Subgroups — C₆₁×#60723 #29564156

Order 15616.

1886 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3511	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3185	C₂	1	2	yes	maximal, complemented
4	C₆₁×#3182	C₂	1	2	yes	maximal, complemented
5	C₆₁×#3178	C₂	1	8	yes	maximal, complemented
6	C₆₁×#3170	C₂	1	2	yes	maximal, complemented
7	C₆₁×#3167	C₂	1	2	yes	maximal, complemented
8	C₆₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#3063	C₂	1	4	yes	maximal, complemented
13	C₆₁×#2947	C₂	1	4	yes	maximal, complemented
14	C₄×#131×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#137×C₆₁	C₂²	1	4	yes	—
16	C₂×#131×C₆₁	C₂²	1	2	yes	—
17	C₆₁×#557	—	2	1	no	—
18	C₆₁×#557	—	2	1	no	—
19	C₆₁×#556	—	2	1	no	—
20	C₆₁×#556	—	2	1	no	—
21	C₆₁×#556	—	2	1	no	—
22	C₆₁×#555	—	2	1	no	—
23	C₆₁×#554	—	2	1	no	—
24	C₆₁×#553	—	2	2	no	—
25	C₆₁×#552	—	2	2	no	—
26	C₆₁×#552	—	2	2	no	—
27	C₆₁×#548	—	2	2	no	—
28	C₆₁×#547	—	2	1	no	—
29	C₆₁×#542	C₂²	1	4	yes	—
30	C₆₁×#541	C₂²	1	4	yes	—
31	C₆₁×#537	C₂²	1	2	yes	—
32	C₆₁×#536	C₂²	1	2	yes	—
33	C₆₁×#533	C₂²	1	2	yes	—
34	C₆₁×#533	C₂²	1	2	yes	—
35	C₆₁×#533	C₂²	1	2	yes	—
36	C₆₁×#533	C₂²	1	1	yes	characteristic
37	C₆₁×#532	C₂²	1	1	yes	characteristic
38	C₆₁×#532	C₂²	1	2	yes	—
39	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₆₁×#520	C₂²	1	2	yes	—
42	C₆₁×#519	C₂²	1	2	yes	—
43	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#501	C₂²	1	4	yes	—
45	C₆₁×#500	C₂²	1	2	yes	—
46	C₆₁×#498	C₂²	1	2	yes	—
47	C₆₁×#497	C₂²	1	4	yes	—
48	C₆₁×#494	C₂²	1	2	yes	—
49	C₆₁×#492	C₂²	1	2	yes	—
50	C₆₁×#488	C₂²	1	4	yes	—
51	C₆₁×#487	C₂²	1	2	yes	—
52	C₆₁×#487	C₂²	1	4	yes	—
53	C₆₁×#487	C₂²	1	2	yes	—
54	C₆₁×#487	C₂²	1	1	yes	characteristic
55	C₆₁×#487	C₂²	1	1	yes	characteristic
56	C₆₁×#486	C₂²	1	1	yes	characteristic
57	C₆₁×#486	C₂²	1	1	yes	characteristic
58	C₆₁×#470	C₂²	1	8	yes	—
59	C₆₁×#468	C₂²	1	8	yes	—
60	C₆₁×#467	C₂²	1	8	yes	—
61	C₆₁×#465	C₂²	1	8	yes	—
62	C₆₁×#441	C₂²	1	8	yes	—
63	C₆₁×#440	C₂²	1	8	yes	—
64	C₄×Q₁₆×C₆₁	C₂²	1	8	yes	—
65	C₄×SD₁₆×C₆₁	C₂²	1	8	yes	—
66	C₆₁×#428	C₂²	1	4	yes	—
67	C₆₁×#416	C₂²	1	4	yes	—
68	C₂×#103×C₆₁	C₂²	1	4	yes	—
69	C₆₁×#405	C₂²	1	4	yes	—
70	C₆₁×#405	C₂²	1	2	yes	—
71	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
72	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
73	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
74	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
75	C₂×#41×C₆₁	—	2	2	no	—
76	C₂×#41×C₆₁	C₂³	1	2	yes	—
77	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₆₁	—	2	1	no	—
79	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
80	#137×C₆₁	—	2	16	no	—
81	C₂×Q₁₆×C₆₁	C₂³	1	8	yes	—
82	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
83	#131×C₆₁	—	2	8	no	—
84	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
85	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
86	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
87	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
88	C₄⋊Q₈×C₆₁	—	2	2	no	—
89	C₄⋊Q₈×C₆₁	—	2	1	no	—
90	C₄⋊Q₈×C₆₁	—	2	1	no	—
91	C₄⋊Q₈×C₆₁	—	2	1	no	—
92	C₄⋊Q₈×C₆₁	—	2	1	no	—
93	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
94	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
95	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
96	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
97	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
98	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
99	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
100	#127×C₆₁	C₂³	1	1	yes	characteristic
101	#126×C₆₁	—	2	4	no	—
102	#126×C₆₁	—	2	4	no	—
103	#126×C₆₁	—	2	4	no	—
104	#125×C₆₁	—	2	4	no	—
105	#125×C₆₁	—	2	4	no	—
106	#125×C₆₁	—	2	2	no	—
107	#125×C₆₁	—	2	2	no	—
108	#125×C₆₁	—	2	2	no	—
109	#125×C₆₁	C₂³	1	2	yes	—
110	#124×C₆₁	—	2	2	no	—
111	#124×C₆₁	—	2	4	no	—
112	#124×C₆₁	—	2	2	no	—
113	#124×C₆₁	C₂³	1	2	yes	—
114	#123×C₆₁	—	2	4	no	—
115	#123×C₆₁	—	2	4	no	—
116	C₂²⋊Q₈×C₆₁	—	2	2	no	—
117	C₂²⋊Q₈×C₆₁	—	2	2	no	—
118	C₂²⋊Q₈×C₆₁	—	2	1	no	—
119	C₂²⋊Q₈×C₆₁	—	2	1	no	—
120	C₂²⋊Q₈×C₆₁	—	2	2	no	—
121	C₂²⋊Q₈×C₆₁	—	2	2	no	—
122	C₂²⋊Q₈×C₆₁	—	2	4	no	—
123	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
124	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
125	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
126	#121×C₆₁	—	2	2	no	—
127	#121×C₆₁	—	2	4	no	—
128	#121×C₆₁	—	2	1	no	—
129	#121×C₆₁	—	2	1	no	—
130	#121×C₆₁	—	2	2	no	—
131	#121×C₆₁	C₂³	1	4	yes	—
132	C₄×Q₈×C₆₁	—	2	2	no	—
133	C₄×Q₈×C₆₁	—	2	2	no	—
134	C₄×Q₈×C₆₁	—	2	2	no	—
135	C₄×Q₈×C₆₁	—	2	2	no	—
136	C₄×Q₈×C₆₁	—	2	2	no	—
137	C₄×Q₈×C₆₁	—	2	1	no	—
138	C₄×Q₈×C₆₁	—	2	1	no	—
139	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
140	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
141	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
142	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
143	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
144	C₄×D₄×C₆₁	—	2	4	no	—
145	C₄×D₄×C₆₁	—	2	1	no	—
146	C₄×D₄×C₆₁	—	2	2	no	—
147	C₄×D₄×C₆₁	—	2	1	no	—
148	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
149	C₄×D₄×C₆₁	D₄	1	2	yes	—
150	C₄×D₄×C₆₁	D₄	1	2	yes	—
151	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
152	C₄×D₄×C₆₁	C₂³	1	4	yes	—
153	#117×C₆₁	—	2	2	no	—
154	#117×C₆₁	—	2	2	no	—
155	#117×C₆₁	—	2	2	no	—
156	#117×C₆₁	—	2	2	no	—
157	#117×C₆₁	—	2	4	no	—
158	#117×C₆₁	C₂³	1	2	yes	—
159	#117×C₆₁	C₂³	1	2	yes	—
160	#117×C₆₁	C₂³	1	1	yes	characteristic
161	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
162	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
163	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
164	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
165	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
166	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
167	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
168	#107×C₆₁	C₂³	1	8	yes	—
169	#106×C₆₁	C₂³	1	8	yes	—
170	#103×C₆₁	C₂³	1	8	yes	—
171	#103×C₆₁	C₂³	1	8	yes	—
172	#103×C₆₁	C₂³	1	4	yes	—
173	#103×C₆₁	C₂³	1	4	yes	—
174	#102×C₆₁	C₂³	1	4	yes	—
175	#102×C₆₁	C₂³	1	4	yes	—
176	#97×C₆₁	C₂³	1	4	yes	—
177	#97×C₆₁	C₂³	1	2	yes	—
178	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
180	C₄×C₈×C₆₁	C₂³	1	2	yes	—
181	C₄×C₈×C₆₁	C₂³	1	4	yes	—
182	C₄×C₈×C₆₁	C₂³	1	2	yes	—
183	#41×C₆₁	—	2	4	no	—
184	#41×C₆₁	—	2	8	no	—
185	#41×C₆₁	—	8	4	no	—
186	C₂×Q₈×C₆₁	—	2	1	no	—
187	C₂×Q₈×C₆₁	—	2	1	no	—
188	C₂×Q₈×C₆₁	—	2	2	no	—
189	C₂×Q₈×C₆₁	—	2	2	no	—
190	C₂×Q₈×C₆₁	—	2	2	no	—
191	C₂×Q₈×C₆₁	—	2	8	no	—
192	C₂×Q₈×C₆₁	—	8	2	no	—
193	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
194	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
195	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
196	C₂×D₄×C₆₁	—	2	2	no	—
197	C₂×D₄×C₆₁	—	2	4	no	—
198	C₂×D₄×C₆₁	—	2	1	no	—
199	C₂×D₄×C₆₁	—	2	1	no	—
200	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
201	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
202	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₆₁	—	2	2	no	—
204	C₂²×C₄×C₆₁	—	2	2	no	—
205	C₂²×C₄×C₆₁	—	2	2	no	—
206	C₂²×C₄×C₆₁	—	2	4	no	—
207	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
210	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
211	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
213	Q₁₆×C₆₁	—	2	16	no	—
214	SD₁₆×C₆₁	—	2	16	no	—
215	M₁₆×C₆₁	—	2	8	no	—
216	M₁₆×C₆₁	#41	1	8	yes	complemented
217	C₂×C₈×C₆₁	—	2	4	no	—
218	C₂×C₈×C₆₁	—	2	8	no	—
219	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
220	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
221	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
222	C₄⋊C₄×C₆₁	—	2	2	no	—
223	C₄⋊C₄×C₆₁	—	2	2	no	—
224	C₄⋊C₄×C₆₁	—	2	2	no	—
225	C₄⋊C₄×C₆₁	—	2	2	no	—
226	C₄⋊C₄×C₆₁	—	2	4	no	—
227	C₄⋊C₄×C₆₁	—	2	2	no	—
228	C₄⋊C₄×C₆₁	—	2	2	no	—
229	C₄⋊C₄×C₆₁	—	2	2	no	—
230	C₄⋊C₄×C₆₁	—	2	1	no	—
231	C₄⋊C₄×C₆₁	—	2	1	no	—
232	C₄⋊C₄×C₆₁	—	2	1	no	—
233	C₄⋊C₄×C₆₁	—	2	4	no	—
234	C₄⋊C₄×C₆₁	—	2	4	no	—
235	C₄⋊C₄×C₆₁	—	2	1	no	—
236	C₄⋊C₄×C₆₁	—	2	4	no	—
237	C₄⋊C₄×C₆₁	—	2	2	no	—
238	C₄⋊C₄×C₆₁	—	2	2	no	—
239	C₄⋊C₄×C₆₁	—	2	4	no	—
240	C₄⋊C₄×C₆₁	—	2	4	no	—
241	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
242	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
243	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
245	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
246	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
247	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
248	C₂²⋊C₄×C₆₁	—	2	2	no	—
249	C₂²⋊C₄×C₆₁	—	2	4	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	—	2	2	no	—
255	C₂²⋊C₄×C₆₁	—	2	2	no	—
256	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
257	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
258	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
259	C₄²×C₆₁	—	2	4	no	—
260	C₄²×C₆₁	—	2	4	no	—
261	C₄²×C₆₁	—	2	4	no	—
262	C₄²×C₆₁	—	2	2	no	—
263	C₄²×C₆₁	C₂×D₄	1	2	yes	—
264	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
265	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
266	C₄²×C₆₁	C₂×D₄	1	2	yes	—
267	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
268	C₄²×C₆₁	C₂⁴	1	2	yes	—
269	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
270	C₂³×C₆₁	—	2	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	—	2	8	no	—
275	Q₈×C₆₁	—	2	4	no	—
276	Q₈×C₆₁	—	4	4	no	—
277	Q₈×C₆₁	—	4	2	no	—
278	Q₈×C₆₁	—	4	4	no	—
279	Q₈×C₆₁	—	4	2	no	—
280	Q₈×C₆₁	—	8	2	no	—
281	D₄×C₆₁	—	2	4	no	—
282	D₄×C₆₁	—	2	2	no	—
283	D₄×C₆₁	—	2	4	no	—
284	D₄×C₆₁	—	4	4	no	—
285	D₄×C₆₁	—	4	2	no	—
286	D₄×C₆₁	—	4	2	no	—
287	D₄×C₆₁	—	8	4	no	—
288	C₂×C₄×C₆₁	—	2	2	no	—
289	C₂×C₄×C₆₁	—	2	4	no	—
290	C₂×C₄×C₆₁	—	2	2	no	—
291	C₂×C₄×C₆₁	—	2	4	no	—
292	C₂×C₄×C₆₁	—	2	4	no	—
293	C₂×C₄×C₆₁	—	2	2	no	—
294	C₂×C₄×C₆₁	—	2	2	no	—
295	C₂×C₄×C₆₁	—	2	2	no	—
296	C₂×C₄×C₆₁	—	2	4	no	—
297	C₂×C₄×C₆₁	—	2	4	no	—
298	C₂×C₄×C₆₁	—	4	4	no	—
299	C₂×C₄×C₆₁	—	4	4	no	—
300	C₂×C₄×C₆₁	—	8	4	no	—
301	C₂×C₄×C₆₁	—	8	2	no	—
302	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
303	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
311	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
312	C₈×C₆₁	—	2	2	no	—
313	C₈×C₆₁	—	2	2	no	—
314	C₈×C₆₁	C₂×#41	1	8	yes	—
315	#60723	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂²×C₆₁	—	2	1	no	—
317	C₂²×C₆₁	—	2	1	no	—
318	C₂²×C₆₁	—	2	2	no	—
319	C₂²×C₆₁	—	2	2	no	—
320	C₂²×C₆₁	—	2	2	no	—
321	C₂²×C₆₁	—	4	2	no	—
322	C₂²×C₆₁	—	4	2	no	—
323	C₂²×C₆₁	—	4	2	no	—
324	C₂²×C₆₁	#532	1	2	yes	—
325	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
326	C₄×C₆₁	—	2	2	no	—
327	C₄×C₆₁	—	2	2	no	—
328	C₄×C₆₁	—	2	1	no	—
329	C₄×C₆₁	—	2	2	no	—
330	C₄×C₆₁	—	2	1	no	—
331	C₄×C₆₁	—	4	2	no	—
332	C₄×C₆₁	—	4	4	no	—
333	C₄×C₆₁	—	4	4	no	—
334	C₄×C₆₁	—	4	2	no	—
335	C₄×C₆₁	—	4	2	no	—
336	C₄×C₆₁	#532	1	2	yes	—
337	C₄×C₆₁	#577	1	2	yes	—
338	C₄×C₆₁	#578	1	2	yes	—
339	C₄×C₆₁	C₂²×#41	1	2	yes	—
340	#3511	C₂×C₆₁	1	1	yes	characteristic
341	#3509	C₂×C₆₁	1	1	yes	characteristic, complemented
342	#3185	C₂×C₆₁	1	2	yes	complemented
343	#3182	C₂×C₆₁	1	2	yes	complemented
344	#3178	C₂×C₆₁	1	8	yes	complemented
345	#3170	C₂×C₆₁	1	2	yes	complemented
346	#3167	C₂×C₆₁	1	2	yes	complemented
347	#3165	C₂×C₆₁	1	1	yes	characteristic, complemented
348	#3152	C₂×C₆₁	1	1	yes	characteristic, complemented
349	#3131	C₂×C₆₁	1	1	yes	characteristic, complemented
350	C₂×#487	C₂×C₆₁	1	1	yes	characteristic, complemented
351	#3063	C₂×C₆₁	1	4	yes	complemented
352	#2947	C₂×C₆₁	1	4	yes	complemented
353	C₄×#131	C₂×C₆₁	1	1	yes	characteristic, complemented
354	C₂×C₆₁	—	2	2	no	—
355	C₂×C₆₁	—	4	2	no	—
356	C₂×C₆₁	—	8	2	no	—
357	C₂×C₆₁	C₂×#532	1	1	yes	characteristic
358	C₂×C₆₁	C₂×#577	1	1	yes	characteristic
359	C₂×C₆₁	C₂×#578	1	1	yes	characteristic
360	C₂×#137	C₂²×C₆₁	1	4	yes	—
361	C₂×#131	C₂²×C₆₁	1	2	yes	—
362	#557	—	2	1	no	—
363	#557	—	2	1	no	—
364	#556	—	2	1	no	—
365	#556	—	2	1	no	—
366	#556	—	2	1	no	—
367	#555	—	2	1	no	—
368	#554	—	2	1	no	—
369	#553	—	2	2	no	—
370	#552	—	2	2	no	—
371	#552	—	2	2	no	—
372	#548	—	2	2	no	—
373	#547	—	2	1	no	—
374	#542	C₂²×C₆₁	1	4	yes	—
375	#541	C₂²×C₆₁	1	4	yes	—
376	#537	C₂²×C₆₁	1	2	yes	—
377	#536	C₂²×C₆₁	1	2	yes	—
378	#533	C₂²×C₆₁	1	2	yes	—
379	#533	C₂²×C₆₁	1	2	yes	—
380	#533	C₂²×C₆₁	1	2	yes	—
381	#533	C₂²×C₆₁	1	1	yes	characteristic
382	#532	C₂²×C₆₁	1	1	yes	characteristic
383	#532	C₂²×C₆₁	1	2	yes	—
384	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
385	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
386	#520	C₂²×C₆₁	1	2	yes	—
387	#519	C₂²×C₆₁	1	2	yes	—
388	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
389	#501	C₂²×C₆₁	1	4	yes	—
390	#500	C₂²×C₆₁	1	2	yes	—
391	#498	C₂²×C₆₁	1	2	yes	—
392	#497	C₂²×C₆₁	1	4	yes	—
393	#494	C₂²×C₆₁	1	2	yes	—
394	#492	C₂²×C₆₁	1	2	yes	—
395	#488	C₂²×C₆₁	1	4	yes	—
396	#487	C₂²×C₆₁	1	2	yes	—
397	#487	C₂²×C₆₁	1	4	yes	—
398	#487	C₂²×C₆₁	1	2	yes	—
399	#487	C₂²×C₆₁	1	1	yes	characteristic
400	#487	C₂²×C₆₁	1	1	yes	characteristic
401	#486	C₂²×C₆₁	1	1	yes	characteristic
402	#486	C₂²×C₆₁	1	1	yes	characteristic
403	#470	C₂²×C₆₁	1	8	yes	—
404	#468	C₂²×C₆₁	1	8	yes	—
405	#467	C₂²×C₆₁	1	8	yes	—
406	#465	C₂²×C₆₁	1	8	yes	—
407	#441	C₂²×C₆₁	1	8	yes	—
408	#440	C₂²×C₆₁	1	8	yes	—
409	C₄×Q₁₆	C₂²×C₆₁	1	8	yes	—
410	C₄×SD₁₆	C₂²×C₆₁	1	8	yes	—
411	#428	C₂²×C₆₁	1	4	yes	—
412	#416	C₂²×C₆₁	1	4	yes	—
413	C₂×#103	C₂²×C₆₁	1	4	yes	—
414	#405	C₂²×C₆₁	1	4	yes	—
415	#405	C₂²×C₆₁	1	2	yes	—
416	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
417	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
418	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
419	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
420	C₆₁	#60723	1	1	yes	characteristic, direct product factor, Sylow
421	C₂×#41	—	2	2	no	—
422	C₂×#41	C₂³×C₆₁	1	2	yes	—
423	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
424	C₂²×Q₈	—	2	1	no	—
425	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
426	#137	—	2	16	no	—
427	C₂×Q₁₆	C₂³×C₆₁	1	8	yes	—
428	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
429	#131	—	2	8	no	—
430	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
431	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
432	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
433	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
434	C₄⋊Q₈	—	2	2	no	—
435	C₄⋊Q₈	—	2	1	no	—
436	C₄⋊Q₈	—	2	1	no	—
437	C₄⋊Q₈	—	2	1	no	—
438	C₄⋊Q₈	—	2	1	no	—
439	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
440	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
441	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
442	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
443	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
444	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
445	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
446	#127	C₂³×C₆₁	1	1	yes	characteristic
447	#126	—	2	4	no	—
448	#126	—	2	4	no	—
449	#126	—	2	4	no	—
450	#125	—	2	4	no	—
451	#125	—	2	4	no	—
452	#125	—	2	2	no	—
453	#125	—	2	2	no	—
454	#125	—	2	2	no	—
455	#125	C₂³×C₆₁	1	2	yes	—
456	#124	—	2	2	no	—
457	#124	—	2	4	no	—
458	#124	—	2	2	no	—
459	#124	C₂³×C₆₁	1	2	yes	—
460	#123	—	2	4	no	—
461	#123	—	2	4	no	—
462	C₂²⋊Q₈	—	2	2	no	—
463	C₂²⋊Q₈	—	2	2	no	—
464	C₂²⋊Q₈	—	2	1	no	—
465	C₂²⋊Q₈	—	2	1	no	—
466	C₂²⋊Q₈	—	2	2	no	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	—	2	4	no	—
469	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
470	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
471	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
472	#121	—	2	2	no	—
473	#121	—	2	4	no	—
474	#121	—	2	1	no	—
475	#121	—	2	1	no	—
476	#121	—	2	2	no	—
477	#121	C₂³×C₆₁	1	4	yes	—
478	C₄×Q₈	—	2	2	no	—
479	C₄×Q₈	—	2	2	no	—
480	C₄×Q₈	—	2	2	no	—
481	C₄×Q₈	—	2	2	no	—
482	C₄×Q₈	—	2	2	no	—
483	C₄×Q₈	—	2	1	no	—
484	C₄×Q₈	—	2	1	no	—
485	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
486	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
487	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
488	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
489	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
490	C₄×D₄	—	2	4	no	—
491	C₄×D₄	—	2	1	no	—
492	C₄×D₄	—	2	2	no	—
493	C₄×D₄	—	2	1	no	—
494	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
495	C₄×D₄	D₄×C₆₁	1	2	yes	—
496	C₄×D₄	D₄×C₆₁	1	2	yes	—
497	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
498	C₄×D₄	C₂³×C₆₁	1	4	yes	—
499	#117	—	2	2	no	—
500	#117	—	2	2	no	—
501	#117	—	2	2	no	—
502	#117	—	2	2	no	—
503	#117	—	2	4	no	—
504	#117	C₂³×C₆₁	1	2	yes	—
505	#117	C₂³×C₆₁	1	1	yes	characteristic
506	#117	C₂³×C₆₁	1	2	yes	—
507	C₂×C₄⋊C₄	—	2	1	no	—
508	C₂×C₄⋊C₄	—	2	1	no	—
509	C₂×C₄⋊C₄	—	2	1	no	—
510	C₂×C₄⋊C₄	—	2	2	no	—
511	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
512	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
513	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
514	#107	C₂³×C₆₁	1	8	yes	—
515	#106	C₂³×C₆₁	1	8	yes	—
516	#103	C₂³×C₆₁	1	8	yes	—
517	#103	C₂³×C₆₁	1	8	yes	—
518	#103	C₂³×C₆₁	1	4	yes	—
519	#103	C₂³×C₆₁	1	4	yes	—
520	#102	C₂³×C₆₁	1	4	yes	—
521	#102	C₂³×C₆₁	1	4	yes	—
522	#97	C₂³×C₆₁	1	4	yes	—
523	#97	C₂³×C₆₁	1	2	yes	—
524	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
525	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
526	C₄×C₈	C₂³×C₆₁	1	2	yes	—
527	C₄×C₈	C₂³×C₆₁	1	4	yes	—
528	C₄×C₈	C₂³×C₆₁	1	2	yes	—
529	#41	—	2	4	no	—
530	#41	—	2	8	no	—
531	#41	—	8	4	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	2	2	no	—
536	C₂×Q₈	—	2	2	no	—
537	C₂×Q₈	—	2	8	no	—
538	C₂×Q₈	—	8	2	no	—
539	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
540	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
541	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
542	C₂×D₄	—	2	2	no	—
543	C₂×D₄	—	2	4	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	1	no	—
546	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
547	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
548	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
554	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
556	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
557	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
558	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
559	Q₁₆	—	2	16	no	—
560	SD₁₆	—	2	16	no	—
561	M₁₆	—	2	8	no	—
562	M₁₆	#41×C₆₁	1	8	yes	complemented
563	C₂×C₈	—	2	4	no	—
564	C₂×C₈	—	2	8	no	—
565	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
566	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
567	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	4	no	—
587	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
588	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
589	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
590	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
591	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
592	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
593	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
594	C₂²⋊C₄	—	2	2	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	4	no	—
597	C₂²⋊C₄	—	2	4	no	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	4	no	—
600	C₂²⋊C₄	—	2	2	no	—
601	C₂²⋊C₄	—	2	2	no	—
602	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
603	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
604	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
605	C₄²	—	2	4	no	—
606	C₄²	—	2	4	no	—
607	C₄²	—	2	4	no	—
608	C₄²	—	2	2	no	—
609	C₄²	C₂×D₄×C₆₁	1	2	yes	—
610	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
611	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
612	C₄²	C₂×D₄×C₆₁	1	2	yes	—
613	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
614	C₄²	C₂⁴×C₆₁	1	2	yes	—
615	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
616	C₂³	—	2	2	no	—
617	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
618	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
619	Q₈	—	2	2	no	—
620	Q₈	—	2	8	no	—
621	Q₈	—	2	4	no	—
622	Q₈	—	4	4	no	—
623	Q₈	—	4	2	no	—
624	Q₈	—	4	4	no	—
625	Q₈	—	4	2	no	—
626	Q₈	—	8	2	no	—
627	D₄	—	2	4	no	—
628	D₄	—	2	2	no	—
629	D₄	—	2	4	no	—
630	D₄	—	4	4	no	—
631	D₄	—	4	2	no	—
632	D₄	—	4	2	no	—
633	D₄	—	8	4	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	4	no	—
638	C₂×C₄	—	2	4	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	4	no	—
643	C₂×C₄	—	2	4	no	—
644	C₂×C₄	—	4	4	no	—
645	C₂×C₄	—	4	4	no	—
646	C₂×C₄	—	8	4	no	—
647	C₂×C₄	—	8	2	no	—
648	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
649	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
650	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
651	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
652	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
653	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
654	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
655	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
656	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
657	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
658	C₈	—	2	2	no	—
659	C₈	—	2	2	no	—
660	C₈	C₂×#41×C₆₁	1	8	yes	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	1	no	—
663	C₂²	—	2	2	no	—
664	C₂²	—	2	2	no	—
665	C₂²	—	2	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	—	4	2	no	—
668	C₂²	—	4	2	no	—
669	C₂²	C₆₁×#532	1	2	yes	—
670	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
671	C₄	—	2	2	no	—
672	C₄	—	2	1	no	—
673	C₄	—	2	2	no	—
674	C₄	—	2	1	no	—
675	C₄	—	2	2	no	—
676	C₄	—	4	4	no	—
677	C₄	—	4	2	no	—
678	C₄	—	4	2	no	—
679	C₄	—	4	2	no	—
680	C₄	—	4	4	no	—
681	C₄	C₆₁×#532	1	2	yes	—
682	C₄	C₆₁×#577	1	2	yes	—
683	C₄	C₆₁×#578	1	2	yes	—
684	C₄	C₂²×#41×C₆₁	1	2	yes	—
685	C₂	—	2	2	no	—
686	C₂	—	4	2	no	—
687	C₂	—	8	2	no	—
688	C₂	C₂×C₆₁×#532	1	1	yes	characteristic
689	C₂	C₂×C₆₁×#577	1	1	yes	characteristic
690	C₂	C₂×C₆₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₆₁×#3511	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×#3509	C₂	semidirect
3. C₆₁×#3185	C₂	semidirect
4. C₆₁×#3182	C₂	semidirect
5. C₆₁×#3178	C₂	semidirect
6. C₆₁×#3170	C₂	semidirect
7. C₆₁×#3167	C₂	semidirect
8. C₆₁×#3165	C₂	semidirect
9. C₆₁×#3152	C₂	semidirect
10. C₆₁×#3131	C₂	semidirect
11. C₂×C₆₁×#487	C₂	semidirect
12. C₆₁×#3063	C₂	semidirect
13. C₆₁×#2947	C₂	semidirect
14. C₄×#131×C₆₁	C₂	semidirect
216. M₁₆×C₆₁	#41	semidirect
315. #60723	C₆₁	direct (N × H)
341. #3509	C₂×C₆₁	semidirect
342. #3185	C₂×C₆₁	semidirect
343. #3182	C₂×C₆₁	semidirect
344. #3178	C₂×C₆₁	semidirect
345. #3170	C₂×C₆₁	semidirect
346. #3167	C₂×C₆₁	semidirect
347. #3165	C₂×C₆₁	semidirect
348. #3152	C₂×C₆₁	semidirect
349. #3131	C₂×C₆₁	semidirect
350. C₂×#487	C₂×C₆₁	semidirect
351. #3063	C₂×C₆₁	semidirect
352. #2947	C₂×C₆₁	semidirect
353. C₄×#131	C₂×C₆₁	semidirect
420. C₆₁	#60723	direct (N × H)
562. M₁₆	#41×C₆₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₁×#3511	2
C₆₁×#3509	2
C₆₁×#3185	2
C₆₁×#3182	2
C₆₁×#3178	2
C₆₁×#3170	2
C₆₁×#3167	2
C₆₁×#3165	2
C₆₁×#3152	2
C₆₁×#3131	2
C₂×C₆₁×#487	2
C₆₁×#3063	2
C₆₁×#2947	2
C₄×#131×C₆₁	2
#60723	61

Subgroups — C61×#60723 #29564156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60723 #29564156