Subgroups — C₆₁×#61822 #29565255

Order 15616.

1694 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3507	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3383	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3381	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3365	C₂	1	4	yes	maximal, complemented
6	C₆₁×#3364	C₂	1	4	yes	maximal, complemented
7	C₆₁×#3232	C₂	1	4	yes	maximal, complemented
8	C₆₁×#3229	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#3226	C₂	1	1	yes	maximal, characteristic
10	C₆₁×#3108	C₂	1	4	yes	maximal, complemented
11	C₆₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#3103	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#559	C₂²	1	2	yes	—
15	C₆₁×#557	—	2	2	no	—
16	C₆₁×#557	—	2	2	no	—
17	C₆₁×#557	C₂²	1	4	yes	—
18	C₆₁×#557	C₂²	1	4	yes	—
19	C₆₁×#557	C₂²	1	2	yes	—
20	C₆₁×#556	—	2	2	no	—
21	C₆₁×#556	C₂²	1	2	yes	—
22	C₆₁×#554	C₂²	1	2	yes	—
23	C₆₁×#552	—	2	2	no	—
24	C₆₁×#539	—	2	2	no	—
25	C₆₁×#536	—	2	1	no	—
26	C₆₁×#533	C₂²	1	1	yes	characteristic
27	C₆₁×#532	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
29	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#529	C₂²	1	4	yes	—
31	C₂×#125×C₆₁	—	2	2	no	—
32	C₂×#125×C₆₁	C₂²	1	2	yes	—
33	C₆₁×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
35	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
37	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#500	C₂²	1	2	yes	—
39	C₆₁×#499	C₂²	1	8	yes	—
40	C₆₁×#498	C₂²	1	2	yes	—
41	C₆₁×#488	C₂²	1	8	yes	—
42	C₆₁×#487	C₂²	1	2	yes	—
43	C₆₁×#486	C₂²	1	2	yes	—
44	C₆₁×#484	C₂²	1	2	yes	—
45	C₆₁×#483	C₂²	1	4	yes	—
46	C₆₁×#481	C₂²	1	2	yes	—
47	C₆₁×#480	C₂²	1	4	yes	—
48	C₆₁×#479	C₂²	1	8	yes	—
49	C₆₁×#479	C₂²	1	8	yes	—
50	C₆₁×#478	C₂²	1	8	yes	—
51	C₆₁×#477	C₂²	1	2	yes	—
52	C₆₁×#475	C₂²	1	2	yes	—
53	C₆₁×#475	C₂²	1	4	yes	—
54	C₆₁×#474	C₂²	1	4	yes	—
55	C₆₁×#464	C₂²	1	8	yes	—
56	C₆₁×#462	C₂²	1	2	yes	—
57	C₆₁×#460	C₂²	1	2	yes	—
58	C₆₁×#451	C₂²	1	2	yes	—
59	C₆₁×#448	C₂²	1	2	yes	—
60	C₈×Q₈×C₆₁	C₂²	1	4	yes	—
61	C₈×Q₈×C₆₁	C₂²	1	4	yes	—
62	C₄×Q₁₆×C₆₁	C₂²	1	4	yes	—
63	C₄×Q₁₆×C₆₁	C₂²	1	8	yes	—
64	C₄×SD₁₆×C₆₁	C₂²	1	4	yes	—
65	C₄×SD₁₆×C₆₁	C₂²	1	8	yes	—
66	C₆₁×#431	C₂²	1	1	yes	characteristic
67	C₆₁×#431	C₂²	1	1	yes	characteristic
68	C₆₁×#431	C₂²	1	4	yes	—
69	C₂×#41×C₆₁	—	2	1	no	—
70	C₂²×Q₈×C₆₁	—	2	1	no	—
71	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
72	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
73	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
75	C₄⋊Q₈×C₆₁	—	2	4	no	—
76	C₄⋊Q₈×C₆₁	—	2	4	no	—
77	C₄⋊Q₈×C₆₁	—	2	2	no	—
78	C₄⋊Q₈×C₆₁	—	2	1	no	—
79	C₄⋊Q₈×C₆₁	—	2	1	no	—
80	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
82	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
83	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
84	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
85	#127×C₆₁	C₂³	1	1	yes	characteristic
86	#126×C₆₁	—	2	4	no	—
87	#126×C₆₁	—	2	4	no	—
88	#125×C₆₁	—	2	4	no	—
89	#125×C₆₁	—	2	8	no	—
90	#125×C₆₁	—	2	8	no	—
91	#125×C₆₁	—	2	1	no	—
92	#125×C₆₁	—	2	4	no	—
93	#125×C₆₁	C₂³	1	4	yes	—
94	#125×C₆₁	C₂³	1	8	yes	—
95	#124×C₆₁	—	2	2	no	—
96	#124×C₆₁	—	2	1	no	—
97	#124×C₆₁	C₂³	1	4	yes	—
98	#123×C₆₁	—	2	4	no	—
99	#123×C₆₁	—	2	4	no	—
100	C₂²⋊Q₈×C₆₁	—	2	2	no	—
101	C₂²⋊Q₈×C₆₁	—	2	2	no	—
102	C₂²⋊Q₈×C₆₁	—	2	1	no	—
103	C₂²⋊Q₈×C₆₁	—	2	4	no	—
104	C₂²⋊Q₈×C₆₁	C₂³	1	2	yes	—
105	#121×C₆₁	—	2	1	no	—
106	#121×C₆₁	C₂³	1	2	yes	—
107	C₄×Q₈×C₆₁	—	2	2	no	—
108	C₄×Q₈×C₆₁	—	2	4	no	—
109	C₄×Q₈×C₆₁	—	2	4	no	—
110	C₄×Q₈×C₆₁	—	2	1	no	—
111	C₄×Q₈×C₆₁	—	2	4	no	—
112	C₄×Q₈×C₆₁	—	2	1	no	—
113	C₄×Q₈×C₆₁	—	2	1	no	—
114	C₄×Q₈×C₆₁	D₄	1	2	yes	—
115	C₄×Q₈×C₆₁	D₄	1	2	yes	—
116	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
117	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
118	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
119	C₄×Q₈×C₆₁	C₂³	1	8	yes	—
120	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
121	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
122	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
123	C₄×D₄×C₆₁	—	2	4	no	—
124	C₄×D₄×C₆₁	—	2	2	no	—
125	C₄×D₄×C₆₁	—	2	1	no	—
126	C₄×D₄×C₆₁	C₂³	1	2	yes	—
127	C₄×D₄×C₆₁	C₂³	1	4	yes	—
128	#117×C₆₁	—	2	1	no	—
129	#117×C₆₁	—	2	2	no	—
130	#117×C₆₁	—	2	4	no	—
131	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
132	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
133	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
134	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
135	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
136	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
137	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
138	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
139	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
140	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₆₁	—	2	1	no	—
142	C₂×C₄²×C₆₁	—	2	2	no	—
143	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
145	#107×C₆₁	C₂³	1	8	yes	—
146	#107×C₆₁	C₂³	1	4	yes	—
147	#106×C₆₁	C₂³	1	8	yes	—
148	#106×C₆₁	C₂³	1	4	yes	—
149	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
150	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
151	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
152	#103×C₆₁	C₂³	1	4	yes	—
153	#103×C₆₁	C₂³	1	4	yes	—
154	#103×C₆₁	C₂³	1	4	yes	—
155	#103×C₆₁	C₂³	1	8	yes	—
156	#102×C₆₁	C₂³	1	4	yes	—
157	#102×C₆₁	C₂³	1	8	yes	—
158	C₂²⋊C₈×C₆₁	C₂³	1	2	yes	—
159	C₂²⋊C₈×C₆₁	C₂³	1	2	yes	—
160	C₄×C₈×C₆₁	C₂³	1	2	yes	—
161	C₄×C₈×C₆₁	C₂³	1	8	yes	—
162	C₄×C₈×C₆₁	C₂³	1	2	yes	—
163	#41×C₆₁	—	4	4	no	—
164	C₂×Q₈×C₆₁	—	2	4	no	—
165	C₂×Q₈×C₆₁	—	2	4	no	—
166	C₂×Q₈×C₆₁	—	2	1	no	—
167	C₂×Q₈×C₆₁	—	2	1	no	—
168	C₂×Q₈×C₆₁	—	2	2	no	—
169	C₂×Q₈×C₆₁	—	2	2	no	—
170	C₂×Q₈×C₆₁	—	2	1	no	—
171	C₂×Q₈×C₆₁	—	4	4	no	—
172	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
173	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
174	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
175	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
176	C₂×D₄×C₆₁	—	2	2	no	—
177	C₂×D₄×C₆₁	—	2	1	no	—
178	C₂×D₄×C₆₁	—	2	1	no	—
179	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₁	—	2	1	no	—
181	C₂²×C₄×C₆₁	—	2	1	no	—
182	C₂²×C₄×C₆₁	—	2	2	no	—
183	C₂²×C₄×C₆₁	—	2	2	no	—
184	C₂²×C₄×C₆₁	—	2	4	no	—
185	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₆₁	—	2	8	no	—
191	SD₁₆×C₆₁	—	2	8	no	—
192	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
193	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₆₁	—	2	1	no	—
195	C₄⋊C₄×C₆₁	—	2	4	no	—
196	C₄⋊C₄×C₆₁	—	2	8	no	—
197	C₄⋊C₄×C₆₁	—	2	2	no	—
198	C₄⋊C₄×C₆₁	—	2	2	no	—
199	C₄⋊C₄×C₆₁	—	2	8	no	—
200	C₄⋊C₄×C₆₁	—	2	2	no	—
201	C₄⋊C₄×C₆₁	—	2	2	no	—
202	C₄⋊C₄×C₆₁	—	2	8	no	—
203	C₄⋊C₄×C₆₁	—	2	1	no	—
204	C₄⋊C₄×C₆₁	—	2	2	no	—
205	C₄⋊C₄×C₆₁	—	2	1	no	—
206	C₄⋊C₄×C₆₁	—	2	2	no	—
207	C₄⋊C₄×C₆₁	—	2	4	no	—
208	C₄⋊C₄×C₆₁	—	2	1	no	—
209	C₄⋊C₄×C₆₁	—	2	1	no	—
210	C₄⋊C₄×C₆₁	—	2	2	no	—
211	C₄⋊C₄×C₆₁	—	2	4	no	—
212	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
213	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
214	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
215	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
216	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
217	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
218	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
219	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
220	C₂²⋊C₄×C₆₁	—	2	2	no	—
221	C₂²⋊C₄×C₆₁	—	2	2	no	—
222	C₂²⋊C₄×C₆₁	—	2	4	no	—
223	C₂²⋊C₄×C₆₁	—	2	4	no	—
224	C₂²⋊C₄×C₆₁	—	2	2	no	—
225	C₂²⋊C₄×C₆₁	—	2	4	no	—
226	C₄²×C₆₁	—	2	8	no	—
227	C₄²×C₆₁	—	2	2	no	—
228	C₄²×C₆₁	—	2	2	no	—
229	C₄²×C₆₁	—	2	4	no	—
230	C₄²×C₆₁	—	2	2	no	—
231	C₄²×C₆₁	C₂×D₄	1	4	yes	—
232	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
233	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
235	C₄²×C₆₁	C₂⁴	1	4	yes	—
236	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
237	C₂³×C₆₁	—	2	2	no	—
238	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
239	Q₈×C₆₁	—	2	4	no	—
240	Q₈×C₆₁	—	2	4	no	—
241	Q₈×C₆₁	—	2	4	no	—
242	Q₈×C₆₁	—	2	2	no	—
243	Q₈×C₆₁	—	2	4	no	—
244	Q₈×C₆₁	—	4	4	no	—
245	Q₈×C₆₁	—	4	4	no	—
246	Q₈×C₆₁	—	4	2	no	—
247	Q₈×C₆₁	—	8	1	no	—
248	Q₈×C₆₁	2₋¹⁺⁴	1	4	yes	—
249	D₄×C₆₁	—	2	4	no	—
250	D₄×C₆₁	—	4	4	no	—
251	D₄×C₆₁	—	4	2	no	—
252	D₄×C₆₁	—	8	1	no	—
253	C₂×C₄×C₆₁	—	2	4	no	—
254	C₂×C₄×C₆₁	—	2	4	no	—
255	C₂×C₄×C₆₁	—	2	2	no	—
256	C₂×C₄×C₆₁	—	2	2	no	—
257	C₂×C₄×C₆₁	—	2	2	no	—
258	C₂×C₄×C₆₁	—	2	2	no	—
259	C₂×C₄×C₆₁	—	2	2	no	—
260	C₂×C₄×C₆₁	—	2	4	no	—
261	C₂×C₄×C₆₁	—	2	2	no	—
262	C₂×C₄×C₆₁	—	4	2	no	—
263	C₂×C₄×C₆₁	—	4	1	no	—
264	C₂×C₄×C₆₁	—	4	2	no	—
265	C₂×C₄×C₆₁	—	4	4	no	—
266	C₂×C₄×C₆₁	—	4	4	no	—
267	C₂×C₄×C₆₁	—	8	4	no	—
268	C₂×C₄×C₆₁	#135	1	4	yes	—
269	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
270	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
272	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
277	C₈×C₆₁	—	2	4	no	—
278	C₈×C₆₁	—	2	4	no	—
279	#61822	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂²×C₆₁	—	2	1	no	—
281	C₂²×C₆₁	—	2	1	no	—
282	C₂²×C₆₁	—	4	2	no	—
283	C₂²×C₆₁	—	4	2	no	—
284	C₂²×C₆₁	—	4	2	no	—
285	C₂²×C₆₁	C₂×#135	1	2	yes	—
286	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
287	C₄×C₆₁	—	2	4	no	—
288	C₄×C₆₁	—	2	1	no	—
289	C₄×C₆₁	—	2	1	no	—
290	C₄×C₆₁	—	2	4	no	—
291	C₄×C₆₁	—	2	1	no	—
292	C₄×C₆₁	—	2	4	no	—
293	C₄×C₆₁	—	4	2	no	—
294	C₄×C₆₁	—	4	2	no	—
295	C₄×C₆₁	—	4	2	no	—
296	C₄×C₆₁	—	4	2	no	—
297	C₄×C₆₁	—	4	4	no	—
298	C₄×C₆₁	C₂×#135	1	2	yes	—
299	C₄×C₆₁	C₂×#135	1	2	yes	—
300	C₄×C₆₁	C₂×2₋¹⁺⁴	1	2	yes	—
301	#3507	C₂×C₆₁	1	1	yes	characteristic
302	C₂×#557	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#3383	C₂×C₆₁	1	4	yes	complemented
304	#3381	C₂×C₆₁	1	4	yes	complemented
305	#3365	C₂×C₆₁	1	4	yes	complemented
306	#3364	C₂×C₆₁	1	4	yes	complemented
307	#3232	C₂×C₆₁	1	4	yes	complemented
308	#3229	C₂×C₆₁	1	1	yes	characteristic, complemented
309	#3226	C₂×C₆₁	1	1	yes	characteristic
310	#3108	C₂×C₆₁	1	4	yes	complemented
311	#3104	C₂×C₆₁	1	1	yes	characteristic, complemented
312	#3103	C₂×C₆₁	1	1	yes	characteristic
313	#2982	C₂×C₆₁	1	1	yes	characteristic, complemented
314	C₂×C₆₁	—	2	2	no	—
315	C₂×C₆₁	—	8	2	no	—
316	C₂×C₆₁	#3486	1	1	yes	characteristic
317	C₂×C₆₁	C₂²×#135	1	1	yes	characteristic
318	C₂×C₆₁	#3586	1	1	yes	characteristic
319	#559	C₂²×C₆₁	1	2	yes	—
320	#557	—	2	2	no	—
321	#557	—	2	2	no	—
322	#557	C₂²×C₆₁	1	4	yes	—
323	#557	C₂²×C₆₁	1	4	yes	—
324	#557	C₂²×C₆₁	1	2	yes	—
325	#556	—	2	2	no	—
326	#556	C₂²×C₆₁	1	2	yes	—
327	#554	C₂²×C₆₁	1	2	yes	—
328	#552	—	2	2	no	—
329	#539	—	2	2	no	—
330	#536	—	2	1	no	—
331	#533	C₂²×C₆₁	1	1	yes	characteristic
332	#532	C₂²×C₆₁	1	1	yes	characteristic
333	C₂×C₄⋊Q₈	—	2	1	no	—
334	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
335	#529	C₂²×C₆₁	1	4	yes	—
336	C₂×#125	—	2	2	no	—
337	C₂×#125	C₂²×C₆₁	1	2	yes	—
338	#520	—	2	1	no	—
339	C₂×C₄×Q₈	—	2	1	no	—
340	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
341	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
342	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
343	#500	C₂²×C₆₁	1	2	yes	—
344	#499	C₂²×C₆₁	1	8	yes	—
345	#498	C₂²×C₆₁	1	2	yes	—
346	#488	C₂²×C₆₁	1	8	yes	—
347	#487	C₂²×C₆₁	1	2	yes	—
348	#486	C₂²×C₆₁	1	2	yes	—
349	#484	C₂²×C₆₁	1	2	yes	—
350	#483	C₂²×C₆₁	1	4	yes	—
351	#481	C₂²×C₆₁	1	2	yes	—
352	#480	C₂²×C₆₁	1	4	yes	—
353	#479	C₂²×C₆₁	1	8	yes	—
354	#479	C₂²×C₆₁	1	8	yes	—
355	#478	C₂²×C₆₁	1	8	yes	—
356	#477	C₂²×C₆₁	1	2	yes	—
357	#475	C₂²×C₆₁	1	2	yes	—
358	#475	C₂²×C₆₁	1	4	yes	—
359	#474	C₂²×C₆₁	1	4	yes	—
360	#464	C₂²×C₆₁	1	8	yes	—
361	#462	C₂²×C₆₁	1	2	yes	—
362	#460	C₂²×C₆₁	1	2	yes	—
363	#451	C₂²×C₆₁	1	2	yes	—
364	#448	C₂²×C₆₁	1	2	yes	—
365	C₈×Q₈	C₂²×C₆₁	1	4	yes	—
366	C₈×Q₈	C₂²×C₆₁	1	4	yes	—
367	C₄×Q₁₆	C₂²×C₆₁	1	4	yes	—
368	C₄×Q₁₆	C₂²×C₆₁	1	8	yes	—
369	C₄×SD₁₆	C₂²×C₆₁	1	4	yes	—
370	C₄×SD₁₆	C₂²×C₆₁	1	8	yes	—
371	#431	C₂²×C₆₁	1	1	yes	characteristic
372	#431	C₂²×C₆₁	1	1	yes	characteristic
373	#431	C₂²×C₆₁	1	4	yes	—
374	C₆₁	#61822	1	1	yes	characteristic, direct product factor, Sylow
375	C₂×#41	—	2	1	no	—
376	C₂²×Q₈	—	2	1	no	—
377	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
378	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
379	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
380	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
381	C₄⋊Q₈	—	2	4	no	—
382	C₄⋊Q₈	—	2	4	no	—
383	C₄⋊Q₈	—	2	2	no	—
384	C₄⋊Q₈	—	2	1	no	—
385	C₄⋊Q₈	—	2	1	no	—
386	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
387	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
388	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
389	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
390	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
391	#127	C₂³×C₆₁	1	1	yes	characteristic
392	#126	—	2	4	no	—
393	#126	—	2	4	no	—
394	#125	—	2	8	no	—
395	#125	—	2	8	no	—
396	#125	—	2	4	no	—
397	#125	—	2	4	no	—
398	#125	—	2	1	no	—
399	#125	C₂³×C₆₁	1	8	yes	—
400	#125	C₂³×C₆₁	1	4	yes	—
401	#124	—	2	2	no	—
402	#124	—	2	1	no	—
403	#124	C₂³×C₆₁	1	4	yes	—
404	#123	—	2	4	no	—
405	#123	—	2	4	no	—
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	1	no	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	C₂³×C₆₁	1	2	yes	—
411	#121	—	2	1	no	—
412	#121	C₂³×C₆₁	1	2	yes	—
413	C₄×Q₈	—	2	4	no	—
414	C₄×Q₈	—	2	4	no	—
415	C₄×Q₈	—	2	2	no	—
416	C₄×Q₈	—	2	4	no	—
417	C₄×Q₈	—	2	1	no	—
418	C₄×Q₈	—	2	1	no	—
419	C₄×Q₈	—	2	1	no	—
420	C₄×Q₈	D₄×C₆₁	1	2	yes	—
421	C₄×Q₈	D₄×C₆₁	1	2	yes	—
422	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
423	C₄×Q₈	C₂³×C₆₁	1	8	yes	—
424	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
425	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
426	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
427	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
428	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
429	C₄×D₄	—	2	4	no	—
430	C₄×D₄	—	2	2	no	—
431	C₄×D₄	—	2	1	no	—
432	C₄×D₄	C₂³×C₆₁	1	4	yes	—
433	C₄×D₄	C₂³×C₆₁	1	2	yes	—
434	#117	—	2	1	no	—
435	#117	—	2	2	no	—
436	#117	—	2	4	no	—
437	C₂×C₄⋊C₄	—	2	2	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	—	2	2	no	—
440	C₂×C₄⋊C₄	—	2	1	no	—
441	C₂×C₄⋊C₄	—	2	2	no	—
442	C₂×C₄⋊C₄	—	2	2	no	—
443	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
444	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
445	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
447	C₂×C₄²	—	2	2	no	—
448	C₂×C₄²	—	2	1	no	—
449	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
450	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
451	#107	C₂³×C₆₁	1	8	yes	—
452	#107	C₂³×C₆₁	1	4	yes	—
453	#106	C₂³×C₆₁	1	8	yes	—
454	#106	C₂³×C₆₁	1	4	yes	—
455	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
456	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
457	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
458	#103	C₂³×C₆₁	1	4	yes	—
459	#103	C₂³×C₆₁	1	4	yes	—
460	#103	C₂³×C₆₁	1	4	yes	—
461	#103	C₂³×C₆₁	1	8	yes	—
462	#102	C₂³×C₆₁	1	4	yes	—
463	#102	C₂³×C₆₁	1	8	yes	—
464	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
465	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
466	C₄×C₈	C₂³×C₆₁	1	2	yes	—
467	C₄×C₈	C₂³×C₆₁	1	8	yes	—
468	C₄×C₈	C₂³×C₆₁	1	2	yes	—
469	#41	—	4	4	no	—
470	C₂×Q₈	—	2	4	no	—
471	C₂×Q₈	—	2	4	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	—	2	2	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	4	4	no	—
478	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
479	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
480	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
481	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
492	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
493	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
496	Q₁₆	—	2	8	no	—
497	SD₁₆	—	2	8	no	—
498	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
499	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	8	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	8	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	8	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	4	no	—
518	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
519	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
520	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
521	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
522	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
523	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
524	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
525	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	2	no	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₄²	—	2	8	no	—
533	C₄²	—	2	2	no	—
534	C₄²	—	2	2	no	—
535	C₄²	—	2	4	no	—
536	C₄²	—	2	2	no	—
537	C₄²	C₂×D₄×C₆₁	1	4	yes	—
538	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
539	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
540	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
541	C₄²	C₂⁴×C₆₁	1	4	yes	—
542	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
543	C₂³	—	2	2	no	—
544	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
545	Q₈	—	2	4	no	—
546	Q₈	—	2	4	no	—
547	Q₈	—	2	4	no	—
548	Q₈	—	2	2	no	—
549	Q₈	—	2	4	no	—
550	Q₈	—	4	4	no	—
551	Q₈	—	4	4	no	—
552	Q₈	—	4	2	no	—
553	Q₈	—	8	1	no	—
554	Q₈	2₋¹⁺⁴×C₆₁	1	4	yes	—
555	D₄	—	2	4	no	—
556	D₄	—	4	4	no	—
557	D₄	—	4	2	no	—
558	D₄	—	8	1	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	2	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	8	4	no	—
574	C₂×C₄	#135×C₆₁	1	4	yes	—
575	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
576	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
577	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
578	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
579	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
580	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
581	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
582	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
583	C₈	—	2	4	no	—
584	C₈	—	2	4	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	C₂×#135×C₆₁	1	2	yes	—
591	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
592	C₄	—	2	4	no	—
593	C₄	—	2	4	no	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	4	no	—
597	C₄	—	2	1	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	4	no	—
600	C₄	—	4	2	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	2	no	—
603	C₄	C₂×#135×C₆₁	1	2	yes	—
604	C₄	C₂×#135×C₆₁	1	2	yes	—
605	C₄	C₂×2₋¹⁺⁴×C₆₁	1	2	yes	—
606	C₂	—	2	2	no	—
607	C₂	—	8	2	no	—
608	C₂	C₆₁×#3486	1	1	yes	characteristic
609	C₂	C₂²×#135×C₆₁	1	1	yes	characteristic
610	C₂	C₆₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#557	C₂	semidirect
3. C₆₁×#3383	C₂	semidirect
4. C₆₁×#3381	C₂	semidirect
5. C₆₁×#3365	C₂	semidirect
6. C₆₁×#3364	C₂	semidirect
7. C₆₁×#3232	C₂	semidirect
8. C₆₁×#3229	C₂	semidirect
10. C₆₁×#3108	C₂	semidirect
11. C₆₁×#3104	C₂	semidirect
13. C₆₁×#2982	C₂	semidirect
279. #61822	C₆₁	direct (N × H)
302. C₂×#557	C₂×C₆₁	semidirect
303. #3383	C₂×C₆₁	semidirect
304. #3381	C₂×C₆₁	semidirect
305. #3365	C₂×C₆₁	semidirect
306. #3364	C₂×C₆₁	semidirect
307. #3232	C₂×C₆₁	semidirect
308. #3229	C₂×C₆₁	semidirect
310. #3108	C₂×C₆₁	semidirect
311. #3104	C₂×C₆₁	semidirect
313. #2982	C₂×C₆₁	semidirect
374. C₆₁	#61822	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×#3507	2
C₂×C₆₁×#557	2
C₆₁×#3383	2
C₆₁×#3381	2
C₆₁×#3365	2
C₆₁×#3364	2
C₆₁×#3232	2
C₆₁×#3229	2
C₆₁×#3226	2
C₆₁×#3108	2
C₆₁×#3104	2
C₆₁×#3103	2
C₆₁×#2982	2
#61822	61

Subgroups — C61×#61822 #29565255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#61822 #29565255