Subgroups — C₆₁×#10830 #29514263

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2274	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#344	C₂	1	2	yes	maximal
3	C₂×C₆₁×#344	C₂	1	2	yes	maximal
4	C₂×C₆₁×#343	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
8	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
12	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
13	C₄×C₂²⋊C₄×C₆₁	C₄	1	4	yes	—
14	C₆₁×#344	—	2	4	no	—
15	C₆₁×#344	C₄	1	8	yes	complemented
16	C₆₁×#343	—	2	2	no	—
17	C₆₁×#343	—	2	2	no	—
18	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₆₁	—	2	2	no	—
21	C₂×M₁₆×C₆₁	—	2	2	no	—
22	C₂×M₁₆×C₆₁	—	2	2	no	—
23	C₂×M₁₆×C₆₁	—	2	2	no	—
24	C₂×M₁₆×C₆₁	—	2	2	no	—
25	C₂×M₁₆×C₆₁	—	2	2	no	—
26	C₂×M₁₆×C₆₁	—	2	2	no	—
27	C₂×M₁₆×C₆₁	—	2	2	no	—
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂×M₁₆×C₆₁	—	2	2	no	—
30	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₆₁	—	2	2	no	—
33	C₂²×C₈×C₆₁	—	2	2	no	—
34	#117×C₆₁	—	2	2	no	—
35	#117×C₆₁	—	2	1	no	—
36	#117×C₆₁	—	2	1	no	—
37	#117×C₆₁	C₂×C₄	1	4	yes	—
38	#117×C₆₁	D₄	1	2	yes	—
39	#117×C₆₁	Q₈	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₁	—	2	1	no	—
49	C₂×C₄²×C₆₁	—	2	1	no	—
50	C₂×C₄²×C₆₁	—	2	4	no	—
51	#95×C₆₁	—	2	4	no	—
52	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	—	2	2	no	—
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	—	2	2	no	—
56	C₂²×C₄×C₆₁	—	2	2	no	—
57	C₂²×C₄×C₆₁	—	8	2	no	—
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	#41	1	2	yes	—
67	C₂²×C₄×C₆₁	#41	1	2	yes	—
68	C₂²×C₄×C₆₁	#41	1	2	yes	—
69	M₁₆×C₆₁	—	2	4	no	—
70	M₁₆×C₆₁	—	2	4	no	—
71	M₁₆×C₆₁	—	2	4	no	—
72	M₁₆×C₆₁	—	2	4	no	—
73	M₁₆×C₆₁	—	4	2	no	—
74	M₁₆×C₆₁	—	4	2	no	—
75	M₁₆×C₆₁	—	4	2	no	—
76	M₁₆×C₆₁	—	4	2	no	—
77	C₂×C₈×C₆₁	—	2	4	no	—
78	C₂×C₈×C₆₁	—	2	4	no	—
79	C₂×C₈×C₆₁	—	4	2	no	—
80	C₂×C₈×C₆₁	—	4	2	no	—
81	C₂×C₈×C₆₁	—	4	2	no	—
82	C₂×C₈×C₆₁	—	4	2	no	—
83	C₄⋊C₄×C₆₁	—	2	4	no	—
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₄⋊C₄×C₆₁	—	2	2	no	—
86	C₄⋊C₄×C₆₁	—	2	4	no	—
87	C₄⋊C₄×C₆₁	—	2	4	no	—
88	C₂²⋊C₄×C₆₁	—	2	2	no	—
89	C₂²⋊C₄×C₆₁	—	2	2	no	—
90	C₂²⋊C₄×C₆₁	C₄²	1	4	yes	—
91	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
92	C₄²×C₆₁	—	2	4	no	—
93	C₄²×C₆₁	—	2	2	no	—
94	C₄²×C₆₁	—	2	2	no	—
95	C₄²×C₆₁	—	8	4	no	—
96	C₂³×C₆₁	—	4	1	no	—
97	C₂³×C₆₁	—	4	1	no	—
98	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₆₁	#117	1	1	yes	characteristic
102	C₂³×C₆₁	#117	1	1	yes	characteristic
103	C₂³×C₆₁	#117	1	1	yes	characteristic
104	C₂³×C₆₁	#117	1	1	yes	characteristic
105	C₂×C₄×C₆₁	—	2	4	no	—
106	C₂×C₄×C₆₁	—	2	4	no	—
107	C₂×C₄×C₆₁	—	2	4	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	4	2	no	—
110	C₂×C₄×C₆₁	—	4	2	no	—
111	C₂×C₄×C₆₁	—	4	2	no	—
112	C₂×C₄×C₆₁	—	4	4	no	—
113	C₂×C₄×C₆₁	—	8	2	no	—
114	C₂×C₄×C₆₁	—	8	2	no	—
115	C₂×C₄×C₆₁	#117	1	2	yes	—
116	C₂×C₄×C₆₁	#117	1	2	yes	—
117	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#123	1	2	yes	—
126	C₂×C₄×C₆₁	#123	1	2	yes	—
127	C₂×C₄×C₆₁	#124	1	2	yes	—
128	C₂×C₄×C₆₁	#125	1	2	yes	—
129	C₂×C₄×C₆₁	#126	1	2	yes	—
130	C₂×C₄×C₆₁	#126	1	2	yes	—
131	C₈×C₆₁	—	4	4	no	—
132	C₈×C₆₁	—	4	4	no	—
133	#10830	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₆₁	—	2	1	no	—
135	C₂²×C₆₁	—	2	1	no	—
136	C₂²×C₆₁	—	2	1	no	—
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	—	2	1	no	—
139	C₂²×C₆₁	—	2	1	no	—
140	C₂²×C₆₁	—	4	1	no	—
141	C₂²×C₆₁	—	4	1	no	—
142	C₂²×C₆₁	—	4	1	no	—
143	C₂²×C₆₁	—	4	1	no	—
144	C₂²×C₆₁	#376	1	1	yes	characteristic
145	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₆₁	#380	1	1	yes	characteristic
148	C₂²×C₆₁	#381	1	1	yes	characteristic
149	C₂²×C₆₁	#382	1	1	yes	characteristic
150	C₂²×C₆₁	#383	1	1	yes	characteristic
151	C₄×C₆₁	—	4	4	no	—
152	C₄×C₆₁	—	4	4	no	—
153	C₄×C₆₁	—	4	4	no	—
154	C₄×C₆₁	#387	1	2	yes	—
155	C₄×C₆₁	#387	1	2	yes	—
156	C₄×C₆₁	#388	1	2	yes	—
157	C₄×C₆₁	#388	1	2	yes	—
158	#2274	C₂×C₆₁	1	1	yes	characteristic
159	C₂×#344	C₂×C₆₁	1	2	yes	—
160	C₂×#344	C₂×C₆₁	1	2	yes	—
161	C₂×#343	C₂×C₆₁	1	1	yes	characteristic
162	C₂×#343	C₂×C₆₁	1	1	yes	characteristic
163	C₂×C₆₁	—	2	1	no	—
164	C₂×C₆₁	—	2	1	no	—
165	C₂×C₆₁	—	2	1	no	—
166	C₂×C₆₁	—	2	1	no	—
167	C₂×C₆₁	—	2	1	no	—
168	C₂×C₆₁	—	2	1	no	—
169	C₂×C₆₁	#1434	1	1	yes	characteristic
170	C₂×C₆₁	#1829	1	1	yes	characteristic
171	C₂×C₆₁	#1830	1	1	yes	characteristic
172	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
173	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
174	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
175	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
176	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
177	C₄×C₄⋊C₄	—	2	2	no	—
178	C₄×C₄⋊C₄	—	2	2	no	—
179	C₄×C₂²⋊C₄	C₄×C₆₁	1	4	yes	—
180	#344	—	2	4	no	—
181	#344	C₄×C₆₁	1	8	yes	complemented
182	#343	—	2	2	no	—
183	#343	—	2	2	no	—
184	C₆₁	#10830	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
186	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
198	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	1	no	—
203	#117	—	2	1	no	—
204	#117	C₂×C₄×C₆₁	1	4	yes	—
205	#117	D₄×C₆₁	1	2	yes	—
206	#117	Q₈×C₆₁	1	2	yes	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
212	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
213	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
214	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	—	2	4	no	—
218	#95	—	2	4	no	—
219	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	8	2	no	—
225	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
231	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
232	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
233	C₂²×C₄	#41×C₆₁	1	2	yes	—
234	C₂²×C₄	#41×C₆₁	1	2	yes	—
235	C₂²×C₄	#41×C₆₁	1	2	yes	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	C₄²×C₆₁	1	4	yes	—
258	C₂²⋊C₄	C₄⋊C₄×C₆₁	1	4	yes	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	8	4	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
266	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
267	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
268	C₂³	#117×C₆₁	1	1	yes	characteristic
269	C₂³	#117×C₆₁	1	1	yes	characteristic
270	C₂³	#117×C₆₁	1	1	yes	characteristic
271	C₂³	#117×C₆₁	1	1	yes	characteristic
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	#117×C₆₁	1	2	yes	—
283	C₂×C₄	#117×C₆₁	1	2	yes	—
284	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
285	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
287	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
288	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
289	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
290	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
291	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
292	C₂×C₄	#123×C₆₁	1	2	yes	—
293	C₂×C₄	#123×C₆₁	1	2	yes	—
294	C₂×C₄	#124×C₆₁	1	2	yes	—
295	C₂×C₄	#125×C₆₁	1	2	yes	—
296	C₂×C₄	#126×C₆₁	1	2	yes	—
297	C₂×C₄	#126×C₆₁	1	2	yes	—
298	C₈	—	4	4	no	—
299	C₈	—	4	4	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	C₆₁×#376	1	1	yes	characteristic
311	C₂²	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
312	C₂²	C₄×C₄⋊C₄×C₆₁	1	1	yes	characteristic
313	C₂²	C₆₁×#380	1	1	yes	characteristic
314	C₂²	C₆₁×#381	1	1	yes	characteristic
315	C₂²	C₆₁×#382	1	1	yes	characteristic
316	C₂²	C₆₁×#383	1	1	yes	characteristic
317	C₄	—	4	4	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	4	no	—
320	C₄	C₆₁×#387	1	2	yes	—
321	C₄	C₆₁×#387	1	2	yes	—
322	C₄	C₆₁×#388	1	2	yes	—
323	C₄	C₆₁×#388	1	2	yes	—
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	C₆₁×#1434	1	1	yes	characteristic
331	C₂	C₆₁×#1829	1	1	yes	characteristic
332	C₂	C₆₁×#1830	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#381
commutator	C₂³	C₂×C₄²×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₆₁×#344	C₄	semidirect
133. #10830	C₆₁	direct (N × H)
181. #344	C₄×C₆₁	semidirect
184. C₆₁	#10830	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×#2274	2
C₂×C₆₁×#344	2
C₂×C₆₁×#343	2
#10830	61

Subgroups — C61×#10830 #29514263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10830 #29514263