Subgroups — C₆₁×#61719 #29565152

Order 15616.

1798 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3506	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3357	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3356	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3333	C₂	1	4	yes	maximal, complemented
6	C₆₁×#3331	C₂	1	4	yes	maximal, complemented
7	C₆₁×#3230	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#3196	C₂	1	4	yes	maximal, complemented
10	C₆₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#3092	C₂	1	4	yes	maximal, complemented
13	C₆₁×#2975	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#559	—	2	2	no	—
15	C₆₁×#556	C₂²	1	2	yes	—
16	C₆₁×#555	C₂²	1	2	yes	—
17	C₆₁×#554	—	2	2	no	—
18	C₆₁×#554	—	2	2	no	—
19	C₆₁×#554	C₂²	1	4	yes	—
20	C₆₁×#554	C₂²	1	4	yes	—
21	C₆₁×#553	—	2	2	no	—
22	C₆₁×#550	C₂²	1	2	yes	—
23	C₆₁×#547	C₂²	1	2	yes	—
24	C₆₁×#538	—	2	2	no	—
25	C₆₁×#535	—	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	C₂×#125×C₆₁	—	2	1	no	—
30	C₆₁×#525	C₂²	1	4	yes	—
31	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
32	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
33	C₆₁×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₆₁	—	2	1	no	—
36	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
38	C₆₁×#500	C₂²	1	2	yes	complemented
39	C₆₁×#498	C₂²	1	2	yes	complemented
40	C₆₁×#487	C₂²	1	2	yes	complemented
41	C₆₁×#486	C₂²	1	2	yes	complemented
42	C₆₁×#485	C₂²	1	8	yes	complemented
43	C₆₁×#484	C₂²	1	2	yes	—
44	C₆₁×#484	C₂²	1	4	yes	—
45	C₆₁×#483	C₂²	1	8	yes	complemented
46	C₆₁×#481	C₂²	1	2	yes	—
47	C₆₁×#481	C₂²	1	4	yes	—
48	C₆₁×#480	C₂²	1	8	yes	complemented
49	C₆₁×#477	C₂²	1	4	yes	complemented
50	C₆₁×#477	C₂²	1	2	yes	complemented
51	C₆₁×#476	C₂²	1	4	yes	complemented
52	C₆₁×#475	C₂²	1	2	yes	complemented
53	C₆₁×#470	C₂²	1	8	yes	complemented
54	C₆₁×#468	C₂²	1	8	yes	complemented
55	C₆₁×#462	C₂²	1	2	yes	complemented
56	C₆₁×#460	C₂²	1	2	yes	complemented
57	C₆₁×#452	C₂²	1	8	yes	complemented
58	C₆₁×#451	C₂²	1	2	yes	—
59	C₆₁×#448	C₂²	1	2	yes	—
60	C₄×Q₁₆×C₆₁	C₂²	1	4	yes	complemented
61	C₄×SD₁₆×C₆₁	C₂²	1	4	yes	complemented
62	C₆₁×#436	C₂²	1	4	yes	—
63	C₆₁×#436	C₂²	1	4	yes	—
64	C₆₁×#431	C₂²	1	1	yes	characteristic
65	C₆₁×#431	C₂²	1	1	yes	characteristic
66	C₆₁×#428	C₂²	1	8	yes	complemented
67	C₆₁×#417	C₂²	1	8	yes	complemented
68	C₆₁×#407	C₂²	1	4	yes	—
69	C₂×#41×C₆₁	—	2	1	no	—
70	C₂×#41×C₆₁	C₂³	1	4	yes	—
71	C₂²×Q₈×C₆₁	—	2	1	no	—
72	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
74	C₂³×C₄×C₆₁	—	2	2	no	—
75	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
76	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
78	C₂×M₁₆×C₆₁	C₂³	1	8	yes	—
79	C₄⋊Q₈×C₆₁	—	2	1	no	—
80	C₄⋊Q₈×C₆₁	—	2	1	no	—
81	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
83	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
84	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
86	#127×C₆₁	—	2	2	no	—
87	#127×C₆₁	C₂³	1	1	yes	characteristic
88	#126×C₆₁	—	2	4	no	—
89	#125×C₆₁	—	2	4	no	—
90	#125×C₆₁	—	2	4	no	—
91	#125×C₆₁	—	2	1	no	—
92	#124×C₆₁	—	2	2	no	—
93	#124×C₆₁	—	2	4	no	—
94	#124×C₆₁	—	2	1	no	—
95	#123×C₆₁	—	2	4	no	—
96	#123×C₆₁	—	2	4	no	—
97	C₂²⋊Q₈×C₆₁	—	2	1	no	—
98	C₂²⋊Q₈×C₆₁	—	2	4	no	—
99	C₂²⋊Q₈×C₆₁	—	2	8	no	—
100	C₂²⋊Q₈×C₆₁	—	2	8	no	—
101	C₂²⋊Q₈×C₆₁	C₂³	1	2	yes	—
102	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
103	C₂²⋊Q₈×C₆₁	C₂³	1	8	yes	—
104	#121×C₆₁	—	2	2	no	—
105	#121×C₆₁	—	2	2	no	—
106	#121×C₆₁	—	2	1	no	—
107	#121×C₆₁	—	2	4	no	—
108	#121×C₆₁	C₂³	1	2	yes	—
109	#121×C₆₁	C₂³	1	4	yes	—
110	#120×C₆₁	—	2	4	no	—
111	C₄×Q₈×C₆₁	—	2	2	no	—
112	C₄×Q₈×C₆₁	—	2	1	no	—
113	C₄×Q₈×C₆₁	—	2	1	no	—
114	C₄×Q₈×C₆₁	—	2	1	no	—
115	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
116	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
117	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
118	C₄×D₄×C₆₁	—	2	4	no	—
119	C₄×D₄×C₆₁	—	2	4	no	—
120	C₄×D₄×C₆₁	—	2	4	no	—
121	C₄×D₄×C₆₁	—	2	4	no	—
122	C₄×D₄×C₆₁	—	2	2	no	—
123	C₄×D₄×C₆₁	—	2	1	no	—
124	C₄×D₄×C₆₁	D₄	1	2	yes	—
125	C₄×D₄×C₆₁	D₄	1	2	yes	—
126	C₄×D₄×C₆₁	C₂³	1	2	yes	—
127	C₄×D₄×C₆₁	C₂³	1	2	yes	—
128	C₄×D₄×C₆₁	C₂³	1	2	yes	—
129	#117×C₆₁	—	2	1	no	—
130	#117×C₆₁	—	2	2	no	—
131	#117×C₆₁	C₂³	1	4	yes	—
132	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
133	C₂×C₄⋊C₄×C₆₁	—	2	4	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	—	2	2	no	—
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	8	yes	—
141	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
142	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
143	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
144	C₂×C₄²×C₆₁	—	2	1	no	—
145	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
146	#107×C₆₁	C₂³	1	4	yes	—
147	#107×C₆₁	C₂³	1	8	yes	—
148	#106×C₆₁	C₂³	1	4	yes	—
149	#106×C₆₁	C₂³	1	8	yes	—
150	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
151	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
152	#103×C₆₁	C₂³	1	4	yes	—
153	#103×C₆₁	C₂³	1	8	yes	—
154	#103×C₆₁	C₂³	1	4	yes	—
155	#103×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	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
161	C₄×C₈×C₆₁	C₂³	1	2	yes	—
162	C₄×C₈×C₆₁	C₂³	1	2	yes	—
163	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
164	#41×C₆₁	—	4	8	no	—
165	#41×C₆₁	—	4	4	no	—
166	C₂×Q₈×C₆₁	—	2	1	no	—
167	C₂×Q₈×C₆₁	—	2	1	no	—
168	C₂×Q₈×C₆₁	—	2	1	no	—
169	C₂×Q₈×C₆₁	—	2	2	no	—
170	C₂×Q₈×C₆₁	—	2	2	no	—
171	C₂×Q₈×C₆₁	—	4	2	no	—
172	C₂×Q₈×C₆₁	—	8	2	no	—
173	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
174	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
175	C₂×D₄×C₆₁	—	2	2	no	—
176	C₂×D₄×C₆₁	—	2	4	no	—
177	C₂×D₄×C₆₁	—	2	4	no	—
178	C₂×D₄×C₆₁	—	2	4	no	—
179	C₂×D₄×C₆₁	—	2	1	no	—
180	C₂×D₄×C₆₁	—	2	1	no	—
181	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
182	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
183	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₆₁	—	2	1	no	—
185	C₂²×C₄×C₆₁	—	2	1	no	—
186	C₂²×C₄×C₆₁	—	2	2	no	—
187	C₂²×C₄×C₆₁	—	2	8	no	—
188	C₂²×C₄×C₆₁	—	2	2	no	—
189	C₂²×C₄×C₆₁	—	2	4	no	—
190	C₂²×C₄×C₆₁	—	4	4	no	—
191	C₂²×C₄×C₆₁	—	8	4	no	—
192	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
198	Q₁₆×C₆₁	—	4	4	no	—
199	SD₁₆×C₆₁	—	4	4	no	—
200	M₁₆×C₆₁	—	4	8	no	—
201	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
202	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
203	C₄⋊C₄×C₆₁	—	2	1	no	—
204	C₄⋊C₄×C₆₁	—	2	8	no	—
205	C₄⋊C₄×C₆₁	—	2	2	no	—
206	C₄⋊C₄×C₆₁	—	2	1	no	—
207	C₄⋊C₄×C₆₁	—	2	2	no	—
208	C₄⋊C₄×C₆₁	—	2	2	no	—
209	C₄⋊C₄×C₆₁	—	2	4	no	—
210	C₄⋊C₄×C₆₁	—	2	2	no	—
211	C₄⋊C₄×C₆₁	—	2	2	no	—
212	C₄⋊C₄×C₆₁	—	2	8	no	—
213	C₄⋊C₄×C₆₁	—	2	1	no	—
214	C₄⋊C₄×C₆₁	—	2	2	no	—
215	C₄⋊C₄×C₆₁	—	2	1	no	—
216	C₄⋊C₄×C₆₁	—	2	1	no	—
217	C₄⋊C₄×C₆₁	—	2	2	no	—
218	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
219	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
220	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
222	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
223	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
224	C₂²⋊C₄×C₆₁	—	2	8	no	—
225	C₂²⋊C₄×C₆₁	—	2	2	no	—
226	C₂²⋊C₄×C₆₁	—	2	4	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	2	no	—
231	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
232	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
233	C₄²×C₆₁	—	2	2	no	—
234	C₄²×C₆₁	—	2	2	no	—
235	C₄²×C₆₁	—	2	4	no	—
236	C₄²×C₆₁	—	2	2	no	—
237	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
238	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
239	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
240	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
241	C₂³×C₆₁	—	2	4	no	—
242	C₂³×C₆₁	—	2	2	no	—
243	C₂³×C₆₁	—	4	2	no	—
244	C₂³×C₆₁	—	4	2	no	—
245	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
246	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
247	Q₈×C₆₁	—	2	2	no	—
248	Q₈×C₆₁	—	2	2	no	—
249	Q₈×C₆₁	—	2	4	no	—
250	Q₈×C₆₁	—	4	2	no	—
251	Q₈×C₆₁	—	4	2	no	—
252	Q₈×C₆₁	—	8	1	no	—
253	Q₈×C₆₁	—	8	2	no	—
254	Q₈×C₆₁	—	8	2	no	—
255	D₄×C₆₁	—	2	4	no	—
256	D₄×C₆₁	—	2	4	no	—
257	D₄×C₆₁	—	2	4	no	—
258	D₄×C₆₁	—	4	4	no	—
259	D₄×C₆₁	—	4	2	no	—
260	D₄×C₆₁	—	8	1	no	—
261	D₄×C₆₁	—	8	4	no	—
262	D₄×C₆₁	—	8	4	no	—
263	C₂×C₄×C₆₁	—	2	8	no	—
264	C₂×C₄×C₆₁	—	2	2	no	—
265	C₂×C₄×C₆₁	—	2	2	no	—
266	C₂×C₄×C₆₁	—	2	2	no	—
267	C₂×C₄×C₆₁	—	2	2	no	—
268	C₂×C₄×C₆₁	—	2	2	no	—
269	C₂×C₄×C₆₁	—	2	2	no	—
270	C₂×C₄×C₆₁	—	2	4	no	—
271	C₂×C₄×C₆₁	—	4	4	no	—
272	C₂×C₄×C₆₁	—	4	2	no	—
273	C₂×C₄×C₆₁	—	4	1	no	—
274	C₂×C₄×C₆₁	—	4	2	no	—
275	C₂×C₄×C₆₁	—	4	4	no	—
276	C₂×C₄×C₆₁	—	4	2	no	—
277	C₂×C₄×C₆₁	—	4	4	no	—
278	C₂×C₄×C₆₁	—	8	4	no	—
279	C₂×C₄×C₆₁	—	8	8	no	—
280	C₂×C₄×C₆₁	#137	1	4	yes	—
281	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
282	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
288	C₈×C₆₁	—	2	4	no	—
289	C₈×C₆₁	—	2	4	no	—
290	#61719	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂²×C₆₁	—	2	1	no	—
292	C₂²×C₆₁	—	2	1	no	—
293	C₂²×C₆₁	—	2	4	no	—
294	C₂²×C₆₁	—	2	4	no	—
295	C₂²×C₆₁	—	2	4	no	—
296	C₂²×C₆₁	—	4	8	no	—
297	C₂²×C₆₁	—	4	2	no	—
298	C₂²×C₆₁	—	4	2	no	—
299	C₂²×C₆₁	—	4	2	no	—
300	C₂²×C₆₁	C₂×#137	1	2	yes	—
301	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
302	C₄×C₆₁	—	2	1	no	—
303	C₄×C₆₁	—	2	1	no	—
304	C₄×C₆₁	—	2	1	no	—
305	C₄×C₆₁	—	2	4	no	—
306	C₄×C₆₁	—	4	2	no	—
307	C₄×C₆₁	—	4	4	no	—
308	C₄×C₆₁	—	4	2	no	—
309	C₄×C₆₁	—	4	2	no	—
310	C₄×C₆₁	—	4	2	no	—
311	C₄×C₆₁	—	4	2	no	—
312	C₄×C₆₁	C₂×#137	1	2	yes	—
313	C₄×C₆₁	C₂×#137	1	2	yes	—
314	C₄×C₆₁	#585	1	2	yes	—
315	#3506	C₂×C₆₁	1	1	yes	characteristic
316	C₂×#554	C₂×C₆₁	1	1	yes	characteristic, complemented
317	#3357	C₂×C₆₁	1	4	yes	complemented
318	#3356	C₂×C₆₁	1	4	yes	complemented
319	#3333	C₂×C₆₁	1	4	yes	complemented
320	#3331	C₂×C₆₁	1	4	yes	complemented
321	#3230	C₂×C₆₁	1	1	yes	characteristic, complemented
322	#3227	C₂×C₆₁	1	1	yes	characteristic, complemented
323	#3196	C₂×C₆₁	1	4	yes	complemented
324	#3104	C₂×C₆₁	1	1	yes	characteristic, complemented
325	#3103	C₂×C₆₁	1	1	yes	characteristic, complemented
326	#3092	C₂×C₆₁	1	4	yes	complemented
327	#2975	C₂×C₆₁	1	1	yes	characteristic, complemented
328	C₂×C₆₁	—	2	2	no	—
329	C₂×C₆₁	—	4	4	no	—
330	C₂×C₆₁	—	8	2	no	—
331	C₂×C₆₁	#3485	1	1	yes	characteristic
332	C₂×C₆₁	C₂²×#137	1	1	yes	characteristic
333	C₂×C₆₁	#3585	1	1	yes	characteristic
334	#559	—	2	2	no	—
335	#556	C₂²×C₆₁	1	2	yes	—
336	#555	C₂²×C₆₁	1	2	yes	—
337	#554	—	2	2	no	—
338	#554	—	2	2	no	—
339	#554	C₂²×C₆₁	1	4	yes	—
340	#554	C₂²×C₆₁	1	4	yes	—
341	#553	—	2	2	no	—
342	#550	C₂²×C₆₁	1	2	yes	—
343	#547	C₂²×C₆₁	1	2	yes	—
344	#538	—	2	2	no	—
345	#535	—	2	1	no	—
346	#533	C₂²×C₆₁	1	1	yes	characteristic
347	#532	C₂²×C₆₁	1	1	yes	characteristic
348	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
349	C₂×#125	—	2	1	no	—
350	#525	C₂²×C₆₁	1	4	yes	—
351	C₂×C₂²⋊Q₈	—	2	2	no	—
352	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
353	#520	—	2	1	no	—
354	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
355	C₂×C₄×D₄	—	2	1	no	—
356	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
357	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
358	#500	C₂²×C₆₁	1	2	yes	complemented
359	#498	C₂²×C₆₁	1	2	yes	complemented
360	#487	C₂²×C₆₁	1	2	yes	complemented
361	#486	C₂²×C₆₁	1	2	yes	complemented
362	#485	C₂²×C₆₁	1	8	yes	complemented
363	#484	C₂²×C₆₁	1	2	yes	—
364	#484	C₂²×C₆₁	1	4	yes	—
365	#483	C₂²×C₆₁	1	8	yes	complemented
366	#481	C₂²×C₆₁	1	2	yes	—
367	#481	C₂²×C₆₁	1	4	yes	—
368	#480	C₂²×C₆₁	1	8	yes	complemented
369	#477	C₂²×C₆₁	1	4	yes	complemented
370	#477	C₂²×C₆₁	1	2	yes	complemented
371	#476	C₂²×C₆₁	1	4	yes	complemented
372	#475	C₂²×C₆₁	1	2	yes	complemented
373	#470	C₂²×C₆₁	1	8	yes	complemented
374	#468	C₂²×C₆₁	1	8	yes	complemented
375	#462	C₂²×C₆₁	1	2	yes	complemented
376	#460	C₂²×C₆₁	1	2	yes	complemented
377	#452	C₂²×C₆₁	1	8	yes	complemented
378	#451	C₂²×C₆₁	1	2	yes	—
379	#448	C₂²×C₆₁	1	2	yes	—
380	C₄×Q₁₆	C₂²×C₆₁	1	4	yes	complemented
381	C₄×SD₁₆	C₂²×C₆₁	1	4	yes	complemented
382	#436	C₂²×C₆₁	1	4	yes	—
383	#436	C₂²×C₆₁	1	4	yes	—
384	#431	C₂²×C₆₁	1	1	yes	characteristic
385	#431	C₂²×C₆₁	1	1	yes	characteristic
386	#428	C₂²×C₆₁	1	8	yes	complemented
387	#417	C₂²×C₆₁	1	8	yes	complemented
388	#407	C₂²×C₆₁	1	4	yes	—
389	C₆₁	#61719	1	1	yes	characteristic, direct product factor, Sylow
390	C₂×#41	—	2	1	no	—
391	C₂×#41	C₂³×C₆₁	1	4	yes	—
392	C₂²×Q₈	—	2	1	no	—
393	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
394	C₂²×D₄	D₄×C₆₁	1	1	yes	characteristic
395	C₂³×C₄	—	2	2	no	—
396	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
397	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
398	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
399	C₂×M₁₆	C₂³×C₆₁	1	8	yes	—
400	C₄⋊Q₈	—	2	1	no	—
401	C₄⋊Q₈	—	2	1	no	—
402	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
403	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
404	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
405	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
406	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
407	#127	—	2	2	no	—
408	#127	C₂³×C₆₁	1	1	yes	characteristic
409	#126	—	2	4	no	—
410	#125	—	2	4	no	—
411	#125	—	2	4	no	—
412	#125	—	2	1	no	—
413	#124	—	2	2	no	—
414	#124	—	2	4	no	—
415	#124	—	2	1	no	—
416	#123	—	2	4	no	—
417	#123	—	2	4	no	—
418	C₂²⋊Q₈	—	2	4	no	—
419	C₂²⋊Q₈	—	2	1	no	—
420	C₂²⋊Q₈	—	2	8	no	—
421	C₂²⋊Q₈	—	2	8	no	—
422	C₂²⋊Q₈	C₂³×C₆₁	1	8	yes	—
423	C₂²⋊Q₈	C₂³×C₆₁	1	2	yes	—
424	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
425	#121	—	2	2	no	—
426	#121	—	2	2	no	—
427	#121	—	2	1	no	—
428	#121	—	2	4	no	—
429	#121	C₂³×C₆₁	1	2	yes	—
430	#121	C₂³×C₆₁	1	4	yes	—
431	#120	—	2	4	no	—
432	C₄×Q₈	—	2	2	no	—
433	C₄×Q₈	—	2	1	no	—
434	C₄×Q₈	—	2	1	no	—
435	C₄×Q₈	—	2	1	no	—
436	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
437	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
438	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
439	C₄×D₄	—	2	4	no	—
440	C₄×D₄	—	2	4	no	—
441	C₄×D₄	—	2	4	no	—
442	C₄×D₄	—	2	4	no	—
443	C₄×D₄	—	2	2	no	—
444	C₄×D₄	—	2	1	no	—
445	C₄×D₄	D₄×C₆₁	1	2	yes	—
446	C₄×D₄	D₄×C₆₁	1	2	yes	—
447	C₄×D₄	C₂³×C₆₁	1	2	yes	—
448	C₄×D₄	C₂³×C₆₁	1	2	yes	—
449	C₄×D₄	C₂³×C₆₁	1	2	yes	—
450	#117	—	2	1	no	—
451	#117	—	2	2	no	—
452	#117	C₂³×C₆₁	1	4	yes	—
453	C₂×C₄⋊C₄	—	2	2	no	—
454	C₂×C₄⋊C₄	—	2	4	no	—
455	C₂×C₄⋊C₄	—	2	1	no	—
456	C₂×C₄⋊C₄	—	2	1	no	—
457	C₂×C₄⋊C₄	—	2	2	no	—
458	C₂×C₄⋊C₄	—	2	2	no	—
459	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
460	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
461	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
462	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
463	C₂×C₂²⋊C₄	—	2	2	no	—
464	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
465	C₂×C₄²	—	2	1	no	—
466	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
467	#107	C₂³×C₆₁	1	4	yes	—
468	#107	C₂³×C₆₁	1	8	yes	—
469	#106	C₂³×C₆₁	1	4	yes	—
470	#106	C₂³×C₆₁	1	8	yes	—
471	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
472	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
473	#103	C₂³×C₆₁	1	4	yes	—
474	#103	C₂³×C₆₁	1	8	yes	—
475	#103	C₂³×C₆₁	1	4	yes	—
476	#103	C₂³×C₆₁	1	4	yes	—
477	#102	C₂³×C₆₁	1	4	yes	—
478	#102	C₂³×C₆₁	1	8	yes	—
479	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
480	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
481	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
482	C₄×C₈	C₂³×C₆₁	1	2	yes	—
483	C₄×C₈	C₂³×C₆₁	1	2	yes	—
484	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
485	#41	—	4	8	no	—
486	#41	—	4	4	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	2	no	—
491	C₂×Q₈	—	2	2	no	—
492	C₂×Q₈	—	4	2	no	—
493	C₂×Q₈	—	8	2	no	—
494	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
495	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	4	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
503	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
504	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	2	no	—
508	C₂²×C₄	—	2	8	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	4	4	no	—
512	C₂²×C₄	—	8	4	no	—
513	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
514	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
515	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
516	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
518	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
519	Q₁₆	—	4	4	no	—
520	SD₁₆	—	4	4	no	—
521	M₁₆	—	4	8	no	—
522	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
523	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	8	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	2	no	—
532	C₄⋊C₄	—	2	2	no	—
533	C₄⋊C₄	—	2	8	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	2	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
540	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
541	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
542	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
543	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
544	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
545	C₂²⋊C₄	—	2	8	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₂²⋊C₄	—	2	4	no	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	2	no	—
551	C₂²⋊C₄	—	2	2	no	—
552	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
553	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
554	C₄²	—	2	2	no	—
555	C₄²	—	2	2	no	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	2	no	—
558	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
559	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
560	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
561	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
562	C₂³	—	2	4	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	4	2	no	—
565	C₂³	—	4	2	no	—
566	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
567	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
568	Q₈	—	2	2	no	—
569	Q₈	—	2	2	no	—
570	Q₈	—	2	4	no	—
571	Q₈	—	4	2	no	—
572	Q₈	—	4	2	no	—
573	Q₈	—	8	1	no	—
574	Q₈	—	8	2	no	—
575	Q₈	—	8	2	no	—
576	D₄	—	2	4	no	—
577	D₄	—	2	4	no	—
578	D₄	—	2	4	no	—
579	D₄	—	4	4	no	—
580	D₄	—	4	2	no	—
581	D₄	—	8	1	no	—
582	D₄	—	8	4	no	—
583	D₄	—	8	4	no	—
584	C₂×C₄	—	2	8	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	4	4	no	—
593	C₂×C₄	—	4	2	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	4	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	4	no	—
599	C₂×C₄	—	8	4	no	—
600	C₂×C₄	—	8	8	no	—
601	C₂×C₄	#137×C₆₁	1	4	yes	—
602	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
603	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
604	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
605	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
606	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
607	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
608	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
609	C₈	—	2	4	no	—
610	C₈	—	2	4	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	1	no	—
613	C₂²	—	2	4	no	—
614	C₂²	—	2	4	no	—
615	C₂²	—	2	4	no	—
616	C₂²	—	4	8	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	2	no	—
620	C₂²	C₂×#137×C₆₁	1	2	yes	—
621	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
622	C₄	—	2	1	no	—
623	C₄	—	2	1	no	—
624	C₄	—	2	1	no	—
625	C₄	—	2	4	no	—
626	C₄	—	4	2	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	4	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	2	no	—
632	C₄	C₂×#137×C₆₁	1	2	yes	—
633	C₄	C₂×#137×C₆₁	1	2	yes	—
634	C₄	C₆₁×#585	1	2	yes	—
635	C₂	—	2	2	no	—
636	C₂	—	4	4	no	—
637	C₂	—	8	2	no	—
638	C₂	C₆₁×#3485	1	1	yes	characteristic
639	C₂	C₂²×#137×C₆₁	1	1	yes	characteristic
640	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×#3506	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄×D₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#554	C₂	semidirect
3. C₆₁×#3357	C₂	semidirect
4. C₆₁×#3356	C₂	semidirect
5. C₆₁×#3333	C₂	semidirect
6. C₆₁×#3331	C₂	semidirect
7. C₆₁×#3230	C₂	semidirect
8. C₆₁×#3227	C₂	semidirect
9. C₆₁×#3196	C₂	semidirect
10. C₆₁×#3104	C₂	semidirect
11. C₆₁×#3103	C₂	semidirect
12. C₆₁×#3092	C₂	semidirect
13. C₆₁×#2975	C₂	semidirect
38. C₆₁×#500	C₂²	semidirect
39. C₆₁×#498	C₂²	semidirect
40. C₆₁×#487	C₂²	semidirect
41. C₆₁×#486	C₂²	semidirect
42. C₆₁×#485	C₂²	semidirect
45. C₆₁×#483	C₂²	semidirect
48. C₆₁×#480	C₂²	semidirect
49. C₆₁×#477	C₂²	semidirect
50. C₆₁×#477	C₂²	semidirect
51. C₆₁×#476	C₂²	semidirect
52. C₆₁×#475	C₂²	semidirect
53. C₆₁×#470	C₂²	semidirect
54. C₆₁×#468	C₂²	semidirect
55. C₆₁×#462	C₂²	semidirect
56. C₆₁×#460	C₂²	semidirect
57. C₆₁×#452	C₂²	semidirect
60. C₄×Q₁₆×C₆₁	C₂²	semidirect
61. C₄×SD₁₆×C₆₁	C₂²	semidirect
66. C₆₁×#428	C₂²	semidirect
67. C₆₁×#417	C₂²	semidirect
290. #61719	C₆₁	direct (N × H)
316. C₂×#554	C₂×C₆₁	semidirect
317. #3357	C₂×C₆₁	semidirect
318. #3356	C₂×C₆₁	semidirect
319. #3333	C₂×C₆₁	semidirect
320. #3331	C₂×C₆₁	semidirect
321. #3230	C₂×C₆₁	semidirect
322. #3227	C₂×C₆₁	semidirect
323. #3196	C₂×C₆₁	semidirect
324. #3104	C₂×C₆₁	semidirect
325. #3103	C₂×C₆₁	semidirect
326. #3092	C₂×C₆₁	semidirect
327. #2975	C₂×C₆₁	semidirect
358. #500	C₂²×C₆₁	semidirect
359. #498	C₂²×C₆₁	semidirect
360. #487	C₂²×C₆₁	semidirect
361. #486	C₂²×C₆₁	semidirect
362. #485	C₂²×C₆₁	semidirect
365. #483	C₂²×C₆₁	semidirect
368. #480	C₂²×C₆₁	semidirect
369. #477	C₂²×C₆₁	semidirect
370. #477	C₂²×C₆₁	semidirect
371. #476	C₂²×C₆₁	semidirect
372. #475	C₂²×C₆₁	semidirect
373. #470	C₂²×C₆₁	semidirect
374. #468	C₂²×C₆₁	semidirect
375. #462	C₂²×C₆₁	semidirect
376. #460	C₂²×C₆₁	semidirect
377. #452	C₂²×C₆₁	semidirect
380. C₄×Q₁₆	C₂²×C₆₁	semidirect
381. C₄×SD₁₆	C₂²×C₆₁	semidirect
386. #428	C₂²×C₆₁	semidirect
387. #417	C₂²×C₆₁	semidirect
389. C₆₁	#61719	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×#3506	2
C₂×C₆₁×#554	2
C₆₁×#3357	2
C₆₁×#3356	2
C₆₁×#3333	2
C₆₁×#3331	2
C₆₁×#3230	2
C₆₁×#3227	2
C₆₁×#3196	2
C₆₁×#3104	2
C₆₁×#3103	2
C₆₁×#3092	2
C₆₁×#2975	2
#61719	61

Subgroups — C61×#61719 #29565152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#61719 #29565152