Subgroups — C₆₁×#10292 #29513725

Order 15616.

640 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×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3208	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2858	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1545	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#533	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×#124×C₆₁	—	2	2	no	—
12	C₆₁×#500	—	2	1	no	—
13	C₆₁×#492	—	2	1	no	—
14	C₆₁×#487	—	2	1	no	—
15	C₆₁×#486	—	2	1	no	—
16	C₆₁×#484	—	2	2	no	—
17	C₆₁×#477	—	2	2	no	—
18	C₆₁×#460	—	2	2	no	—
19	C₆₁×#450	—	2	2	no	—
20	C₆₁×#431	C₂²	1	1	yes	characteristic
21	C₆₁×#431	C₂²	1	1	yes	characteristic
22	C₆₁×#423	C₂²	1	1	yes	characteristic
23	C₆₁×#390	—	2	2	no	—
24	C₆₁×#390	—	2	2	no	—
25	C₄³×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#339	—	2	2	no	—
27	C₆₁×#332	—	2	2	no	—
28	C₆₁×#331	—	2	2	no	—
29	C₂²×Q₈×C₆₁	—	2	1	no	—
30	C₂²×D₄×C₆₁	—	2	1	no	—
31	C₂²×D₄×C₆₁	—	2	1	no	—
32	C₂²×D₄×C₆₁	D₄	1	2	yes	—
33	C₂×SD₁₆×C₆₁	—	2	4	no	—
34	C₂×M₁₆×C₆₁	—	2	1	no	—
35	C₄⋊Q₈×C₆₁	—	2	1	no	—
36	C₄⋊Q₈×C₆₁	—	2	1	no	—
37	C₄⋊Q₈×C₆₁	—	2	1	no	—
38	C₄⋊Q₈×C₆₁	—	2	1	no	—
39	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
41	#127×C₆₁	—	2	1	no	—
42	#127×C₆₁	—	2	1	no	—
43	#127×C₆₁	—	2	1	no	—
44	#127×C₆₁	—	2	1	no	—
45	#125×C₆₁	—	2	1	no	—
46	#124×C₆₁	—	4	8	no	—
47	C₂²⋊Q₈×C₆₁	—	2	1	no	—
48	C₂²⋊Q₈×C₆₁	—	2	1	no	—
49	C₄×Q₈×C₆₁	—	2	1	no	—
50	C₄×Q₈×C₆₁	—	2	1	no	—
51	#117×C₆₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₆₁	—	2	8	no	—
55	C₂×C₄²×C₆₁	—	2	1	no	—
56	C₂×C₄²×C₆₁	—	2	1	no	—
57	C₂×C₄²×C₆₁	D₄	1	2	yes	—
58	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
59	#107×C₆₁	—	2	4	no	—
60	C₄⋊C₈×C₆₁	—	2	1	no	—
61	C₄⋊C₈×C₆₁	—	2	1	no	—
62	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
63	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
64	#103×C₆₁	—	2	4	no	—
65	#102×C₆₁	—	2	4	no	—
66	C₂²⋊C₈×C₆₁	—	2	1	no	—
67	C₂²⋊C₈×C₆₁	—	2	1	no	—
68	C₄×C₈×C₆₁	—	2	1	no	—
69	C₄×C₈×C₆₁	—	2	1	no	—
70	C₂⁴×C₆₁	—	2	2	no	—
71	C₂×Q₈×C₆₁	—	2	1	no	—
72	C₂×Q₈×C₆₁	—	2	1	no	—
73	C₂×Q₈×C₆₁	—	4	4	no	—
74	C₂×Q₈×C₆₁	—	4	1	no	—
75	C₂×Q₈×C₆₁	—	4	1	no	—
76	C₂×D₄×C₆₁	—	2	2	no	—
77	C₂×D₄×C₆₁	—	2	2	no	—
78	C₂×D₄×C₆₁	—	4	4	no	—
79	C₂×D₄×C₆₁	—	4	4	no	—
80	C₂×D₄×C₆₁	—	4	1	no	—
81	C₂×D₄×C₆₁	—	4	1	no	—
82	C₂×D₄×C₆₁	—	4	1	no	—
83	C₂×D₄×C₆₁	—	4	1	no	—
84	C₂×D₄×C₆₁	—	4	2	no	—
85	C₂×D₄×C₆₁	—	4	2	no	—
86	C₂²×C₄×C₆₁	—	2	2	no	—
87	C₂²×C₄×C₆₁	—	2	2	no	—
88	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
91	SD₁₆×C₆₁	—	8	4	no	—
92	M₁₆×C₆₁	—	4	2	no	—
93	C₂×C₈×C₆₁	—	2	2	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	4	1	no	—
96	C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	2	1	no	—
99	C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₄⋊C₄×C₆₁	—	2	1	no	—
101	C₄⋊C₄×C₆₁	—	2	1	no	—
102	C₄⋊C₄×C₆₁	—	4	1	no	—
103	C₄⋊C₄×C₆₁	—	4	1	no	—
104	C₄⋊C₄×C₆₁	—	4	1	no	—
105	C₄⋊C₄×C₆₁	—	4	1	no	—
106	C₂²⋊C₄×C₆₁	—	4	16	no	—
107	C₂²⋊C₄×C₆₁	—	4	1	no	—
108	C₄²×C₆₁	—	2	4	no	—
109	C₄²×C₆₁	—	2	2	no	—
110	C₄²×C₆₁	—	2	2	no	—
111	C₄²×C₆₁	—	4	2	no	—
112	C₄²×C₆₁	—	4	1	no	—
113	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₁	#41	1	1	yes	characteristic
117	C₂³×C₆₁	—	4	4	no	—
118	C₂³×C₆₁	—	4	4	no	—
119	C₂³×C₆₁	—	4	2	no	—
120	C₂³×C₆₁	—	4	4	no	—
121	C₂³×C₆₁	#120	1	1	yes	characteristic
122	Q₈×C₆₁	—	4	2	no	—
123	Q₈×C₆₁	—	4	2	no	—
124	Q₈×C₆₁	—	8	2	no	—
125	Q₈×C₆₁	—	8	2	no	—
126	D₄×C₆₁	—	4	4	no	—
127	D₄×C₆₁	—	8	2	no	—
128	D₄×C₆₁	—	8	2	no	—
129	D₄×C₆₁	—	8	2	no	—
130	D₄×C₆₁	—	8	2	no	—
131	D₄×C₆₁	—	8	1	no	—
132	D₄×C₆₁	—	8	1	no	—
133	C₂×C₄×C₆₁	—	2	2	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	4	no	—
136	C₂×C₄×C₆₁	—	4	2	no	—
137	C₂×C₄×C₆₁	—	4	2	no	—
138	C₂×C₄×C₆₁	—	4	4	no	—
139	C₂×C₄×C₆₁	—	4	1	no	—
140	C₂×C₄×C₆₁	—	4	1	no	—
141	C₂×C₄×C₆₁	—	4	2	no	—
142	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
143	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
144	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
146	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
147	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
148	C₂×C₄×C₆₁	#135	1	2	yes	—
149	C₂×C₄×C₆₁	#137	1	2	yes	—
150	C₈×C₆₁	—	4	2	no	—
151	C₈×C₆₁	—	4	2	no	—
152	C₈×C₆₁	—	8	1	no	—
153	#10292	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	—	8	8	no	—
157	C₂²×C₆₁	—	8	2	no	—
158	C₂²×C₆₁	—	8	2	no	—
159	C₂²×C₆₁	—	8	2	no	—
160	C₂²×C₆₁	#392	1	1	yes	characteristic
161	C₂²×C₆₁	#452	1	2	yes	—
162	C₄×C₆₁	—	2	1	no	—
163	C₄×C₆₁	—	2	1	no	—
164	C₄×C₆₁	—	4	4	no	—
165	C₄×C₆₁	—	8	2	no	—
166	C₄×C₆₁	—	8	1	no	—
167	C₄×C₆₁	—	8	1	no	—
168	C₄×C₆₁	#453	1	2	yes	—
169	C₄×C₆₁	#454	1	2	yes	—
170	C₄×C₆₁	#464	1	2	yes	—
171	C₄×C₆₁	#495	1	1	yes	characteristic
172	C₄×C₆₁	#497	1	1	yes	characteristic
173	#3230	C₂×C₆₁	1	1	yes	characteristic, complemented
174	#3208	C₂×C₆₁	1	1	yes	characteristic
175	#2858	C₂×C₆₁	1	1	yes	characteristic
176	#1994	C₂×C₆₁	1	1	yes	characteristic, complemented
177	#1549	C₂×C₆₁	1	1	yes	characteristic, complemented
178	#1545	C₂×C₆₁	1	1	yes	characteristic
179	#1276	C₂×C₆₁	1	1	yes	characteristic, complemented
180	C₂×C₆₁	—	2	2	no	—
181	C₂×C₆₁	—	16	2	no	—
182	C₂×C₆₁	#2002	1	1	yes	characteristic
183	C₂×C₆₁	#2003	1	1	yes	characteristic
184	C₂×C₆₁	#2015	1	1	yes	characteristic
185	#533	C₂²×C₆₁	1	1	yes	characteristic
186	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
187	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
188	C₂×#124	—	2	2	no	—
189	#500	—	2	1	no	—
190	#492	—	2	1	no	—
191	#487	—	2	1	no	—
192	#486	—	2	1	no	—
193	#484	—	2	2	no	—
194	#477	—	2	2	no	—
195	#460	—	2	2	no	—
196	#450	—	2	2	no	—
197	#431	C₂²×C₆₁	1	1	yes	characteristic
198	#431	C₂²×C₆₁	1	1	yes	characteristic
199	#423	C₂²×C₆₁	1	1	yes	characteristic
200	#390	—	2	2	no	—
201	#390	—	2	2	no	—
202	C₄³	C₂²×C₆₁	1	1	yes	characteristic
203	#339	—	2	2	no	—
204	#332	—	2	2	no	—
205	#331	—	2	2	no	—
206	C₆₁	#10292	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂²×D₄	D₄×C₆₁	1	2	yes	—
211	C₂×SD₁₆	—	2	4	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
218	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
219	#127	—	2	1	no	—
220	#127	—	2	1	no	—
221	#127	—	2	1	no	—
222	#127	—	2	1	no	—
223	#125	—	2	1	no	—
224	#124	—	4	8	no	—
225	C₂²⋊Q₈	—	2	1	no	—
226	C₂²⋊Q₈	—	2	1	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	#117	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
232	C₂×C₂²⋊C₄	—	2	8	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	—	2	1	no	—
235	C₂×C₄²	D₄×C₆₁	1	2	yes	—
236	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
237	#107	—	2	4	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
241	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
242	#103	—	2	4	no	—
243	#102	—	2	4	no	—
244	C₂²⋊C₈	—	2	1	no	—
245	C₂²⋊C₈	—	2	1	no	—
246	C₄×C₈	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	C₂⁴	—	2	2	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	4	no	—
257	C₂×D₄	—	4	4	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
269	SD₁₆	—	8	4	no	—
270	M₁₆	—	4	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	16	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	4	2	no	—
290	C₄²	—	4	1	no	—
291	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
292	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
294	C₄²	#41×C₆₁	1	1	yes	characteristic
295	C₂³	—	4	4	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	4	no	—
299	C₂³	#120×C₆₁	1	1	yes	characteristic
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	8	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	4	4	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	2	no	—
307	D₄	—	8	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	1	no	—
310	D₄	—	8	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
321	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
322	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
323	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
324	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
325	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
326	C₂×C₄	#135×C₆₁	1	2	yes	—
327	C₂×C₄	#137×C₆₁	1	2	yes	—
328	C₈	—	4	2	no	—
329	C₈	—	4	2	no	—
330	C₈	—	8	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	8	8	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	C₆₁×#392	1	1	yes	characteristic
338	C₂²	C₆₁×#452	1	2	yes	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	4	4	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	C₆₁×#453	1	2	yes	—
346	C₄	C₆₁×#454	1	2	yes	—
347	C₄	C₆₁×#464	1	2	yes	—
348	C₄	C₆₁×#495	1	1	yes	characteristic
349	C₄	C₆₁×#497	1	1	yes	characteristic
350	C₂	—	2	2	no	—
351	C₂	—	16	2	no	—
352	C₂	C₆₁×#2002	1	1	yes	characteristic
353	C₂	C₆₁×#2003	1	1	yes	characteristic
354	C₂	C₆₁×#2015	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂×#127×C₆₁	C₂²
LCS	G ▷ C₂×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×#3230	C₂	semidirect
4. C₆₁×#1994	C₂	semidirect
5. C₆₁×#1549	C₂	semidirect
7. C₆₁×#1276	C₂	semidirect
153. #10292	C₆₁	direct (N × H)
173. #3230	C₂×C₆₁	semidirect
176. #1994	C₂×C₆₁	semidirect
177. #1549	C₂×C₆₁	semidirect
179. #1276	C₂×C₆₁	semidirect
206. C₆₁	#10292	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×#3230	2
C₆₁×#3208	2
C₆₁×#2858	2
C₆₁×#1994	2
C₆₁×#1549	2
C₆₁×#1545	2
C₆₁×#1276	2
#10292	61

Subgroups — C61×#10292 #29513725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10292 #29513725