Subgroups — C₆₁×#61819 #29565252

Order 15616.

1770 conjugacy classes of subgroups.

The subgroup lattice has 634 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, complemented
2	C₂×C₆₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3381	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3380	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3361	C₂	1	4	yes	maximal, complemented
6	C₆₁×#3360	C₂	1	4	yes	maximal, complemented
7	C₆₁×#3232	C₂	1	4	yes	maximal, complemented
8	C₆₁×#3230	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#3105	C₂	1	4	yes	maximal, complemented
11	C₆₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#3100	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#559	C₂²	1	2	yes	complemented
15	C₆₁×#557	C₂²	1	2	yes	complemented
16	C₆₁×#556	—	2	2	no	—
17	C₆₁×#556	C₂²	1	2	yes	complemented
18	C₆₁×#554	—	2	2	no	—
19	C₆₁×#554	—	2	2	no	—
20	C₆₁×#554	C₂²	1	4	yes	—
21	C₆₁×#554	C₂²	1	4	yes	—
22	C₆₁×#554	C₂²	1	2	yes	complemented
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	C₂²	1	1	yes	characteristic
29	C₆₁×#529	C₂²	1	4	yes	—
30	C₂×#125×C₆₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
32	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
33	C₆₁×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
38	C₆₁×#500	C₂²	1	2	yes	complemented
39	C₆₁×#499	C₂²	1	8	yes	complemented
40	C₆₁×#498	C₂²	1	2	yes	complemented
41	C₆₁×#488	C₂²	1	8	yes	complemented
42	C₆₁×#487	C₂²	1	2	yes	complemented
43	C₆₁×#486	C₂²	1	2	yes	complemented
44	C₆₁×#484	C₂²	1	4	yes	—
45	C₆₁×#483	C₂²	1	2	yes	—
46	C₆₁×#481	C₂²	1	4	yes	—
47	C₆₁×#480	C₂²	1	2	yes	—
48	C₆₁×#479	C₂²	1	8	yes	complemented
49	C₆₁×#478	C₂²	1	8	yes	—
50	C₆₁×#478	C₂²	1	8	yes	complemented
51	C₆₁×#477	C₂²	1	4	yes	complemented
52	C₆₁×#476	C₂²	1	2	yes	—
53	C₆₁×#476	C₂²	1	4	yes	complemented
54	C₆₁×#474	C₂²	1	2	yes	—
55	C₆₁×#463	C₂²	1	8	yes	—
56	C₆₁×#461	C₂²	1	2	yes	—
57	C₆₁×#459	C₂²	1	2	yes	—
58	C₆₁×#450	C₂²	1	2	yes	—
59	C₆₁×#447	C₂²	1	2	yes	—
60	C₈×Q₈×C₆₁	C₂²	1	4	yes	complemented
61	C₈×Q₈×C₆₁	C₂²	1	4	yes	complemented
62	C₄×SD₁₆×C₆₁	C₂²	1	4	yes	—
63	C₄×SD₁₆×C₆₁	C₂²	1	8	yes	—
64	C₄×D₈×C₆₁	C₂²	1	4	yes	—
65	C₄×D₈×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×D₄×C₆₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₁	—	2	1	no	—
74	C₂³×C₄×C₆₁	—	2	2	no	—
75	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
76	C₂×D₈×C₆₁	C₂³	1	4	yes	—
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	1	no	—
90	#125×C₆₁	—	2	4	no	—
91	#125×C₆₁	C₂³	1	8	yes	—
92	#125×C₆₁	C₂³	1	4	yes	—
93	#124×C₆₁	—	2	2	no	—
94	#124×C₆₁	—	2	1	no	—
95	#124×C₆₁	C₂³	1	4	yes	—
96	#123×C₆₁	—	2	4	no	—
97	#123×C₆₁	—	2	4	no	—
98	C₂²⋊Q₈×C₆₁	—	2	2	no	—
99	C₂²⋊Q₈×C₆₁	—	2	2	no	—
100	C₂²⋊Q₈×C₆₁	—	2	4	no	—
101	C₂²⋊Q₈×C₆₁	—	2	4	no	—
102	C₂²⋊Q₈×C₆₁	—	2	1	no	—
103	C₂²⋊Q₈×C₆₁	—	2	4	no	—
104	C₂²⋊Q₈×C₆₁	—	2	8	no	—
105	C₂²⋊Q₈×C₆₁	—	2	8	no	—
106	C₂²⋊Q₈×C₆₁	C₂³	1	2	yes	—
107	#121×C₆₁	—	2	1	no	—
108	#121×C₆₁	C₂³	1	2	yes	—
109	C₄×Q₈×C₆₁	—	2	2	no	—
110	C₄×Q₈×C₆₁	—	2	1	no	—
111	C₄×Q₈×C₆₁	D₄	1	2	yes	—
112	C₄×Q₈×C₆₁	D₄	1	2	yes	—
113	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
114	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
115	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
116	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
117	C₄×D₄×C₆₁	—	2	4	no	—
118	C₄×D₄×C₆₁	—	2	2	no	—
119	C₄×D₄×C₆₁	—	2	1	no	—
120	C₄×D₄×C₆₁	—	2	4	no	—
121	C₄×D₄×C₆₁	—	2	1	no	—
122	C₄×D₄×C₆₁	—	2	1	no	—
123	C₄×D₄×C₆₁	C₂³	1	2	yes	—
124	C₄×D₄×C₆₁	C₂³	1	2	yes	—
125	C₄×D₄×C₆₁	C₂³	1	8	yes	—
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	1	no	—
132	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
133	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
134	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
135	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
136	C₂×C₄⋊C₄×C₆₁	—	2	4	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×C₂²⋊C₄×C₆₁	—	2	1	no	—
142	C₂×C₂²⋊C₄×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	4	yes	—
146	#107×C₆₁	C₂³	1	8	yes	—
147	#106×C₆₁	C₂³	1	4	yes	—
148	#106×C₆₁	C₂³	1	8	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	8	yes	—
154	#102×C₆₁	C₂³	1	4	yes	—
155	#102×C₆₁	C₂³	1	4	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	C₂⁴×C₆₁	—	2	1	no	—
164	#41×C₆₁	—	4	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	—	8	2	no	—
171	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
172	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
173	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
174	C₂×D₄×C₆₁	—	2	4	no	—
175	C₂×D₄×C₆₁	—	2	2	no	—
176	C₂×D₄×C₆₁	—	2	1	no	—
177	C₂×D₄×C₆₁	—	2	1	no	—
178	C₂×D₄×C₆₁	—	2	1	no	—
179	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
180	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
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	2	no	—
185	C₂²×C₄×C₆₁	—	2	2	no	—
186	C₂²×C₄×C₆₁	—	2	8	no	—
187	C₂²×C₄×C₆₁	—	2	4	no	—
188	C₂²×C₄×C₆₁	—	4	4	no	—
189	C₂²×C₄×C₆₁	—	8	4	no	—
190	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
193	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
195	SD₁₆×C₆₁	—	2	8	no	—
196	D₈×C₆₁	—	2	8	no	—
197	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
198	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
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	1	no	—
203	C₄⋊C₄×C₆₁	—	2	4	no	—
204	C₄⋊C₄×C₆₁	—	2	8	no	—
205	C₄⋊C₄×C₆₁	—	2	1	no	—
206	C₄⋊C₄×C₆₁	—	2	2	no	—
207	C₄⋊C₄×C₆₁	—	2	1	no	—
208	C₄⋊C₄×C₆₁	—	2	2	no	—
209	C₄⋊C₄×C₆₁	—	2	4	no	—
210	C₄⋊C₄×C₆₁	—	2	1	no	—
211	C₄⋊C₄×C₆₁	—	2	1	no	—
212	C₄⋊C₄×C₆₁	—	2	2	no	—
213	C₄⋊C₄×C₆₁	—	2	4	no	—
214	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
215	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
217	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
218	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
219	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
220	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
222	C₂²⋊C₄×C₆₁	—	2	2	no	—
223	C₂²⋊C₄×C₆₁	—	2	2	no	—
224	C₂²⋊C₄×C₆₁	—	2	2	no	—
225	C₂²⋊C₄×C₆₁	—	2	8	no	—
226	C₂²⋊C₄×C₆₁	—	2	2	no	—
227	C₂²⋊C₄×C₆₁	—	2	4	no	—
228	C₂²⋊C₄×C₆₁	—	2	4	no	—
229	C₂²⋊C₄×C₆₁	—	2	2	no	—
230	C₂²⋊C₄×C₆₁	—	2	4	no	—
231	C₄²×C₆₁	—	2	4	no	—
232	C₄²×C₆₁	—	2	2	no	—
233	C₄²×C₆₁	C₂×D₄	1	4	yes	—
234	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
237	C₄²×C₆₁	C₂⁴	1	4	yes	—
238	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
239	C₂³×C₆₁	—	2	4	no	—
240	C₂³×C₆₁	—	2	2	no	—
241	C₂³×C₆₁	—	2	2	no	—
242	C₂³×C₆₁	—	4	2	no	—
243	C₂³×C₆₁	—	4	2	no	—
244	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
245	Q₈×C₆₁	—	2	4	no	—
246	Q₈×C₆₁	—	2	4	no	—
247	Q₈×C₆₁	—	2	4	no	—
248	Q₈×C₆₁	—	4	2	no	—
249	Q₈×C₆₁	—	8	1	no	—
250	Q₈×C₆₁	—	8	2	no	—
251	Q₈×C₆₁	—	8	2	no	—
252	D₄×C₆₁	—	2	4	no	—
253	D₄×C₆₁	—	2	2	no	—
254	D₄×C₆₁	—	2	4	no	—
255	D₄×C₆₁	—	4	4	no	—
256	D₄×C₆₁	—	4	2	no	—
257	D₄×C₆₁	—	8	1	no	—
258	D₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
259	C₂×C₄×C₆₁	—	2	4	no	—
260	C₂×C₄×C₆₁	—	2	2	no	—
261	C₂×C₄×C₆₁	—	2	2	no	—
262	C₂×C₄×C₆₁	—	2	2	no	—
263	C₂×C₄×C₆₁	—	2	2	no	—
264	C₂×C₄×C₆₁	—	2	2	no	—
265	C₂×C₄×C₆₁	—	2	4	no	—
266	C₂×C₄×C₆₁	—	4	2	no	—
267	C₂×C₄×C₆₁	—	4	1	no	—
268	C₂×C₄×C₆₁	—	4	4	no	—
269	C₂×C₄×C₆₁	—	4	2	no	—
270	C₂×C₄×C₆₁	—	4	4	no	—
271	C₂×C₄×C₆₁	—	4	4	no	—
272	C₂×C₄×C₆₁	—	4	4	no	—
273	C₂×C₄×C₆₁	—	8	4	no	—
274	C₂×C₄×C₆₁	—	8	8	no	—
275	C₂×C₄×C₆₁	#135	1	4	yes	—
276	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
277	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
279	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
284	C₈×C₆₁	—	2	4	no	—
285	C₈×C₆₁	—	2	4	no	—
286	#61819	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂²×C₆₁	—	2	1	no	—
288	C₂²×C₆₁	—	2	1	no	—
289	C₂²×C₆₁	—	2	4	no	—
290	C₂²×C₆₁	—	4	8	no	—
291	C₂²×C₆₁	—	4	2	no	—
292	C₂²×C₆₁	—	4	2	no	—
293	C₂²×C₆₁	—	4	2	no	—
294	C₂²×C₆₁	—	4	2	no	—
295	C₂²×C₆₁	—	4	2	no	—
296	C₂²×C₆₁	C₂×#135	1	2	yes	—
297	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
298	C₄×C₆₁	—	2	4	no	—
299	C₄×C₆₁	—	2	1	no	—
300	C₄×C₆₁	—	2	1	no	—
301	C₄×C₆₁	—	2	1	no	—
302	C₄×C₆₁	—	2	4	no	—
303	C₄×C₆₁	—	4	4	no	—
304	C₄×C₆₁	—	4	2	no	—
305	C₄×C₆₁	—	4	2	no	—
306	C₄×C₆₁	—	4	2	no	—
307	C₄×C₆₁	—	4	2	no	—
308	C₄×C₆₁	C₂×#135	1	2	yes	—
309	C₄×C₆₁	C₂×#135	1	2	yes	—
310	C₄×C₆₁	C₂×2₋¹⁺⁴	1	2	yes	—
311	#3507	C₂×C₆₁	1	1	yes	characteristic, complemented
312	C₂×#554	C₂×C₆₁	1	1	yes	characteristic, complemented
313	#3381	C₂×C₆₁	1	4	yes	complemented
314	#3380	C₂×C₆₁	1	4	yes	complemented
315	#3361	C₂×C₆₁	1	4	yes	complemented
316	#3360	C₂×C₆₁	1	4	yes	complemented
317	#3232	C₂×C₆₁	1	4	yes	complemented
318	#3230	C₂×C₆₁	1	1	yes	characteristic, complemented
319	#3227	C₂×C₆₁	1	1	yes	characteristic, complemented
320	#3105	C₂×C₆₁	1	4	yes	complemented
321	#3101	C₂×C₆₁	1	1	yes	characteristic, complemented
322	#3100	C₂×C₆₁	1	1	yes	characteristic
323	#2982	C₂×C₆₁	1	1	yes	characteristic, complemented
324	C₂×C₆₁	—	2	2	no	—
325	C₂×C₆₁	—	4	4	no	—
326	C₂×C₆₁	—	8	2	no	—
327	C₂×C₆₁	#3486	1	1	yes	characteristic
328	C₂×C₆₁	C₂²×#135	1	1	yes	characteristic
329	C₂×C₆₁	#3585	1	1	yes	characteristic
330	#559	C₂²×C₆₁	1	2	yes	complemented
331	#557	C₂²×C₆₁	1	2	yes	complemented
332	#556	—	2	2	no	—
333	#556	C₂²×C₆₁	1	2	yes	complemented
334	#554	—	2	2	no	—
335	#554	—	2	2	no	—
336	#554	C₂²×C₆₁	1	4	yes	—
337	#554	C₂²×C₆₁	1	4	yes	—
338	#554	C₂²×C₆₁	1	2	yes	complemented
339	#552	—	2	2	no	—
340	#539	—	2	2	no	—
341	#536	—	2	1	no	—
342	#533	C₂²×C₆₁	1	1	yes	characteristic
343	#532	C₂²×C₆₁	1	1	yes	characteristic
344	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
345	#529	C₂²×C₆₁	1	4	yes	—
346	C₂×#125	C₂²×C₆₁	1	2	yes	—
347	C₂×C₂²⋊Q₈	—	2	1	no	—
348	C₂×C₂²⋊Q₈	—	2	2	no	—
349	#520	—	2	1	no	—
350	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
351	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
352	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
353	C₂²×C₄⋊C₄	—	2	1	no	—
354	#500	C₂²×C₆₁	1	2	yes	complemented
355	#499	C₂²×C₆₁	1	8	yes	complemented
356	#498	C₂²×C₆₁	1	2	yes	complemented
357	#488	C₂²×C₆₁	1	8	yes	complemented
358	#487	C₂²×C₆₁	1	2	yes	complemented
359	#486	C₂²×C₆₁	1	2	yes	complemented
360	#484	C₂²×C₆₁	1	4	yes	—
361	#483	C₂²×C₆₁	1	2	yes	—
362	#481	C₂²×C₆₁	1	4	yes	—
363	#480	C₂²×C₆₁	1	2	yes	—
364	#479	C₂²×C₆₁	1	8	yes	complemented
365	#478	C₂²×C₆₁	1	8	yes	—
366	#478	C₂²×C₆₁	1	8	yes	complemented
367	#477	C₂²×C₆₁	1	4	yes	complemented
368	#476	C₂²×C₆₁	1	2	yes	—
369	#476	C₂²×C₆₁	1	4	yes	complemented
370	#474	C₂²×C₆₁	1	2	yes	—
371	#463	C₂²×C₆₁	1	8	yes	—
372	#461	C₂²×C₆₁	1	2	yes	—
373	#459	C₂²×C₆₁	1	2	yes	—
374	#450	C₂²×C₆₁	1	2	yes	—
375	#447	C₂²×C₆₁	1	2	yes	—
376	C₈×Q₈	C₂²×C₆₁	1	4	yes	complemented
377	C₈×Q₈	C₂²×C₆₁	1	4	yes	complemented
378	C₄×SD₁₆	C₂²×C₆₁	1	4	yes	—
379	C₄×SD₁₆	C₂²×C₆₁	1	8	yes	—
380	C₄×D₈	C₂²×C₆₁	1	4	yes	—
381	C₄×D₈	C₂²×C₆₁	1	8	yes	—
382	#431	C₂²×C₆₁	1	1	yes	characteristic
383	#431	C₂²×C₆₁	1	1	yes	characteristic
384	#431	C₂²×C₆₁	1	4	yes	—
385	C₆₁	#61819	1	1	yes	characteristic, direct product factor, Sylow
386	C₂×#41	—	2	1	no	—
387	C₂²×Q₈	—	2	1	no	—
388	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
389	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	—	2	2	no	—
392	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
393	C₂×D₈	C₂³×C₆₁	1	4	yes	—
394	C₄⋊Q₈	—	2	2	no	—
395	C₄⋊Q₈	—	2	1	no	—
396	C₄⋊Q₈	—	2	1	no	—
397	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
398	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
399	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
400	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
401	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
402	#127	C₂³×C₆₁	1	1	yes	characteristic
403	#126	—	2	4	no	—
404	#126	—	2	4	no	—
405	#125	—	2	4	no	—
406	#125	—	2	4	no	—
407	#125	—	2	1	no	—
408	#125	C₂³×C₆₁	1	8	yes	—
409	#125	C₂³×C₆₁	1	4	yes	—
410	#124	—	2	2	no	—
411	#124	—	2	1	no	—
412	#124	C₂³×C₆₁	1	4	yes	—
413	#123	—	2	4	no	—
414	#123	—	2	4	no	—
415	C₂²⋊Q₈	—	2	4	no	—
416	C₂²⋊Q₈	—	2	4	no	—
417	C₂²⋊Q₈	—	2	2	no	—
418	C₂²⋊Q₈	—	2	2	no	—
419	C₂²⋊Q₈	—	2	1	no	—
420	C₂²⋊Q₈	—	2	8	no	—
421	C₂²⋊Q₈	—	2	8	no	—
422	C₂²⋊Q₈	—	2	4	no	—
423	C₂²⋊Q₈	C₂³×C₆₁	1	2	yes	—
424	#121	—	2	1	no	—
425	#121	C₂³×C₆₁	1	2	yes	—
426	C₄×Q₈	—	2	2	no	—
427	C₄×Q₈	—	2	1	no	—
428	C₄×Q₈	D₄×C₆₁	1	2	yes	—
429	C₄×Q₈	D₄×C₆₁	1	2	yes	—
430	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
431	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
432	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
433	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
434	C₄×D₄	—	2	4	no	—
435	C₄×D₄	—	2	2	no	—
436	C₄×D₄	—	2	4	no	—
437	C₄×D₄	—	2	1	no	—
438	C₄×D₄	—	2	1	no	—
439	C₄×D₄	—	2	1	no	—
440	C₄×D₄	C₂³×C₆₁	1	2	yes	—
441	C₄×D₄	C₂³×C₆₁	1	8	yes	—
442	C₄×D₄	C₂³×C₆₁	1	2	yes	—
443	C₄×D₄	C₂³×C₆₁	1	4	yes	—
444	C₄×D₄	C₂³×C₆₁	1	2	yes	—
445	#117	—	2	1	no	—
446	#117	—	2	2	no	—
447	#117	—	2	4	no	—
448	C₂×C₄⋊C₄	—	2	2	no	—
449	C₂×C₄⋊C₄	—	2	1	no	—
450	C₂×C₄⋊C₄	—	2	2	no	—
451	C₂×C₄⋊C₄	—	2	4	no	—
452	C₂×C₄⋊C₄	—	2	4	no	—
453	C₂×C₄⋊C₄	—	2	2	no	—
454	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
455	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
456	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₂²⋊C₄	—	2	2	no	—
460	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
461	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
462	#107	C₂³×C₆₁	1	4	yes	—
463	#107	C₂³×C₆₁	1	8	yes	—
464	#106	C₂³×C₆₁	1	4	yes	—
465	#106	C₂³×C₆₁	1	8	yes	—
466	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
467	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
468	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
469	#103	C₂³×C₆₁	1	4	yes	—
470	#103	C₂³×C₆₁	1	8	yes	—
471	#102	C₂³×C₆₁	1	4	yes	—
472	#102	C₂³×C₆₁	1	4	yes	—
473	#102	C₂³×C₆₁	1	4	yes	—
474	#102	C₂³×C₆₁	1	8	yes	—
475	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
476	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
477	C₄×C₈	C₂³×C₆₁	1	2	yes	—
478	C₄×C₈	C₂³×C₆₁	1	8	yes	—
479	C₄×C₈	C₂³×C₆₁	1	2	yes	—
480	C₂⁴	—	2	1	no	—
481	#41	—	4	4	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	2	no	—
486	C₂×Q₈	—	2	2	no	—
487	C₂×Q₈	—	8	2	no	—
488	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
489	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
490	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
491	C₂×D₄	—	2	4	no	—
492	C₂×D₄	—	2	2	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
497	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	8	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	—	4	4	no	—
506	C₂²×C₄	—	8	4	no	—
507	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
508	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
509	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
510	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
512	SD₁₆	—	2	8	no	—
513	D₈	—	2	8	no	—
514	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
515	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
516	C₄⋊C₄	—	2	8	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	8	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	2	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
532	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
533	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
534	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
535	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
536	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
537	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
538	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	8	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	4	no	—
545	C₂²⋊C₄	—	2	4	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₄²	—	2	4	no	—
549	C₄²	—	2	2	no	—
550	C₄²	C₂×D₄×C₆₁	1	4	yes	—
551	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
552	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
553	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
554	C₄²	C₂⁴×C₆₁	1	4	yes	—
555	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
556	C₂³	—	2	4	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	4	2	no	—
560	C₂³	—	4	2	no	—
561	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
562	Q₈	—	2	4	no	—
563	Q₈	—	2	4	no	—
564	Q₈	—	2	4	no	—
565	Q₈	—	4	2	no	—
566	Q₈	—	8	1	no	—
567	Q₈	—	8	2	no	—
568	Q₈	—	8	2	no	—
569	D₄	—	2	4	no	—
570	D₄	—	2	2	no	—
571	D₄	—	2	4	no	—
572	D₄	—	4	4	no	—
573	D₄	—	4	2	no	—
574	D₄	—	8	1	no	—
575	D₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	8	4	no	—
591	C₂×C₄	—	8	8	no	—
592	C₂×C₄	#135×C₆₁	1	4	yes	—
593	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
594	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
595	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
596	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
599	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
600	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
601	C₈	—	2	4	no	—
602	C₈	—	2	4	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	4	8	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	C₂×#135×C₆₁	1	2	yes	—
613	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
614	C₄	—	2	4	no	—
615	C₄	—	2	4	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	4	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	C₂×#135×C₆₁	1	2	yes	—
625	C₄	C₂×#135×C₆₁	1	2	yes	—
626	C₄	C₂×2₋¹⁺⁴×C₆₁	1	2	yes	—
627	C₂	—	2	2	no	—
628	C₂	—	4	4	no	—
629	C₂	—	8	2	no	—
630	C₂	C₆₁×#3486	1	1	yes	characteristic
631	C₂	C₂²×#135×C₆₁	1	1	yes	characteristic
632	C₂	C₆₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₆₁×#3100	C₂
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
1. C₆₁×#3507	C₂	semidirect
2. C₂×C₆₁×#554	C₂	semidirect
3. C₆₁×#3381	C₂	semidirect
4. C₆₁×#3380	C₂	semidirect
5. C₆₁×#3361	C₂	semidirect
6. C₆₁×#3360	C₂	semidirect
7. C₆₁×#3232	C₂	semidirect
8. C₆₁×#3230	C₂	semidirect
9. C₆₁×#3227	C₂	semidirect
10. C₆₁×#3105	C₂	semidirect
11. C₆₁×#3101	C₂	semidirect
13. C₆₁×#2982	C₂	semidirect
14. C₆₁×#559	C₂²	semidirect
15. C₆₁×#557	C₂²	semidirect
17. C₆₁×#556	C₂²	semidirect
22. C₆₁×#554	C₂²	semidirect
38. C₆₁×#500	C₂²	semidirect
39. C₆₁×#499	C₂²	semidirect
40. C₆₁×#498	C₂²	semidirect
41. C₆₁×#488	C₂²	semidirect
42. C₆₁×#487	C₂²	semidirect
43. C₆₁×#486	C₂²	semidirect
48. C₆₁×#479	C₂²	semidirect
50. C₆₁×#478	C₂²	semidirect
51. C₆₁×#477	C₂²	semidirect
53. C₆₁×#476	C₂²	semidirect
60. C₈×Q₈×C₆₁	C₂²	semidirect
61. C₈×Q₈×C₆₁	C₂²	semidirect
286. #61819	C₆₁	direct (N × H)
311. #3507	C₂×C₆₁	semidirect
312. C₂×#554	C₂×C₆₁	semidirect
313. #3381	C₂×C₆₁	semidirect
314. #3380	C₂×C₆₁	semidirect
315. #3361	C₂×C₆₁	semidirect
316. #3360	C₂×C₆₁	semidirect
317. #3232	C₂×C₆₁	semidirect
318. #3230	C₂×C₆₁	semidirect
319. #3227	C₂×C₆₁	semidirect
320. #3105	C₂×C₆₁	semidirect
321. #3101	C₂×C₆₁	semidirect
323. #2982	C₂×C₆₁	semidirect
330. #559	C₂²×C₆₁	semidirect
331. #557	C₂²×C₆₁	semidirect
333. #556	C₂²×C₆₁	semidirect
338. #554	C₂²×C₆₁	semidirect
354. #500	C₂²×C₆₁	semidirect
355. #499	C₂²×C₆₁	semidirect
356. #498	C₂²×C₆₁	semidirect
357. #488	C₂²×C₆₁	semidirect
358. #487	C₂²×C₆₁	semidirect
359. #486	C₂²×C₆₁	semidirect
364. #479	C₂²×C₆₁	semidirect
366. #478	C₂²×C₆₁	semidirect
367. #477	C₂²×C₆₁	semidirect
369. #476	C₂²×C₆₁	semidirect
376. C₈×Q₈	C₂²×C₆₁	semidirect
377. C₈×Q₈	C₂²×C₆₁	semidirect
385. C₆₁	#61819	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×#554	2
C₆₁×#3381	2
C₆₁×#3380	2
C₆₁×#3361	2
C₆₁×#3360	2
C₆₁×#3232	2
C₆₁×#3230	2
C₆₁×#3227	2
C₆₁×#3105	2
C₆₁×#3101	2
C₆₁×#3100	2
C₆₁×#2982	2
#61819	61

Subgroups — C61×#61819 #29565252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#61819 #29565252