	—
91	C₂×Q₈×C₆₁	—	2	2	no	—
92	C₂×Q₈×C₆₁	—	2	2	no	—
93	C₂×Q₈×C₆₁	—	2	2	no	—
94	C₂×Q₈×C₆₁	—	2	4	no	—
95	C₂×Q₈×C₆₁	—	2	4	no	—
96	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×D₄×C₆₁	—	2	1	no	—
98	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₁	—	2	2	no	—
100	C₂²×C₄×C₆₁	—	2	1	no	—
101	C₂²×C₄×C₆₁	—	2	2	no	—
102	C₂²×C₄×C₆₁	—	2	2	no	—
103	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
104	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
106	M₁₆×C₆₁	—	2	2	no	—
107	M₁₆×C₆₁	—	2	4	no	—
108	M₁₆×C₆₁	#41	1	4	yes	—
109	C₂×C₈×C₆₁	—	2	2	no	—
110	C₂×C₈×C₆₁	—	2	2	no	—
111	C₄⋊C₄×C₆₁	—	2	8	no	—
112	C₄⋊C₄×C₆₁	—	2	8	no	—
113	C₄⋊C₄×C₆₁	—	2	4	no	—
114	C₄⋊C₄×C₆₁	—	2	4	no	—
115	C₄⋊C₄×C₆₁	—	2	2	no	—
116	C₄⋊C₄×C₆₁	—	2	2	no	—
117	C₄⋊C₄×C₆₁	—	2	4	no	—
118	C₄⋊C₄×C₆₁	—	2	2	no	—
119	C₄⋊C₄×C₆₁	—	2	2	no	—
120	C₄⋊C₄×C₆₁	—	4	2	no	—
121	C₄⋊C₄×C₆₁	—	4	2	no	—
122	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
124	C₂²⋊C₄×C₆₁	—	2	8	no	—
125	C₂²⋊C₄×C₆₁	—	2	1	no	—
126	C₂²⋊C₄×C₆₁	—	2	2	no	—
127	C₂²⋊C₄×C₆₁	—	2	2	no	—
128	C₄²×C₆₁	—	2	8	no	—
129	C₄²×C₆₁	—	2	2	no	—
130	C₄²×C₆₁	—	2	1	no	—
131	C₄²×C₆₁	—	2	2	no	—
132	C₄²×C₆₁	—	2	2	no	—
133	C₄²×C₆₁	—	2	2	no	—
134	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₆₁	C₂×D₄	1	2	yes	—
137	C₂³×C₆₁	—	2	1	no	—
138	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	Q₈×C₆₁	—	2	1	no	—
140	Q₈×C₆₁	—	2	1	no	—
141	Q₈×C₆₁	—	4	4	no	—
142	Q₈×C₆₁	—	4	2	no	—
143	Q₈×C₆₁	—	4	2	no	—
144	Q₈×C₆₁	—	4	4	no	—
145	D₄×C₆₁	—	2	1	no	—
146	D₄×C₆₁	—	2	1	no	—
147	D₄×C₆₁	—	4	2	no	—
148	C₂×C₄×C₆₁	—	2	2	no	—
149	C₂×C₄×C₆₁	—	2	4	no	—
150	C₂×C₄×C₆₁	—	2	4	no	—
151	C₂×C₄×C₆₁	—	2	1	no	—
152	C₂×C₄×C₆₁	—	2	8	no	—
153	C₂×C₄×C₆₁	—	4	2	no	—
154	C₂×C₄×C₆₁	—	4	4	no	—
155	C₂×C₄×C₆₁	—	4	2	no	—
156	C₂×C₄×C₆₁	—	4	4	no	—
157	C₂×C₄×C₆₁	—	4	4	no	—
158	C₂×C₄×C₆₁	—	4	2	no	—
159	C₂×C₄×C₆₁	—	4	2	no	—
160	C₂×C₄×C₆₁	#120	1	2	yes	—
161	C₂×C₄×C₆₁	#120	1	2	yes	—
162	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
165	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
166	C₈×C₆₁	—	2	4	no	—
167	C₈×C₆₁	—	4	2	no	—
168	#25691	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂²×C₆₁	—	2	2	no	—
170	C₂²×C₆₁	—	4	1	no	—
171	C₂²×C₆₁	—	4	2	no	—
172	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
173	C₂²×C₆₁	#547	1	2	yes	—
174	C₄×C₆₁	—	2	2	no	—
175	C₄×C₆₁	—	2	2	no	—
176	C₄×C₆₁	—	4	4	no	—
177	C₄×C₆₁	—	4	1	no	—
178	C₄×C₆₁	—	4	2	no	—
179	C₄×C₆₁	—	4	4	no	—
180	C₄×C₆₁	—	8	2	no	—
181	C₄×C₆₁	—	8	2	no	—
182	C₄×C₆₁	#532	1	1	yes	characteristic
183	C₄×C₆₁	#536	1	1	yes	characteristic
184	C₄×C₆₁	#548	1	2	yes	—
185	#3509	C₂×C₆₁	1	1	yes	characteristic
186	#3166	C₂×C₆₁	1	1	yes	characteristic, complemented
187	#3080	C₂×C₆₁	1	1	yes	characteristic
188	#2642	C₂×C₆₁	1	1	yes	characteristic, complemented
189	#2038	C₂×C₆₁	1	4	yes	—
190	#1995	C₂×C₆₁	1	1	yes	characteristic, complemented
191	#1966	C₂×C₆₁	1	1	yes	characteristic, complemented
192	#1857	C₂×C₆₁	1	4	yes	complemented
193	#1759	C₂×C₆₁	1	1	yes	characteristic
194	C₂×C₆₁	—	2	2	no	—
195	C₂×C₆₁	—	8	1	no	—
196	C₂×C₆₁	#2432	1	1	yes	characteristic
197	C₂×C₆₁	#3020	1	2	yes	—
198	#557	—	2	2	no	—
199	#556	—	2	2	no	—
200	#552	—	2	4	no	—
201	#541	—	2	2	no	—
202	#536	C₂²×C₆₁	1	2	yes	—
203	#533	—	2	2	no	—
204	#533	C₂²×C₆₁	1	1	yes	characteristic
205	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
206	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
207	#519	C₂²×C₆₁	1	2	yes	—
208	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
209	#501	—	2	2	no	—
210	#499	—	2	2	no	—
211	#479	—	2	2	no	—
212	#478	—	2	2	no	—
213	#428	C₂²×C₆₁	1	2	yes	—
214	#428	C₂²×C₆₁	1	4	yes	—
215	#423	C₂²×C₆₁	1	1	yes	characteristic
216	C₂×#104	C₂²×C₆₁	1	2	yes	—
217	#417	C₂²×C₆₁	1	2	yes	—
218	C₂×#101	C₂²×C₆₁	1	4	yes	—
219	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
220	#400	—	2	4	no	—
221	#391	C₂²×C₆₁	1	4	yes	—
222	#376	C₂²×C₆₁	1	1	yes	characteristic
223	#339	C₂²×C₆₁	1	2	yes	—
224	#339	C₂²×C₆₁	1	4	yes	—
225	C₆₁	#25691	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
227	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
228	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
229	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
230	C₄⋊Q₈	—	2	2	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	2	no	—
234	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
235	#126	—	2	8	no	—
236	#125	—	2	4	no	—
237	#125	—	2	8	no	—
238	#125	—	2	2	no	—
239	#125	—	2	1	no	—
240	#125	—	2	1	no	—
241	#125	D₄×C₆₁	1	2	yes	—
242	#124	—	2	2	no	—
243	#123	—	2	4	no	—
244	C₂²⋊Q₈	—	2	4	no	—
245	C₂²⋊Q₈	—	2	2	no	—
246	C₂²⋊Q₈	—	2	4	no	—
247	C₄×Q₈	—	2	4	no	—
248	C₄×Q₈	—	2	4	no	—
249	C₄×Q₈	—	2	4	no	—
250	C₄×Q₈	—	2	2	no	—
251	C₄×Q₈	D₄×C₆₁	1	2	yes	—
252	C₄×D₄	—	2	2	no	—
253	C₄×D₄	D₄×C₆₁	1	2	yes	—
254	#117	—	2	4	no	—
255	#117	—	2	4	no	—
256	#117	—	2	1	no	—
257	#117	C₂³×C₆₁	1	1	yes	characteristic
258	#117	C₂³×C₆₁	1	2	yes	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	D₄×C₆₁	1	4	yes	—
261	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
262	C₂×C₄²	—	2	1	no	—
263	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
264	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
265	#107	—	2	2	no	—
266	#107	—	2	4	no	—
267	#106	—	2	2	no	—
268	#106	—	2	4	no	—
269	C₄⋊C₈	—	2	2	no	—
270	#104	—	2	4	no	—
271	#103	—	2	2	no	—
272	#102	—	2	2	no	—
273	#101	—	2	8	no	—
274	#97	—	2	1	no	—
275	C₄×C₈	—	2	1	no	—
276	#95	—	2	2	no	—
277	#95	—	2	8	no	—
278	#41	—	2	2	no	—
279	#41	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	2	2	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	2	4	no	—
284	C₂×Q₈	—	2	4	no	—
285	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
293	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
294	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
295	M₁₆	—	2	2	no	—
296	M₁₆	—	2	4	no	—
297	M₁₆	#41×C₆₁	1	4	yes	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₄⋊C₄	—	2	8	no	—
301	C₄⋊C₄	—	2	8	no	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	—	2	4	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	4	2	no	—
310	C₄⋊C₄	—	4	2	no	—
311	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
312	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
313	C₂²⋊C₄	—	2	8	no	—
314	C₂²⋊C₄	—	2	1	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₄²	—	2	8	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	1	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
324	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
325	C₄²	C₂×D₄×C₆₁	1	2	yes	—
326	C₂³	—	2	1	no	—
327	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
328	Q₈	—	2	1	no	—
329	Q₈	—	2	1	no	—
330	Q₈	—	4	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	4	no	—
334	D₄	—	2	1	no	—
335	D₄	—	2	1	no	—
336	D₄	—	4	2	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	#120×C₆₁	1	2	yes	—
350	C₂×C₄	#120×C₆₁	1	2	yes	—
351	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
352	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
353	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
354	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
355	C₈	—	2	4	no	—
356	C₈	—	4	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
361	C₂²	C₆₁×#547	1	2	yes	—
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	4	4	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	4	no	—
368	C₄	—	8	2	no	—
369	C₄	—	8	2	no	—
370	C₄	C₆₁×#532	1	1	yes	characteristic
371	C₄	C₆₁×#536	1	1	yes	characteristic
372	C₄	C₆₁×#548	1	2	yes	—
373	C₂	—	2	2	no	—
374	C₂	—	8	1	no	—
375	C₂	C₆₁×#2432	1	1	yes	characteristic
376	C₂	C₆₁×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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
2. C₆₁×#3166	C₂	semidirect
4. C₆₁×#2642	C₂	semidirect
6. C₆₁×#1995	C₂	semidirect
7. C₆₁×#1966	C₂	semidirect
8. C₆₁×#1857	C₂	semidirect
168. #25691	C₆₁	direct (N × H)
186. #3166	C₂×C₆₁	semidirect
188. #2642	C₂×C₆₁	semidirect
190. #1995	C₂×C₆₁	semidirect
191. #1966	C₂×C₆₁	semidirect
192. #1857	C₂×C₆₁	semidirect
225. C₆₁	#25691	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×#3509	2
C₆₁×#3166	2
C₆₁×#3080	2
C₆₁×#2642	2
C₆₁×#2038	2
C₆₁×#1995	2
C₆₁×#1966	2
C₆₁×#1857	2
C₆₁×#1759	2
#25691	61

Subgroups — C61×#25691 #29529124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25691 #29529124