Subgroups — C₆₁×#61269 #29564702

Order 15616.

1750 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3495	C₂	1	2	yes	maximal
2	C₆₁×#3356	C₂	1	4	yes	maximal, complemented
3	C₆₁×#3352	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3349	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3344	C₂	1	4	yes	maximal, complemented
6	C₆₁×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#3097	C₂	1	2	yes	maximal, complemented
8	C₆₁×#3095	C₂	1	2	yes	maximal, complemented
9	C₆₁×#3082	C₂	1	2	yes	maximal, complemented
10	C₆₁×#3059	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₁×#2930	C₂	1	1	yes	maximal, characteristic
14	C₆₁×#2912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#2911	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#556	C₂²	1	4	yes	—
20	C₆₁×#554	C₂²	1	4	yes	—
21	C₆₁×#552	—	2	4	no	—
22	C₆₁×#552	C₂²	1	4	yes	—
23	C₆₁×#552	C₂²	1	4	yes	—
24	C₆₁×#551	—	2	4	no	—
25	C₆₁×#544	C₂²	1	2	yes	—
26	C₆₁×#542	—	2	2	no	—
27	C₆₁×#541	—	2	2	no	—
28	C₆₁×#536	C₂²	1	2	yes	—
29	C₆₁×#533	—	2	2	no	—
30	C₆₁×#529	—	2	2	no	—
31	C₆₁×#525	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
33	C₆₁×#520	C₂²	1	2	yes	—
34	C₆₁×#519	C₂²	1	2	yes	—
35	C₂×#117×C₆₁	C₂²	1	2	yes	—
36	C₆₁×#501	C₂²	1	4	yes	complemented
37	C₆₁×#499	C₂²	1	2	yes	complemented
38	C₆₁×#498	C₂²	1	2	yes	complemented
39	C₆₁×#485	C₂²	1	4	yes	—
40	C₆₁×#485	C₂²	1	4	yes	—
41	C₆₁×#485	C₂²	1	4	yes	complemented
42	C₆₁×#485	C₂²	1	4	yes	complemented
43	C₆₁×#484	C₂²	1	4	yes	—
44	C₆₁×#484	C₂²	1	4	yes	complemented
45	C₆₁×#483	C₂²	1	4	yes	—
46	C₆₁×#483	C₂²	1	4	yes	complemented
47	C₆₁×#479	C₂²	1	4	yes	complemented
48	C₆₁×#479	C₂²	1	4	yes	complemented
49	C₆₁×#477	C₂²	1	4	yes	complemented
50	C₆₁×#475	C₂²	1	4	yes	complemented
51	C₆₁×#470	C₂²	1	4	yes	—
52	C₆₁×#470	C₂²	1	4	yes	—
53	C₆₁×#470	C₂²	1	2	yes	complemented
54	C₆₁×#470	C₂²	1	2	yes	complemented
55	C₆₁×#467	C₂²	1	4	yes	complemented
56	C₆₁×#441	C₂²	1	2	yes	complemented
57	C₆₁×#441	C₂²	1	2	yes	complemented
58	C₄×Q₁₆×C₆₁	C₂²	1	4	yes	complemented
59	C₆₁×#436	C₂²	1	4	yes	—
60	C₆₁×#435	C₂²	1	4	yes	—
61	C₆₁×#428	C₂²	1	2	yes	—
62	C₆₁×#428	C₂²	1	2	yes	—
63	C₆₁×#428	C₂²	1	2	yes	—
64	C₆₁×#428	C₂²	1	2	yes	—
65	C₆₁×#428	C₂²	1	2	yes	—
66	C₆₁×#428	C₂²	1	2	yes	—
67	C₆₁×#428	C₂²	1	4	yes	complemented
68	C₆₁×#427	C₂²	1	1	yes	characteristic
69	C₆₁×#427	C₂²	1	2	yes	—
70	C₆₁×#427	C₂²	1	1	yes	characteristic
71	C₆₁×#427	C₂²	1	2	yes	complemented
72	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
73	C₂×#106×C₆₁	C₂²	1	2	yes	complemented
74	C₆₁×#424	C₂²	1	1	yes	characteristic
75	C₆₁×#419	C₂²	1	2	yes	—
76	C₆₁×#419	C₂²	1	2	yes	—
77	C₂×#103×C₆₁	C₂²	1	2	yes	—
78	C₆₁×#408	C₂²	1	1	yes	characteristic
79	C₆₁×#407	C₂²	1	1	yes	characteristic
80	C₆₁×#405	C₂²	1	1	yes	characteristic
81	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
82	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
83	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
84	C₂×Q₁₆×C₆₁	—	2	4	no	—
85	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
86	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
87	#131×C₆₁	—	2	4	no	—
88	C₂×M₁₆×C₆₁	—	2	4	no	—
89	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
90	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
91	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
92	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
93	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
96	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
97	C₄⋊Q₈×C₆₁	—	2	2	no	—
98	C₄⋊Q₈×C₆₁	—	2	2	no	—
99	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
100	#126×C₆₁	—	2	4	no	—
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	2	no	—
106	#125×C₆₁	—	2	4	no	—
107	#125×C₆₁	—	2	4	no	—
108	#125×C₆₁	C₂³	1	4	yes	—
109	#124×C₆₁	—	2	2	no	—
110	#124×C₆₁	—	2	4	no	—
111	#123×C₆₁	—	2	4	no	—
112	#123×C₆₁	—	2	4	no	—
113	#123×C₆₁	—	2	4	no	—
114	#123×C₆₁	D₄	1	2	yes	—
115	C₂²⋊Q₈×C₆₁	—	2	2	no	—
116	C₂²⋊Q₈×C₆₁	—	2	2	no	—
117	C₂²⋊Q₈×C₆₁	—	2	4	no	—
118	C₂²⋊Q₈×C₆₁	—	2	4	no	—
119	C₂²⋊Q₈×C₆₁	—	2	4	no	—
120	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
121	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
122	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
123	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
124	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
125	#121×C₆₁	D₄	1	2	yes	—
126	#120×C₆₁	D₄	1	2	yes	—
127	C₄×Q₈×C₆₁	—	2	2	no	—
128	C₄×Q₈×C₆₁	—	2	2	no	—
129	C₄×Q₈×C₆₁	—	2	4	no	—
130	C₄×Q₈×C₆₁	—	2	2	no	—
131	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
132	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
133	C₄×D₄×C₆₁	—	2	2	no	—
134	C₄×D₄×C₆₁	—	2	4	no	—
135	C₄×D₄×C₆₁	—	2	4	no	—
136	C₄×D₄×C₆₁	C₂³	1	4	yes	—
137	#117×C₆₁	—	2	2	no	—
138	#117×C₆₁	—	2	2	no	—
139	#117×C₆₁	—	2	4	no	—
140	#117×C₆₁	—	2	4	no	—
141	#117×C₆₁	—	2	4	no	—
142	#117×C₆₁	C₂³	1	2	yes	—
143	#117×C₆₁	C₂³	1	2	yes	—
144	#117×C₆₁	C₂³	1	4	yes	—
145	#117×C₆₁	C₂³	1	2	yes	—
146	#117×C₆₁	C₂³	1	2	yes	—
147	#117×C₆₁	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
149	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
150	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
151	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
152	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
153	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
154	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
155	C₂×C₄²×C₆₁	—	2	2	no	—
156	#107×C₆₁	C₂³	1	2	yes	—
157	#107×C₆₁	C₂³	1	2	yes	—
158	#107×C₆₁	C₂³	1	2	yes	—
159	#107×C₆₁	C₂³	1	2	yes	—
160	#107×C₆₁	C₂³	1	4	yes	—
161	#107×C₆₁	C₂³	1	4	yes	—
162	#106×C₆₁	C₂³	1	2	yes	—
163	#106×C₆₁	C₂³	1	2	yes	—
164	#106×C₆₁	C₂³	1	1	yes	characteristic
165	#106×C₆₁	C₂³	1	4	yes	—
166	#106×C₆₁	C₂³	1	1	yes	characteristic
167	#106×C₆₁	C₂³	1	1	yes	characteristic
168	#106×C₆₁	C₂³	1	1	yes	characteristic
169	#106×C₆₁	C₂³	1	2	yes	—
170	#106×C₆₁	C₂³	1	2	yes	—
171	#106×C₆₁	C₂³	1	4	yes	—
172	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
173	#103×C₆₁	C₂³	1	4	yes	—
174	#103×C₆₁	C₂³	1	4	yes	—
175	#103×C₆₁	C₂³	1	4	yes	—
176	#103×C₆₁	C₂³	1	4	yes	—
177	#103×C₆₁	C₂³	1	4	yes	—
178	#103×C₆₁	C₂³	1	4	yes	—
179	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
180	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
181	#97×C₆₁	C₂³	1	2	yes	—
182	C₄×C₈×C₆₁	C₂³	1	2	yes	—
183	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
184	#41×C₆₁	—	2	2	no	—
185	#41×C₆₁	—	2	2	no	—
186	C₂×Q₈×C₆₁	—	2	2	no	—
187	C₂×Q₈×C₆₁	—	2	4	no	—
188	C₂×Q₈×C₆₁	—	2	4	no	—
189	C₂×Q₈×C₆₁	—	2	4	no	—
190	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
192	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
193	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
194	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₆₁	—	2	2	no	—
196	C₂²×C₄×C₆₁	—	2	2	no	—
197	C₂²×C₄×C₆₁	—	2	2	no	—
198	C₂²×C₄×C₆₁	—	2	2	no	—
199	C₂²×C₄×C₆₁	—	2	2	no	—
200	C₂²×C₄×C₆₁	—	2	2	no	—
201	C₂²×C₄×C₆₁	—	2	4	no	—
202	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
207	Q₁₆×C₆₁	—	2	4	no	—
208	Q₁₆×C₆₁	—	2	4	no	—
209	M₁₆×C₆₁	—	2	2	no	—
210	M₁₆×C₆₁	—	2	2	no	—
211	M₁₆×C₆₁	—	2	4	no	—
212	M₁₆×C₆₁	—	2	4	no	—
213	M₁₆×C₆₁	—	2	4	no	—
214	C₂×C₈×C₆₁	—	2	2	no	—
215	C₂×C₈×C₆₁	—	2	4	no	—
216	C₂×C₈×C₆₁	#41	1	4	yes	—
217	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
218	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
219	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
220	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
221	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
222	C₄⋊C₄×C₆₁	—	2	4	no	—
223	C₄⋊C₄×C₆₁	—	2	4	no	—
224	C₄⋊C₄×C₆₁	—	2	2	no	—
225	C₄⋊C₄×C₆₁	—	2	2	no	—
226	C₄⋊C₄×C₆₁	—	2	2	no	—
227	C₄⋊C₄×C₆₁	—	2	4	no	—
228	C₄⋊C₄×C₆₁	—	2	4	no	—
229	C₄⋊C₄×C₆₁	—	2	4	no	—
230	C₄⋊C₄×C₆₁	—	2	4	no	—
231	C₄⋊C₄×C₆₁	—	2	4	no	—
232	C₄⋊C₄×C₆₁	—	2	4	no	—
233	C₄⋊C₄×C₆₁	—	2	4	no	—
234	C₄⋊C₄×C₆₁	—	2	4	no	—
235	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
236	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
237	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
238	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
239	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
240	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
241	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
242	C₂²⋊C₄×C₆₁	—	2	2	no	—
243	C₂²⋊C₄×C₆₁	—	2	2	no	—
244	C₂²⋊C₄×C₆₁	—	2	2	no	—
245	C₂²⋊C₄×C₆₁	—	2	2	no	—
246	C₂²⋊C₄×C₆₁	—	2	4	no	—
247	C₂²⋊C₄×C₆₁	—	2	4	no	—
248	C₂²⋊C₄×C₆₁	—	2	4	no	—
249	C₂²⋊C₄×C₆₁	—	2	4	no	—
250	C₂²⋊C₄×C₆₁	—	2	4	no	—
251	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
252	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
253	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
254	C₄²×C₆₁	—	2	2	no	—
255	C₄²×C₆₁	—	2	2	no	—
256	C₄²×C₆₁	—	2	2	no	—
257	C₄²×C₆₁	—	2	2	no	—
258	C₄²×C₆₁	—	2	4	no	—
259	C₄²×C₆₁	—	2	4	no	—
260	C₄²×C₆₁	—	2	4	no	—
261	C₄²×C₆₁	C₂⁴	1	2	yes	—
262	C₂³×C₆₁	—	2	2	no	—
263	C₂³×C₆₁	—	4	2	no	—
264	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
265	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
266	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
267	Q₈×C₆₁	—	2	2	no	—
268	Q₈×C₆₁	—	2	4	no	—
269	Q₈×C₆₁	—	2	4	no	—
270	Q₈×C₆₁	—	4	4	no	—
271	Q₈×C₆₁	—	8	4	no	—
272	D₄×C₆₁	—	2	4	no	—
273	D₄×C₆₁	—	2	2	no	—
274	D₄×C₆₁	—	4	4	no	—
275	C₂×C₄×C₆₁	—	2	2	no	—
276	C₂×C₄×C₆₁	—	2	4	no	—
277	C₂×C₄×C₆₁	—	2	2	no	—
278	C₂×C₄×C₆₁	—	2	2	no	—
279	C₂×C₄×C₆₁	—	2	2	no	—
280	C₂×C₄×C₆₁	—	2	2	no	—
281	C₂×C₄×C₆₁	—	2	2	no	—
282	C₂×C₄×C₆₁	—	2	2	no	—
283	C₂×C₄×C₆₁	—	2	2	no	—
284	C₂×C₄×C₆₁	—	2	4	no	—
285	C₂×C₄×C₆₁	—	2	2	no	—
286	C₂×C₄×C₆₁	—	2	4	no	—
287	C₂×C₄×C₆₁	—	4	4	no	—
288	C₂×C₄×C₆₁	—	4	4	no	—
289	C₂×C₄×C₆₁	—	8	4	no	—
290	C₂×C₄×C₆₁	—	8	2	no	—
291	C₂×C₄×C₆₁	—	8	2	no	—
292	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
293	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
297	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
298	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
299	C₈×C₆₁	—	2	2	no	—
300	C₈×C₆₁	—	2	2	no	—
301	C₈×C₆₁	—	2	2	no	—
302	C₈×C₆₁	C₂×#41	1	2	yes	—
303	C₈×C₆₁	C₂×#41	1	2	yes	—
304	#61269	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
305	C₂²×C₆₁	—	2	2	no	—
306	C₂²×C₆₁	—	2	2	no	—
307	C₂²×C₆₁	—	2	4	no	—
308	C₂²×C₆₁	—	2	2	no	—
309	C₂²×C₆₁	—	2	2	no	—
310	C₂²×C₆₁	—	2	2	no	—
311	C₂²×C₆₁	—	8	2	no	—
312	C₂²×C₆₁	#547	1	2	yes	—
313	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
314	C₄×C₆₁	—	2	4	no	—
315	C₄×C₆₁	—	2	2	no	—
316	C₄×C₆₁	—	2	2	no	—
317	C₄×C₆₁	—	4	2	no	—
318	C₄×C₆₁	—	4	2	no	—
319	C₄×C₆₁	—	4	2	no	—
320	C₄×C₆₁	—	4	4	no	—
321	C₄×C₆₁	—	4	2	no	—
322	C₄×C₆₁	—	4	4	no	—
323	C₄×C₆₁	#547	1	2	yes	—
324	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
325	C₄×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
326	#3495	C₂×C₆₁	1	2	yes	—
327	#3356	C₂×C₆₁	1	4	yes	complemented
328	#3352	C₂×C₆₁	1	4	yes	complemented
329	#3349	C₂×C₆₁	1	4	yes	complemented
330	#3344	C₂×C₆₁	1	4	yes	complemented
331	#3163	C₂×C₆₁	1	1	yes	characteristic, complemented
332	#3097	C₂×C₆₁	1	2	yes	complemented
333	#3095	C₂×C₆₁	1	2	yes	complemented
334	#3082	C₂×C₆₁	1	2	yes	complemented
335	#3059	C₂×C₆₁	1	1	yes	characteristic, complemented
336	C₂×#470	C₂×C₆₁	1	1	yes	characteristic, complemented
337	#2950	C₂×C₆₁	1	1	yes	characteristic, complemented
338	#2930	C₂×C₆₁	1	1	yes	characteristic
339	#2912	C₂×C₆₁	1	1	yes	characteristic, complemented
340	#2911	C₂×C₆₁	1	1	yes	characteristic, complemented
341	C₂×C₆₁	—	2	2	no	—
342	C₂×C₆₁	—	4	2	no	—
343	C₂×C₆₁	—	4	2	no	—
344	C₂×C₆₁	C₂×#547	1	1	yes	characteristic
345	C₂×C₆₁	#3586	1	2	yes	—
346	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
347	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
348	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
349	#556	C₂²×C₆₁	1	4	yes	—
350	#554	C₂²×C₆₁	1	4	yes	—
351	#552	—	2	4	no	—
352	#552	C₂²×C₆₁	1	4	yes	—
353	#552	C₂²×C₆₁	1	4	yes	—
354	#551	—	2	4	no	—
355	#544	C₂²×C₆₁	1	2	yes	—
356	#542	—	2	2	no	—
357	#541	—	2	2	no	—
358	#536	C₂²×C₆₁	1	2	yes	—
359	#533	—	2	2	no	—
360	#529	—	2	2	no	—
361	#525	C₂²×C₆₁	1	1	yes	characteristic
362	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
363	#520	C₂²×C₆₁	1	2	yes	—
364	#519	C₂²×C₆₁	1	2	yes	—
365	C₂×#117	C₂²×C₆₁	1	2	yes	—
366	#501	C₂²×C₆₁	1	4	yes	complemented
367	#499	C₂²×C₆₁	1	2	yes	complemented
368	#498	C₂²×C₆₁	1	2	yes	complemented
369	#485	C₂²×C₆₁	1	4	yes	—
370	#485	C₂²×C₆₁	1	4	yes	—
371	#485	C₂²×C₆₁	1	4	yes	complemented
372	#485	C₂²×C₆₁	1	4	yes	complemented
373	#484	C₂²×C₆₁	1	4	yes	—
374	#484	C₂²×C₆₁	1	4	yes	complemented
375	#483	C₂²×C₆₁	1	4	yes	—
376	#483	C₂²×C₆₁	1	4	yes	complemented
377	#479	C₂²×C₆₁	1	4	yes	complemented
378	#479	C₂²×C₆₁	1	4	yes	complemented
379	#477	C₂²×C₆₁	1	4	yes	complemented
380	#475	C₂²×C₆₁	1	4	yes	complemented
381	#470	C₂²×C₆₁	1	4	yes	—
382	#470	C₂²×C₆₁	1	4	yes	—
383	#470	C₂²×C₆₁	1	2	yes	complemented
384	#470	C₂²×C₆₁	1	2	yes	complemented
385	#467	C₂²×C₆₁	1	4	yes	complemented
386	#441	C₂²×C₆₁	1	2	yes	complemented
387	#441	C₂²×C₆₁	1	2	yes	complemented
388	C₄×Q₁₆	C₂²×C₆₁	1	4	yes	complemented
389	#436	C₂²×C₆₁	1	4	yes	—
390	#435	C₂²×C₆₁	1	4	yes	—
391	#428	C₂²×C₆₁	1	2	yes	—
392	#428	C₂²×C₆₁	1	2	yes	—
393	#428	C₂²×C₆₁	1	2	yes	—
394	#428	C₂²×C₆₁	1	2	yes	—
395	#428	C₂²×C₆₁	1	2	yes	—
396	#428	C₂²×C₆₁	1	2	yes	—
397	#428	C₂²×C₆₁	1	4	yes	complemented
398	#427	C₂²×C₆₁	1	2	yes	—
399	#427	C₂²×C₆₁	1	1	yes	characteristic
400	#427	C₂²×C₆₁	1	1	yes	characteristic
401	#427	C₂²×C₆₁	1	2	yes	complemented
402	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
403	C₂×#106	C₂²×C₆₁	1	2	yes	complemented
404	#424	C₂²×C₆₁	1	1	yes	characteristic
405	#419	C₂²×C₆₁	1	2	yes	—
406	#419	C₂²×C₆₁	1	2	yes	—
407	C₂×#103	C₂²×C₆₁	1	2	yes	—
408	#408	C₂²×C₆₁	1	1	yes	characteristic
409	#407	C₂²×C₆₁	1	1	yes	characteristic
410	#405	C₂²×C₆₁	1	1	yes	characteristic
411	C₆₁	#61269	1	1	yes	characteristic, direct product factor, Sylow
412	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
413	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
414	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
415	C₂×Q₁₆	—	2	4	no	—
416	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
417	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
418	#131	—	2	4	no	—
419	C₂×M₁₆	—	2	4	no	—
420	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
421	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
422	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
423	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
424	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
425	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
426	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
427	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
428	C₄⋊Q₈	—	2	2	no	—
429	C₄⋊Q₈	—	2	2	no	—
430	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
431	#126	—	2	4	no	—
432	#126	—	2	4	no	—
433	#126	—	2	4	no	—
434	#126	—	2	4	no	—
435	#125	—	2	2	no	—
436	#125	—	2	4	no	—
437	#125	—	2	4	no	—
438	#125	—	2	4	no	—
439	#125	C₂³×C₆₁	1	4	yes	—
440	#124	—	2	2	no	—
441	#124	—	2	4	no	—
442	#123	—	2	4	no	—
443	#123	—	2	4	no	—
444	#123	—	2	4	no	—
445	#123	D₄×C₆₁	1	2	yes	—
446	C₂²⋊Q₈	—	2	2	no	—
447	C₂²⋊Q₈	—	2	2	no	—
448	C₂²⋊Q₈	—	2	4	no	—
449	C₂²⋊Q₈	—	2	4	no	—
450	C₂²⋊Q₈	—	2	4	no	—
451	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
452	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
453	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
454	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
455	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
456	#121	D₄×C₆₁	1	2	yes	—
457	#120	D₄×C₆₁	1	2	yes	—
458	C₄×Q₈	—	2	2	no	—
459	C₄×Q₈	—	2	2	no	—
460	C₄×Q₈	—	2	2	no	—
461	C₄×Q₈	—	2	4	no	—
462	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
463	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
464	C₄×D₄	—	2	2	no	—
465	C₄×D₄	—	2	4	no	—
466	C₄×D₄	—	2	4	no	—
467	C₄×D₄	C₂³×C₆₁	1	4	yes	—
468	#117	—	2	2	no	—
469	#117	—	2	2	no	—
470	#117	—	2	4	no	—
471	#117	—	2	4	no	—
472	#117	—	2	4	no	—
473	#117	C₂³×C₆₁	1	2	yes	—
474	#117	C₂³×C₆₁	1	2	yes	—
475	#117	C₂³×C₆₁	1	2	yes	—
476	#117	C₂³×C₆₁	1	2	yes	—
477	#117	C₂³×C₆₁	1	4	yes	—
478	#117	C₂³×C₆₁	1	1	yes	characteristic
479	C₂×C₄⋊C₄	—	2	2	no	—
480	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
481	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
482	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
483	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
484	C₂×C₂²⋊C₄	—	2	2	no	—
485	C₂×C₂²⋊C₄	—	2	2	no	—
486	C₂×C₄²	—	2	2	no	—
487	#107	C₂³×C₆₁	1	2	yes	—
488	#107	C₂³×C₆₁	1	2	yes	—
489	#107	C₂³×C₆₁	1	2	yes	—
490	#107	C₂³×C₆₁	1	2	yes	—
491	#107	C₂³×C₆₁	1	4	yes	—
492	#107	C₂³×C₆₁	1	4	yes	—
493	#106	C₂³×C₆₁	1	2	yes	—
494	#106	C₂³×C₆₁	1	2	yes	—
495	#106	C₂³×C₆₁	1	1	yes	characteristic
496	#106	C₂³×C₆₁	1	4	yes	—
497	#106	C₂³×C₆₁	1	1	yes	characteristic
498	#106	C₂³×C₆₁	1	1	yes	characteristic
499	#106	C₂³×C₆₁	1	1	yes	characteristic
500	#106	C₂³×C₆₁	1	2	yes	—
501	#106	C₂³×C₆₁	1	2	yes	—
502	#106	C₂³×C₆₁	1	4	yes	—
503	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
504	#103	C₂³×C₆₁	1	4	yes	—
505	#103	C₂³×C₆₁	1	4	yes	—
506	#103	C₂³×C₆₁	1	4	yes	—
507	#103	C₂³×C₆₁	1	4	yes	—
508	#103	C₂³×C₆₁	1	4	yes	—
509	#103	C₂³×C₆₁	1	4	yes	—
510	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
511	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
512	#97	C₂³×C₆₁	1	2	yes	—
513	C₄×C₈	C₂³×C₆₁	1	2	yes	—
514	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
515	#41	—	2	2	no	—
516	#41	—	2	2	no	—
517	C₂×Q₈	—	2	2	no	—
518	C₂×Q₈	—	2	4	no	—
519	C₂×Q₈	—	2	4	no	—
520	C₂×Q₈	—	2	4	no	—
521	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
522	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
523	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
524	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
525	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	4	no	—
533	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
534	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
537	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
538	Q₁₆	—	2	4	no	—
539	Q₁₆	—	2	4	no	—
540	M₁₆	—	2	2	no	—
541	M₁₆	—	2	2	no	—
542	M₁₆	—	2	4	no	—
543	M₁₆	—	2	4	no	—
544	M₁₆	—	2	4	no	—
545	C₂×C₈	—	2	2	no	—
546	C₂×C₈	—	2	4	no	—
547	C₂×C₈	#41×C₆₁	1	4	yes	—
548	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
549	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
550	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
551	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
552	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	—	2	4	no	—
563	C₄⋊C₄	—	2	4	no	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
567	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
568	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
569	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
570	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
571	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
572	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	2	no	—
575	C₂²⋊C₄	—	2	2	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	4	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	4	no	—
582	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
583	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
584	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
585	C₄²	—	2	2	no	—
586	C₄²	—	2	2	no	—
587	C₄²	—	2	2	no	—
588	C₄²	—	2	2	no	—
589	C₄²	—	2	4	no	—
590	C₄²	—	2	4	no	—
591	C₄²	—	2	4	no	—
592	C₄²	C₂⁴×C₆₁	1	2	yes	—
593	C₂³	—	2	2	no	—
594	C₂³	—	4	2	no	—
595	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
596	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
597	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
598	Q₈	—	2	2	no	—
599	Q₈	—	2	4	no	—
600	Q₈	—	2	4	no	—
601	Q₈	—	4	4	no	—
602	Q₈	—	8	4	no	—
603	D₄	—	2	4	no	—
604	D₄	—	2	2	no	—
605	D₄	—	4	4	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	4	4	no	—
619	C₂×C₄	—	4	4	no	—
620	C₂×C₄	—	8	4	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	—	8	2	no	—
623	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
624	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
625	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
627	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
628	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
629	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
630	C₈	—	2	2	no	—
631	C₈	—	2	2	no	—
632	C₈	—	2	2	no	—
633	C₈	C₂×#41×C₆₁	1	2	yes	—
634	C₈	C₂×#41×C₆₁	1	2	yes	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	4	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	8	2	no	—
642	C₂²	C₆₁×#547	1	2	yes	—
643	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
644	C₄	—	2	4	no	—
645	C₄	—	2	2	no	—
646	C₄	—	2	2	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	2	no	—
650	C₄	—	4	4	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	4	no	—
653	C₄	C₆₁×#547	1	2	yes	—
654	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
655	C₄	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
656	C₂	—	2	2	no	—
657	C₂	—	4	2	no	—
658	C₂	—	4	2	no	—
659	C₂	C₂×C₆₁×#547	1	1	yes	characteristic
660	C₂	C₆₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₆₁×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₆₁×#525 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#3356	C₂	semidirect
3. C₆₁×#3352	C₂	semidirect
4. C₆₁×#3349	C₂	semidirect
5. C₆₁×#3344	C₂	semidirect
6. C₆₁×#3163	C₂	semidirect
7. C₆₁×#3097	C₂	semidirect
8. C₆₁×#3095	C₂	semidirect
9. C₆₁×#3082	C₂	semidirect
10. C₆₁×#3059	C₂	semidirect
11. C₂×C₆₁×#470	C₂	semidirect
12. C₆₁×#2950	C₂	semidirect
14. C₆₁×#2912	C₂	semidirect
15. C₆₁×#2911	C₂	semidirect
36. C₆₁×#501	C₂²	semidirect
37. C₆₁×#499	C₂²	semidirect
38. C₆₁×#498	C₂²	semidirect
41. C₆₁×#485	C₂²	semidirect
42. C₆₁×#485	C₂²	semidirect
44. C₆₁×#484	C₂²	semidirect
46. C₆₁×#483	C₂²	semidirect
47. C₆₁×#479	C₂²	semidirect
48. C₆₁×#479	C₂²	semidirect
49. C₆₁×#477	C₂²	semidirect
50. C₆₁×#475	C₂²	semidirect
53. C₆₁×#470	C₂²	semidirect
54. C₆₁×#470	C₂²	semidirect
55. C₆₁×#467	C₂²	semidirect
56. C₆₁×#441	C₂²	semidirect
57. C₆₁×#441	C₂²	semidirect
58. C₄×Q₁₆×C₆₁	C₂²	semidirect
67. C₆₁×#428	C₂²	semidirect
71. C₆₁×#427	C₂²	semidirect
73. C₂×#106×C₆₁	C₂²	semidirect
304. #61269	C₆₁	direct (N × H)
327. #3356	C₂×C₆₁	semidirect
328. #3352	C₂×C₆₁	semidirect
329. #3349	C₂×C₆₁	semidirect
330. #3344	C₂×C₆₁	semidirect
331. #3163	C₂×C₆₁	semidirect
332. #3097	C₂×C₆₁	semidirect
333. #3095	C₂×C₆₁	semidirect
334. #3082	C₂×C₆₁	semidirect
335. #3059	C₂×C₆₁	semidirect
336. C₂×#470	C₂×C₆₁	semidirect
337. #2950	C₂×C₆₁	semidirect
339. #2912	C₂×C₆₁	semidirect
340. #2911	C₂×C₆₁	semidirect
366. #501	C₂²×C₆₁	semidirect
367. #499	C₂²×C₆₁	semidirect
368. #498	C₂²×C₆₁	semidirect
371. #485	C₂²×C₆₁	semidirect
372. #485	C₂²×C₆₁	semidirect
374. #484	C₂²×C₆₁	semidirect
376. #483	C₂²×C₆₁	semidirect
377. #479	C₂²×C₆₁	semidirect
378. #479	C₂²×C₆₁	semidirect
379. #477	C₂²×C₆₁	semidirect
380. #475	C₂²×C₆₁	semidirect
383. #470	C₂²×C₆₁	semidirect
384. #470	C₂²×C₆₁	semidirect
385. #467	C₂²×C₆₁	semidirect
386. #441	C₂²×C₆₁	semidirect
387. #441	C₂²×C₆₁	semidirect
388. C₄×Q₁₆	C₂²×C₆₁	semidirect
397. #428	C₂²×C₆₁	semidirect
401. #427	C₂²×C₆₁	semidirect
403. C₂×#106	C₂²×C₆₁	semidirect
411. C₆₁	#61269	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×#3495	2
C₆₁×#3356	2
C₆₁×#3352	2
C₆₁×#3349	2
C₆₁×#3344	2
C₆₁×#3163	2
C₆₁×#3097	2
C₆₁×#3095	2
C₆₁×#3082	2
C₆₁×#3059	2
C₂×C₆₁×#470	2
C₆₁×#2950	2
C₆₁×#2930	2
C₆₁×#2912	2
C₆₁×#2911	2
#61269	61

Subgroups — C61×#61269 #29564702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#61269 #29564702