Subgroups — C₆₁×#60877 #29564310

Order 15616.

1786 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3508	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3503	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3241	C₂	1	8	yes	maximal, complemented
4	C₆₁×#3228	C₂	1	2	yes	maximal, complemented
5	C₆₁×#3228	C₂	1	2	yes	maximal, complemented
6	C₆₁×#3227	C₂	1	2	yes	maximal, complemented
7	C₆₁×#3226	C₂	1	2	yes	maximal, complemented
8	C₆₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#3096	C₂	1	4	yes	maximal, complemented
13	C₆₁×#3081	C₂	1	4	yes	maximal, complemented
14	C₆₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#559	—	2	1	no	—
16	C₆₁×#559	—	2	1	no	—
17	Q₈²×C₆₁	—	2	1	no	—
18	C₆₁×#557	—	2	1	no	—
19	C₆₁×#554	—	2	2	no	—
20	C₆₁×#554	—	2	1	no	—
21	C₆₁×#550	—	2	1	no	—
22	C₆₁×#550	—	2	1	no	—
23	D₄×Q₈×C₆₁	—	2	2	no	—
24	D₄×Q₈×C₆₁	—	2	1	no	—
25	C₆₁×#546	—	2	2	no	—
26	C₆₁×#546	—	2	2	no	—
27	C₆₁×#544	C₂²	1	2	yes	—
28	C₆₁×#544	C₂²	1	4	yes	complemented
29	C₆₁×#540	C₂²	1	4	yes	—
30	C₆₁×#535	C₂²	1	2	yes	—
31	C₆₁×#533	C₂²	1	1	yes	characteristic
32	C₆₁×#533	C₂²	1	1	yes	characteristic
33	C₆₁×#533	C₂²	1	2	yes	complemented
34	C₆₁×#532	C₂²	1	2	yes	—
35	C₆₁×#532	C₂²	1	2	yes	—
36	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	complemented
38	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
39	C₆₁×#520	C₂²	1	2	yes	—
40	C₆₁×#518	C₂²	1	2	yes	—
41	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#501	C₂²	1	4	yes	complemented
43	C₆₁×#500	C₂²	1	1	yes	characteristic, complemented
44	C₆₁×#500	C₂²	1	1	yes	characteristic, complemented
45	C₆₁×#500	C₂²	1	2	yes	complemented
46	C₆₁×#499	C₂²	1	4	yes	complemented
47	C₆₁×#498	C₂²	1	1	yes	characteristic, complemented
48	C₆₁×#498	C₂²	1	1	yes	characteristic, complemented
49	C₆₁×#498	C₂²	1	2	yes	complemented
50	C₆₁×#490	C₂²	1	4	yes	—
51	C₆₁×#486	C₂²	1	2	yes	—
52	C₆₁×#486	C₂²	1	2	yes	—
53	C₆₁×#486	C₂²	1	4	yes	—
54	C₆₁×#486	C₂²	1	2	yes	—
55	C₆₁×#486	C₂²	1	2	yes	—
56	C₆₁×#482	C₂²	1	8	yes	complemented
57	C₆₁×#482	C₂²	1	8	yes	complemented
58	C₆₁×#481	C₂²	1	8	yes	complemented
59	C₆₁×#480	C₂²	1	8	yes	complemented
60	C₆₁×#478	C₂²	1	8	yes	complemented
61	C₆₁×#478	C₂²	1	8	yes	complemented
62	C₆₁×#476	C₂²	1	8	yes	complemented
63	C₆₁×#474	C₂²	1	8	yes	complemented
64	C₆₁×#433	C₂²	1	4	yes	complemented
65	C₆₁×#431	C₂²	1	2	yes	complemented
66	C₆₁×#431	C₂²	1	2	yes	complemented
67	C₆₁×#428	C₂²	1	4	yes	—
68	C₆₁×#427	C₂²	1	4	yes	—
69	C₆₁×#424	C₂²	1	2	yes	—
70	C₆₁×#418	C₂²	1	4	yes	—
71	C₂×#102×C₆₁	C₂²	1	4	yes	—
72	C₆₁×#408	C₂²	1	2	yes	—
73	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
74	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
75	C₂×#41×C₆₁	—	2	2	no	—
76	C₂×#41×C₆₁	—	2	2	no	—
77	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₆₁	—	2	1	no	—
79	C₂²×Q₈×C₆₁	—	2	2	no	—
80	C₂²×D₄×C₆₁	—	2	1	no	—
81	C₂²×D₄×C₆₁	—	2	1	no	—
82	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
83	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
84	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
85	C₄⋊Q₈×C₆₁	—	2	2	no	—
86	C₄⋊Q₈×C₆₁	—	2	4	no	—
87	C₄⋊Q₈×C₆₁	—	2	1	no	—
88	C₄⋊Q₈×C₆₁	—	2	1	no	—
89	C₄⋊Q₈×C₆₁	—	2	2	no	—
90	C₄⋊Q₈×C₆₁	—	2	2	no	—
91	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
92	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
93	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
94	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
95	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
96	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
97	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
98	#127×C₆₁	—	2	2	no	—
99	#127×C₆₁	—	2	1	no	—
100	#127×C₆₁	—	2	1	no	—
101	#127×C₆₁	C₂³	1	2	yes	—
102	#127×C₆₁	C₂³	1	2	yes	—
103	#125×C₆₁	—	2	2	no	—
104	#125×C₆₁	—	2	4	no	—
105	#125×C₆₁	C₂³	1	2	yes	—
106	#125×C₆₁	C₂³	1	2	yes	—
107	#125×C₆₁	C₂³	1	4	yes	—
108	#124×C₆₁	—	2	2	no	—
109	#124×C₆₁	—	2	2	no	—
110	#124×C₆₁	—	2	2	no	—
111	#123×C₆₁	—	2	4	no	—
112	#123×C₆₁	—	2	4	no	—
113	C₂²⋊Q₈×C₆₁	—	2	1	no	—
114	C₂²⋊Q₈×C₆₁	—	2	1	no	—
115	C₂²⋊Q₈×C₆₁	—	2	1	no	—
116	C₂²⋊Q₈×C₆₁	—	2	1	no	—
117	C₂²⋊Q₈×C₆₁	—	2	2	no	—
118	C₂²⋊Q₈×C₆₁	—	2	2	no	—
119	C₂²⋊Q₈×C₆₁	—	2	4	no	—
120	C₂²⋊Q₈×C₆₁	—	2	4	no	—
121	C₂²⋊Q₈×C₆₁	—	2	4	no	—
122	#121×C₆₁	—	2	2	no	—
123	#121×C₆₁	—	2	2	no	—
124	#121×C₆₁	—	2	4	no	—
125	#121×C₆₁	—	2	4	no	—
126	#121×C₆₁	C₂³	1	4	yes	—
127	#121×C₆₁	C₂³	1	4	yes	—
128	#120×C₆₁	—	2	4	no	—
129	C₄×Q₈×C₆₁	—	2	2	no	—
130	C₄×Q₈×C₆₁	—	2	1	no	—
131	C₄×Q₈×C₆₁	—	2	1	no	—
132	C₄×Q₈×C₆₁	—	2	1	no	—
133	C₄×Q₈×C₆₁	—	2	1	no	—
134	C₄×Q₈×C₆₁	—	2	2	no	—
135	C₄×Q₈×C₆₁	—	2	2	no	—
136	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
137	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
138	C₄×D₄×C₆₁	—	2	4	no	—
139	C₄×D₄×C₆₁	—	2	2	no	—
140	C₄×D₄×C₆₁	—	2	2	no	—
141	C₄×D₄×C₆₁	—	2	2	no	—
142	C₄×D₄×C₆₁	—	2	2	no	—
143	C₄×D₄×C₆₁	—	2	2	no	—
144	C₄×D₄×C₆₁	D₄	1	2	yes	—
145	C₄×D₄×C₆₁	D₄	1	2	yes	—
146	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
147	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
148	C₄×D₄×C₆₁	C₂³	1	4	yes	—
149	C₄×D₄×C₆₁	C₂³	1	4	yes	—
150	#117×C₆₁	C₂³	1	2	yes	—
151	#117×C₆₁	C₂³	1	2	yes	—
152	#117×C₆₁	C₂³	1	4	yes	—
153	#117×C₆₁	C₂³	1	2	yes	—
154	#117×C₆₁	C₂³	1	1	yes	characteristic
155	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
156	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
157	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
159	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
160	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
161	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
162	#107×C₆₁	C₂³	1	4	yes	—
163	#107×C₆₁	C₂³	1	4	yes	—
164	#107×C₆₁	C₂³	1	8	yes	—
165	#106×C₆₁	C₂³	1	4	yes	—
166	#106×C₆₁	C₂³	1	4	yes	—
167	#106×C₆₁	C₂³	1	8	yes	—
168	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
169	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
170	#102×C₆₁	C₂³	1	8	yes	—
171	#102×C₆₁	C₂³	1	8	yes	—
172	#102×C₆₁	C₂³	1	8	yes	—
173	#102×C₆₁	C₂³	1	8	yes	—
174	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
175	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
176	#97×C₆₁	C₂³	1	2	yes	—
177	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
178	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₆₁	C₂³	1	2	yes	—
180	C₄×C₈×C₆₁	C₂³	1	2	yes	—
181	C₂⁴×C₆₁	—	2	2	no	—
182	#41×C₆₁	—	2	4	no	—
183	#41×C₆₁	—	8	4	no	—
184	#41×C₆₁	—	8	4	no	—
185	C₂×Q₈×C₆₁	—	2	1	no	—
186	C₂×Q₈×C₆₁	—	2	1	no	—
187	C₂×Q₈×C₆₁	—	2	1	no	—
188	C₂×Q₈×C₆₁	—	2	1	no	—
189	C₂×Q₈×C₆₁	—	2	2	no	—
190	C₂×Q₈×C₆₁	—	2	4	no	—
191	C₂×Q₈×C₆₁	—	2	2	no	—
192	C₂×Q₈×C₆₁	—	8	2	no	—
193	C₂×Q₈×C₆₁	—	8	4	no	—
194	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
195	C₂×D₄×C₆₁	—	2	4	no	—
196	C₂×D₄×C₆₁	—	2	4	no	—
197	C₂×D₄×C₆₁	—	2	2	no	—
198	C₂×D₄×C₆₁	—	2	2	no	—
199	C₂×D₄×C₆₁	—	2	2	no	—
200	C₂×D₄×C₆₁	—	2	2	no	—
201	C₂×D₄×C₆₁	—	2	2	no	—
202	C₂×D₄×C₆₁	—	2	8	no	—
203	C₂×D₄×C₆₁	—	8	2	no	—
204	C₂×D₄×C₆₁	—	8	2	no	—
205	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
206	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
207	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
208	C₂²×C₄×C₆₁	—	2	4	no	—
209	C₂²×C₄×C₆₁	—	2	2	no	—
210	C₂²×C₄×C₆₁	—	2	4	no	—
211	C₂²×C₄×C₆₁	—	2	4	no	—
212	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
213	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
215	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
216	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
218	M₁₆×C₆₁	—	2	4	no	—
219	C₂×C₈×C₆₁	—	2	4	no	—
220	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
221	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
222	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
223	C₄⋊C₄×C₆₁	—	2	2	no	—
224	C₄⋊C₄×C₆₁	—	2	1	no	—
225	C₄⋊C₄×C₆₁	—	2	4	no	—
226	C₄⋊C₄×C₆₁	—	2	2	no	—
227	C₄⋊C₄×C₆₁	—	2	4	no	—
228	C₄⋊C₄×C₆₁	—	2	2	no	—
229	C₄⋊C₄×C₆₁	—	2	1	no	—
230	C₄⋊C₄×C₆₁	—	2	1	no	—
231	C₄⋊C₄×C₆₁	—	2	1	no	—
232	C₄⋊C₄×C₆₁	—	2	2	no	—
233	C₄⋊C₄×C₆₁	—	2	2	no	—
234	C₄⋊C₄×C₆₁	—	2	4	no	—
235	C₄⋊C₄×C₆₁	—	2	2	no	—
236	C₄⋊C₄×C₆₁	—	2	4	no	—
237	C₄⋊C₄×C₆₁	—	2	2	no	—
238	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
239	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
240	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
241	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
242	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
243	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
244	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
245	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
246	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
247	C₂²⋊C₄×C₆₁	—	2	2	no	—
248	C₂²⋊C₄×C₆₁	—	2	4	no	—
249	C₂²⋊C₄×C₆₁	—	2	2	no	—
250	C₂²⋊C₄×C₆₁	—	2	4	no	—
251	C₂²⋊C₄×C₆₁	—	2	4	no	—
252	C₂²⋊C₄×C₆₁	—	2	4	no	—
253	C₂²⋊C₄×C₆₁	—	2	4	no	—
254	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
255	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
256	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
257	C₄²×C₆₁	—	2	2	no	—
258	C₄²×C₆₁	—	2	4	no	—
259	C₄²×C₆₁	—	2	2	no	—
260	C₄²×C₆₁	C₂×D₄	1	2	yes	—
261	C₄²×C₆₁	C₂×D₄	1	2	yes	—
262	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
263	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
264	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
265	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
266	C₄²×C₆₁	C₂⁴	1	2	yes	—
267	C₂³×C₆₁	—	2	4	no	—
268	C₂³×C₆₁	—	4	4	no	—
269	C₂³×C₆₁	—	8	2	no	—
270	C₂³×C₆₁	—	8	2	no	—
271	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
272	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
273	Q₈×C₆₁	—	2	2	no	—
274	Q₈×C₆₁	—	4	2	no	—
275	Q₈×C₆₁	—	4	4	no	—
276	Q₈×C₆₁	—	4	2	no	—
277	Q₈×C₆₁	—	4	2	no	—
278	Q₈×C₆₁	—	4	2	no	—
279	Q₈×C₆₁	—	8	4	no	—
280	Q₈×C₆₁	—	8	2	no	—
281	D₄×C₆₁	—	2	4	no	—
282	D₄×C₆₁	—	2	2	no	—
283	D₄×C₆₁	—	2	8	no	—
284	D₄×C₆₁	—	4	4	no	—
285	D₄×C₆₁	—	4	4	no	—
286	D₄×C₆₁	—	4	4	no	—
287	D₄×C₆₁	—	8	2	no	—
288	D₄×C₆₁	—	8	2	no	—
289	D₄×C₆₁	—	8	4	no	—
290	D₄×C₆₁	—	8	4	no	—
291	C₂×C₄×C₆₁	—	2	2	no	—
292	C₂×C₄×C₆₁	—	2	4	no	—
293	C₂×C₄×C₆₁	—	2	2	no	—
294	C₂×C₄×C₆₁	—	2	4	no	—
295	C₂×C₄×C₆₁	—	2	4	no	—
296	C₂×C₄×C₆₁	—	2	2	no	—
297	C₂×C₄×C₆₁	—	2	2	no	—
298	C₂×C₄×C₆₁	—	2	2	no	—
299	C₂×C₄×C₆₁	—	2	4	no	—
300	C₂×C₄×C₆₁	—	2	2	no	—
301	C₂×C₄×C₆₁	—	8	4	no	—
302	C₂×C₄×C₆₁	—	8	4	no	—
303	C₂×C₄×C₆₁	—	8	2	no	—
304	C₂×C₄×C₆₁	—	8	4	no	—
305	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
306	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
307	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
308	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
313	C₂×C₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
314	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
315	C₈×C₆₁	—	2	2	no	—
316	C₈×C₆₁	—	2	2	no	—
317	C₈×C₆₁	—	2	4	no	—
318	#60877	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
319	C₂²×C₆₁	—	2	1	no	—
320	C₂²×C₆₁	—	2	1	no	—
321	C₂²×C₆₁	—	2	2	no	—
322	C₂²×C₆₁	—	2	2	no	—
323	C₂²×C₆₁	—	2	2	no	—
324	C₂²×C₆₁	—	4	4	no	—
325	C₂²×C₆₁	—	4	4	no	—
326	C₂²×C₆₁	—	4	4	no	—
327	C₂²×C₆₁	—	16	4	no	—
328	C₂²×C₆₁	#536	1	2	yes	—
329	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
330	C₄×C₆₁	—	2	2	no	—
331	C₄×C₆₁	—	2	2	no	—
332	C₄×C₆₁	—	2	1	no	—
333	C₄×C₆₁	—	2	2	no	—
334	C₄×C₆₁	—	2	1	no	—
335	C₄×C₆₁	—	4	2	no	—
336	C₄×C₆₁	—	4	4	no	—
337	C₄×C₆₁	—	4	2	no	—
338	C₄×C₆₁	—	4	2	no	—
339	C₄×C₆₁	—	4	2	no	—
340	C₄×C₆₁	#536	1	2	yes	—
341	C₄×C₆₁	#576	1	2	yes	—
342	C₄×C₆₁	#577	1	2	yes	—
343	C₄×C₆₁	C₂×2₋¹⁺⁴	1	2	yes	—
344	#3508	C₂×C₆₁	1	1	yes	characteristic
345	#3503	C₂×C₆₁	1	1	yes	characteristic, complemented
346	#3241	C₂×C₆₁	1	8	yes	complemented
347	#3228	C₂×C₆₁	1	2	yes	complemented
348	#3228	C₂×C₆₁	1	2	yes	complemented
349	#3227	C₂×C₆₁	1	2	yes	complemented
350	#3226	C₂×C₆₁	1	2	yes	complemented
351	#3165	C₂×C₆₁	1	1	yes	characteristic, complemented
352	#3160	C₂×C₆₁	1	1	yes	characteristic, complemented
353	#3138	C₂×C₆₁	1	1	yes	characteristic, complemented
354	C₂×#486	C₂×C₆₁	1	1	yes	characteristic, complemented
355	#3096	C₂×C₆₁	1	4	yes	complemented
356	#3081	C₂×C₆₁	1	4	yes	complemented
357	#2922	C₂×C₆₁	1	1	yes	characteristic, complemented
358	C₂×C₆₁	—	2	2	no	—
359	C₂×C₆₁	—	4	2	no	—
360	C₂×C₆₁	—	8	4	no	—
361	C₂×C₆₁	C₂×#536	1	1	yes	characteristic
362	C₂×C₆₁	C₂×#576	1	1	yes	characteristic
363	C₂×C₆₁	C₂×#577	1	1	yes	characteristic
364	#559	—	2	1	no	—
365	#559	—	2	1	no	—
366	Q₈²	—	2	1	no	—
367	#557	—	2	1	no	—
368	#554	—	2	2	no	—
369	#554	—	2	1	no	—
370	#550	—	2	1	no	—
371	#550	—	2	1	no	—
372	D₄×Q₈	—	2	2	no	—
373	D₄×Q₈	—	2	1	no	—
374	#546	—	2	2	no	—
375	#546	—	2	2	no	—
376	#544	C₂²×C₆₁	1	2	yes	—
377	#544	C₂²×C₆₁	1	4	yes	complemented
378	#540	C₂²×C₆₁	1	4	yes	—
379	#535	C₂²×C₆₁	1	2	yes	—
380	#533	C₂²×C₆₁	1	1	yes	characteristic
381	#533	C₂²×C₆₁	1	1	yes	characteristic
382	#533	C₂²×C₆₁	1	2	yes	complemented
383	#532	C₂²×C₆₁	1	2	yes	—
384	#532	C₂²×C₆₁	1	2	yes	—
385	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
386	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	complemented
387	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
388	#520	C₂²×C₆₁	1	2	yes	—
389	#518	C₂²×C₆₁	1	2	yes	—
390	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
391	#501	C₂²×C₆₁	1	4	yes	complemented
392	#500	C₂²×C₆₁	1	1	yes	characteristic, complemented
393	#500	C₂²×C₆₁	1	1	yes	characteristic, complemented
394	#500	C₂²×C₆₁	1	2	yes	complemented
395	#499	C₂²×C₆₁	1	4	yes	complemented
396	#498	C₂²×C₆₁	1	1	yes	characteristic, complemented
397	#498	C₂²×C₆₁	1	1	yes	characteristic, complemented
398	#498	C₂²×C₆₁	1	2	yes	complemented
399	#490	C₂²×C₆₁	1	4	yes	—
400	#486	C₂²×C₆₁	1	2	yes	—
401	#486	C₂²×C₆₁	1	2	yes	—
402	#486	C₂²×C₆₁	1	4	yes	—
403	#486	C₂²×C₆₁	1	2	yes	—
404	#486	C₂²×C₆₁	1	2	yes	—
405	#482	C₂²×C₆₁	1	8	yes	complemented
406	#482	C₂²×C₆₁	1	8	yes	complemented
407	#481	C₂²×C₆₁	1	8	yes	complemented
408	#480	C₂²×C₆₁	1	8	yes	complemented
409	#478	C₂²×C₆₁	1	8	yes	complemented
410	#478	C₂²×C₆₁	1	8	yes	complemented
411	#476	C₂²×C₆₁	1	8	yes	complemented
412	#474	C₂²×C₆₁	1	8	yes	complemented
413	#433	C₂²×C₆₁	1	4	yes	complemented
414	#431	C₂²×C₆₁	1	2	yes	complemented
415	#431	C₂²×C₆₁	1	2	yes	complemented
416	#428	C₂²×C₆₁	1	4	yes	—
417	#427	C₂²×C₆₁	1	4	yes	—
418	#424	C₂²×C₆₁	1	2	yes	—
419	#418	C₂²×C₆₁	1	4	yes	—
420	C₂×#102	C₂²×C₆₁	1	4	yes	—
421	#408	C₂²×C₆₁	1	2	yes	—
422	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
423	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
424	C₆₁	#60877	1	1	yes	characteristic, direct product factor, Sylow
425	C₂×#41	—	2	2	no	—
426	C₂×#41	—	2	2	no	—
427	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
428	C₂²×Q₈	—	2	1	no	—
429	C₂²×Q₈	—	2	2	no	—
430	C₂²×D₄	—	2	1	no	—
431	C₂²×D₄	—	2	1	no	—
432	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
433	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
434	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
435	C₄⋊Q₈	—	2	2	no	—
436	C₄⋊Q₈	—	2	4	no	—
437	C₄⋊Q₈	—	2	1	no	—
438	C₄⋊Q₈	—	2	1	no	—
439	C₄⋊Q₈	—	2	2	no	—
440	C₄⋊Q₈	—	2	2	no	—
441	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
442	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
443	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
444	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
445	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
446	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
447	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
448	#127	—	2	2	no	—
449	#127	—	2	1	no	—
450	#127	—	2	1	no	—
451	#127	C₂³×C₆₁	1	2	yes	—
452	#127	C₂³×C₆₁	1	2	yes	—
453	#125	—	2	2	no	—
454	#125	—	2	4	no	—
455	#125	C₂³×C₆₁	1	2	yes	—
456	#125	C₂³×C₆₁	1	2	yes	—
457	#125	C₂³×C₆₁	1	4	yes	—
458	#124	—	2	2	no	—
459	#124	—	2	2	no	—
460	#124	—	2	2	no	—
461	#123	—	2	4	no	—
462	#123	—	2	4	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	—	2	1	no	—
466	C₂²⋊Q₈	—	2	1	no	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	—	2	2	no	—
469	C₂²⋊Q₈	—	2	4	no	—
470	C₂²⋊Q₈	—	2	4	no	—
471	C₂²⋊Q₈	—	2	4	no	—
472	#121	—	2	2	no	—
473	#121	—	2	2	no	—
474	#121	—	2	4	no	—
475	#121	—	2	4	no	—
476	#121	C₂³×C₆₁	1	4	yes	—
477	#121	C₂³×C₆₁	1	4	yes	—
478	#120	—	2	4	no	—
479	C₄×Q₈	—	2	2	no	—
480	C₄×Q₈	—	2	1	no	—
481	C₄×Q₈	—	2	1	no	—
482	C₄×Q₈	—	2	1	no	—
483	C₄×Q₈	—	2	1	no	—
484	C₄×Q₈	—	2	2	no	—
485	C₄×Q₈	—	2	2	no	—
486	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
487	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
488	C₄×D₄	—	2	4	no	—
489	C₄×D₄	—	2	2	no	—
490	C₄×D₄	—	2	2	no	—
491	C₄×D₄	—	2	2	no	—
492	C₄×D₄	—	2	2	no	—
493	C₄×D₄	—	2	2	no	—
494	C₄×D₄	D₄×C₆₁	1	2	yes	—
495	C₄×D₄	D₄×C₆₁	1	2	yes	—
496	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
497	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
498	C₄×D₄	C₂³×C₆₁	1	4	yes	—
499	C₄×D₄	C₂³×C₆₁	1	4	yes	—
500	#117	C₂³×C₆₁	1	2	yes	—
501	#117	C₂³×C₆₁	1	2	yes	—
502	#117	C₂³×C₆₁	1	4	yes	—
503	#117	C₂³×C₆₁	1	1	yes	characteristic
504	#117	C₂³×C₆₁	1	2	yes	—
505	C₂×C₄⋊C₄	—	2	1	no	—
506	C₂×C₄⋊C₄	—	2	2	no	—
507	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
508	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
509	C₂×C₂²⋊C₄	—	2	4	no	—
510	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
511	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
512	#107	C₂³×C₆₁	1	4	yes	—
513	#107	C₂³×C₆₁	1	4	yes	—
514	#107	C₂³×C₆₁	1	8	yes	—
515	#106	C₂³×C₆₁	1	4	yes	—
516	#106	C₂³×C₆₁	1	4	yes	—
517	#106	C₂³×C₆₁	1	8	yes	—
518	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
519	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
520	#102	C₂³×C₆₁	1	8	yes	—
521	#102	C₂³×C₆₁	1	8	yes	—
522	#102	C₂³×C₆₁	1	8	yes	—
523	#102	C₂³×C₆₁	1	8	yes	—
524	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
525	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
526	#97	C₂³×C₆₁	1	2	yes	—
527	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
528	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
529	C₄×C₈	C₂³×C₆₁	1	2	yes	—
530	C₄×C₈	C₂³×C₆₁	1	2	yes	—
531	C₂⁴	—	2	2	no	—
532	#41	—	2	4	no	—
533	#41	—	8	4	no	—
534	#41	—	8	4	no	—
535	C₂×Q₈	—	2	1	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	2	2	no	—
540	C₂×Q₈	—	2	4	no	—
541	C₂×Q₈	—	2	2	no	—
542	C₂×Q₈	—	8	2	no	—
543	C₂×Q₈	—	8	4	no	—
544	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
545	C₂×D₄	—	2	4	no	—
546	C₂×D₄	—	2	4	no	—
547	C₂×D₄	—	2	2	no	—
548	C₂×D₄	—	2	2	no	—
549	C₂×D₄	—	2	2	no	—
550	C₂×D₄	—	2	2	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	8	no	—
553	C₂×D₄	—	8	2	no	—
554	C₂×D₄	—	8	2	no	—
555	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
556	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
557	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
558	C₂²×C₄	—	2	4	no	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	4	no	—
561	C₂²×C₄	—	2	4	no	—
562	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
563	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
564	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
565	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
566	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
567	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
568	M₁₆	—	2	4	no	—
569	C₂×C₈	—	2	4	no	—
570	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
571	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
572	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	4	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	4	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
589	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
590	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
591	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
592	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
593	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
594	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
595	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
596	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
597	C₂²⋊C₄	—	2	2	no	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	2	no	—
600	C₂²⋊C₄	—	2	4	no	—
601	C₂²⋊C₄	—	2	4	no	—
602	C₂²⋊C₄	—	2	4	no	—
603	C₂²⋊C₄	—	2	4	no	—
604	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
605	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
606	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
607	C₄²	—	2	2	no	—
608	C₄²	—	2	4	no	—
609	C₄²	—	2	2	no	—
610	C₄²	C₂×D₄×C₆₁	1	2	yes	—
611	C₄²	C₂×D₄×C₆₁	1	2	yes	—
612	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
613	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
614	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
615	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
616	C₄²	C₂⁴×C₆₁	1	2	yes	—
617	C₂³	—	2	4	no	—
618	C₂³	—	4	4	no	—
619	C₂³	—	8	2	no	—
620	C₂³	—	8	2	no	—
621	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
622	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
623	Q₈	—	2	2	no	—
624	Q₈	—	4	2	no	—
625	Q₈	—	4	4	no	—
626	Q₈	—	4	2	no	—
627	Q₈	—	4	2	no	—
628	Q₈	—	4	2	no	—
629	Q₈	—	8	4	no	—
630	Q₈	—	8	2	no	—
631	D₄	—	2	4	no	—
632	D₄	—	2	2	no	—
633	D₄	—	2	8	no	—
634	D₄	—	4	4	no	—
635	D₄	—	4	4	no	—
636	D₄	—	4	4	no	—
637	D₄	—	8	2	no	—
638	D₄	—	8	2	no	—
639	D₄	—	8	4	no	—
640	D₄	—	8	4	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	4	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	4	no	—
645	C₂×C₄	—	2	4	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	4	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	8	4	no	—
652	C₂×C₄	—	8	4	no	—
653	C₂×C₄	—	8	2	no	—
654	C₂×C₄	—	8	4	no	—
655	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
656	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
657	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
658	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
659	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
660	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
661	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
662	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
663	C₂×C₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
664	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
665	C₈	—	2	2	no	—
666	C₈	—	2	2	no	—
667	C₈	—	2	4	no	—
668	C₂²	—	2	1	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	2	2	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	2	2	no	—
673	C₂²	—	4	4	no	—
674	C₂²	—	4	4	no	—
675	C₂²	—	4	4	no	—
676	C₂²	—	16	4	no	—
677	C₂²	C₆₁×#536	1	2	yes	—
678	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
679	C₄	—	2	2	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	2	no	—
682	C₄	—	2	1	no	—
683	C₄	—	2	2	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	4	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	2	no	—
689	C₄	C₆₁×#536	1	2	yes	—
690	C₄	C₆₁×#576	1	2	yes	—
691	C₄	C₆₁×#577	1	2	yes	—
692	C₄	C₂×2₋¹⁺⁴×C₆₁	1	2	yes	—
693	C₂	—	2	2	no	—
694	C₂	—	4	2	no	—
695	C₂	—	8	4	no	—
696	C₂	C₂×C₆₁×#536	1	1	yes	characteristic
697	C₂	C₂×C₆₁×#576	1	1	yes	characteristic
698	C₂	C₂×C₆₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₆₁×#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×#41×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#3503	C₂	semidirect
3. C₆₁×#3241	C₂	semidirect
4. C₆₁×#3228	C₂	semidirect
5. C₆₁×#3228	C₂	semidirect
6. C₆₁×#3227	C₂	semidirect
7. C₆₁×#3226	C₂	semidirect
8. C₆₁×#3165	C₂	semidirect
9. C₆₁×#3160	C₂	semidirect
10. C₆₁×#3138	C₂	semidirect
11. C₂×C₆₁×#486	C₂	semidirect
12. C₆₁×#3096	C₂	semidirect
13. C₆₁×#3081	C₂	semidirect
14. C₆₁×#2922	C₂	semidirect
28. C₆₁×#544	C₂²	semidirect
33. C₆₁×#533	C₂²	semidirect
37. C₂×C₄⋊Q₈×C₆₁	C₂²	semidirect
42. C₆₁×#501	C₂²	semidirect
43. C₆₁×#500	C₂²	semidirect
44. C₆₁×#500	C₂²	semidirect
45. C₆₁×#500	C₂²	semidirect
46. C₆₁×#499	C₂²	semidirect
47. C₆₁×#498	C₂²	semidirect
48. C₆₁×#498	C₂²	semidirect
49. C₆₁×#498	C₂²	semidirect
56. C₆₁×#482	C₂²	semidirect
57. C₆₁×#482	C₂²	semidirect
58. C₆₁×#481	C₂²	semidirect
59. C₆₁×#480	C₂²	semidirect
60. C₆₁×#478	C₂²	semidirect
61. C₆₁×#478	C₂²	semidirect
62. C₆₁×#476	C₂²	semidirect
63. C₆₁×#474	C₂²	semidirect
64. C₆₁×#433	C₂²	semidirect
65. C₆₁×#431	C₂²	semidirect
66. C₆₁×#431	C₂²	semidirect
318. #60877	C₆₁	direct (N × H)
345. #3503	C₂×C₆₁	semidirect
346. #3241	C₂×C₆₁	semidirect
347. #3228	C₂×C₆₁	semidirect
348. #3228	C₂×C₆₁	semidirect
349. #3227	C₂×C₆₁	semidirect
350. #3226	C₂×C₆₁	semidirect
351. #3165	C₂×C₆₁	semidirect
352. #3160	C₂×C₆₁	semidirect
353. #3138	C₂×C₆₁	semidirect
354. C₂×#486	C₂×C₆₁	semidirect
355. #3096	C₂×C₆₁	semidirect
356. #3081	C₂×C₆₁	semidirect
357. #2922	C₂×C₆₁	semidirect
377. #544	C₂²×C₆₁	semidirect
382. #533	C₂²×C₆₁	semidirect
386. C₂×C₄⋊Q₈	C₂²×C₆₁	semidirect
391. #501	C₂²×C₆₁	semidirect
392. #500	C₂²×C₆₁	semidirect
393. #500	C₂²×C₆₁	semidirect
394. #500	C₂²×C₆₁	semidirect
395. #499	C₂²×C₆₁	semidirect
396. #498	C₂²×C₆₁	semidirect
397. #498	C₂²×C₆₁	semidirect
398. #498	C₂²×C₆₁	semidirect
405. #482	C₂²×C₆₁	semidirect
406. #482	C₂²×C₆₁	semidirect
407. #481	C₂²×C₆₁	semidirect
408. #480	C₂²×C₆₁	semidirect
409. #478	C₂²×C₆₁	semidirect
410. #478	C₂²×C₆₁	semidirect
411. #476	C₂²×C₆₁	semidirect
412. #474	C₂²×C₆₁	semidirect
413. #433	C₂²×C₆₁	semidirect
414. #431	C₂²×C₆₁	semidirect
415. #431	C₂²×C₆₁	semidirect
424. C₆₁	#60877	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×#3508	2
C₆₁×#3503	2
C₆₁×#3241	2
C₆₁×#3228	2
C₆₁×#3227	2
C₆₁×#3226	2
C₆₁×#3165	2
C₆₁×#3160	2
C₆₁×#3138	2
C₂×C₆₁×#486	2
C₆₁×#3096	2
C₆₁×#3081	2
C₆₁×#2922	2
#60877	61

Subgroups — C61×#60877 #29564310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60877 #29564310