Subgroups — C₆₁×#60861 #29564294

Order 15616.

1790 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂²×#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3236	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3234	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3232	C₂	1	8	yes	maximal, complemented
6	C₆₁×#3166	C₂	1	2	yes	maximal, complemented
7	C₆₁×#3142	C₂	1	2	yes	maximal, complemented
8	C₆₁×#3092	C₂	1	4	yes	maximal, complemented
9	C₆₁×#3080	C₂	1	4	yes	maximal, complemented
10	C₆₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#533	C₂²	1	2	yes	—
12	C₆₁×#533	C₂²	1	1	yes	characteristic
13	C₆₁×#532	C₂²	1	1	yes	characteristic
14	C₆₁×#532	C₂²	1	2	yes	—
15	C₆₁×#529	—	2	2	no	—
16	C₆₁×#529	—	2	2	no	—
17	C₆₁×#529	C₂²	1	4	yes	—
18	C₂×#125×C₆₁	—	2	2	no	—
19	C₂×#125×C₆₁	—	2	2	no	—
20	C₂×#125×C₆₁	—	2	2	no	—
21	C₂×#125×C₆₁	C₂²	1	2	yes	—
22	C₂×#125×C₆₁	C₂²	1	2	yes	—
23	C₂×#125×C₆₁	C₂²	1	4	yes	—
24	C₂×#125×C₆₁	C₂²	1	4	yes	—
25	C₆₁×#525	—	2	2	no	—
26	C₆₁×#525	C₂²	1	2	yes	—
27	C₄×#41×C₆₁	—	2	2	no	—
28	C₄×#41×C₆₁	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#501	C₂²	1	8	yes	complemented
33	C₆₁×#499	C₂²	1	8	yes	complemented
34	C₆₁×#491	C₂²	1	4	yes	complemented
35	C₆₁×#490	C₂²	1	4	yes	complemented
36	C₆₁×#488	C₂²	1	8	yes	complemented
37	C₆₁×#484	C₂²	1	8	yes	—
38	C₆₁×#483	C₂²	1	8	yes	—
39	C₆₁×#481	C₂²	1	8	yes	—
40	C₆₁×#480	C₂²	1	8	yes	—
41	C₆₁×#479	C₂²	1	16	yes	complemented
42	C₆₁×#478	C₂²	1	16	yes	complemented
43	C₆₁×#432	C₂²	1	4	yes	—
44	C₆₁×#431	C₂²	1	4	yes	—
45	C₆₁×#428	C₂²	1	8	yes	—
46	C₆₁×#423	C₂²	1	2	yes	—
47	C₆₁×#417	C₂²	1	8	yes	—
48	C₆₁×#407	C₂²	1	2	yes	—
49	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
50	C₂×#41×C₆₁	—	2	2	no	—
51	C₂×#41×C₆₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₁	—	2	2	no	—
53	C₂³×C₄×C₆₁	D₄	1	2	yes	—
54	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
56	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
58	#127×C₆₁	C₂³	1	1	yes	characteristic
59	#126×C₆₁	—	2	4	no	—
60	#125×C₆₁	—	2	2	no	—
61	#125×C₆₁	—	2	2	no	—
62	#125×C₆₁	—	2	2	no	—
63	#125×C₆₁	—	2	4	no	—
64	#125×C₆₁	—	2	4	no	—
65	#125×C₆₁	—	2	4	no	—
66	#125×C₆₁	—	2	4	no	—
67	#125×C₆₁	—	2	4	no	—
68	#125×C₆₁	C₂³	1	2	yes	—
69	#125×C₆₁	C₂³	1	8	yes	—
70	#125×C₆₁	C₂³	1	4	yes	—
71	#125×C₆₁	C₂³	1	4	yes	—
72	#124×C₆₁	—	2	2	no	—
73	#124×C₆₁	C₂³	1	2	yes	—
74	#123×C₆₁	—	2	4	no	—
75	#123×C₆₁	—	2	2	no	—
76	C₂²⋊Q₈×C₆₁	—	2	2	no	—
77	C₂²⋊Q₈×C₆₁	—	2	2	no	—
78	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
79	#121×C₆₁	—	2	2	no	—
80	#121×C₆₁	—	2	2	no	—
81	#121×C₆₁	C₂³	1	4	yes	—
82	#120×C₆₁	—	2	2	no	—
83	C₄×Q₈×C₆₁	—	2	2	no	—
84	C₄×Q₈×C₆₁	—	2	2	no	—
85	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
86	C₄×D₄×C₆₁	—	2	4	no	—
87	C₄×D₄×C₆₁	—	2	4	no	—
88	C₄×D₄×C₆₁	—	2	4	no	—
89	C₄×D₄×C₆₁	—	2	2	no	—
90	C₄×D₄×C₆₁	—	2	2	no	—
91	C₄×D₄×C₆₁	C₂³	1	4	yes	—
92	#117×C₆₁	—	2	2	no	—
93	#117×C₆₁	—	2	4	no	—
94	#117×C₆₁	—	2	2	no	—
95	#117×C₆₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
97	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
98	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
99	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
100	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
101	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
102	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
103	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
108	C₂×C₄²×C₆₁	—	2	2	no	—
109	C₂×C₄²×C₆₁	D₄	1	2	yes	—
110	C₂×C₄²×C₆₁	D₄	1	2	yes	—
111	C₂×C₄²×C₆₁	D₄	1	2	yes	—
112	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
116	#107×C₆₁	C₂³	1	16	yes	—
117	#106×C₆₁	C₂³	1	16	yes	—
118	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
119	#103×C₆₁	C₂³	1	16	yes	—
120	#102×C₆₁	C₂³	1	16	yes	—
121	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
122	#97×C₆₁	C₂³	1	4	yes	—
123	C₄×C₈×C₆₁	C₂³	1	4	yes	—
124	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
125	#41×C₆₁	—	2	8	no	—
126	#41×C₆₁	—	4	8	no	—
127	C₂×Q₈×C₆₁	—	2	2	no	—
128	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
129	C₂×D₄×C₆₁	—	2	2	no	—
130	C₂×D₄×C₆₁	—	2	4	no	—
131	C₂×D₄×C₆₁	—	2	2	no	—
132	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₆₁	—	2	4	no	—
134	C₂²×C₄×C₆₁	—	2	4	no	—
135	C₂²×C₄×C₆₁	—	2	4	no	—
136	C₂²×C₄×C₆₁	—	2	4	no	—
137	C₂²×C₄×C₆₁	—	2	2	no	—
138	C₂²×C₄×C₆₁	—	2	4	no	—
139	C₂²×C₄×C₆₁	—	4	4	no	—
140	C₂²×C₄×C₆₁	—	4	8	no	—
141	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
151	M₁₆×C₆₁	—	2	8	no	—
152	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
153	C₄⋊C₄×C₆₁	—	2	8	no	—
154	C₄⋊C₄×C₆₁	—	2	8	no	—
155	C₄⋊C₄×C₆₁	—	2	8	no	—
156	C₄⋊C₄×C₆₁	—	2	8	no	—
157	C₄⋊C₄×C₆₁	—	2	4	no	—
158	C₄⋊C₄×C₆₁	—	2	4	no	—
159	C₄⋊C₄×C₆₁	—	2	4	no	—
160	C₄⋊C₄×C₆₁	—	2	2	no	—
161	C₄⋊C₄×C₆₁	—	2	2	no	—
162	C₄⋊C₄×C₆₁	—	2	4	no	—
163	C₄⋊C₄×C₆₁	—	2	4	no	—
164	C₄⋊C₄×C₆₁	—	2	4	no	—
165	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
166	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
167	C₂²⋊C₄×C₆₁	—	2	4	no	—
168	C₂²⋊C₄×C₆₁	—	2	2	no	—
169	C₂²⋊C₄×C₆₁	—	2	4	no	—
170	C₂²⋊C₄×C₆₁	—	2	4	no	—
171	C₂²⋊C₄×C₆₁	—	2	4	no	—
172	C₄²×C₆₁	—	2	4	no	—
173	C₄²×C₆₁	—	2	2	no	—
174	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₆₁	C₂×D₄	1	2	yes	—
176	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₆₁	C₂×D₄	1	2	yes	—
178	C₄²×C₆₁	C₂×D₄	1	2	yes	—
179	C₄²×C₆₁	C₂×D₄	1	2	yes	—
180	C₄²×C₆₁	C₂×D₄	1	2	yes	—
181	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
182	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
183	C₄²×C₆₁	C₂⁴	1	2	yes	—
184	C₂³×C₆₁	—	2	2	no	—
185	C₂³×C₆₁	—	2	4	no	—
186	C₂³×C₆₁	—	2	2	no	—
187	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
188	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
189	Q₈×C₆₁	—	2	4	no	—
190	Q₈×C₆₁	—	4	4	no	—
191	D₄×C₆₁	—	2	4	no	—
192	D₄×C₆₁	—	4	4	no	—
193	D₄×C₆₁	—	4	4	no	—
194	D₄×C₆₁	—	4	8	no	—
195	C₂×C₄×C₆₁	—	2	2	no	—
196	C₂×C₄×C₆₁	—	2	4	no	—
197	C₂×C₄×C₆₁	—	2	4	no	—
198	C₂×C₄×C₆₁	—	2	4	no	—
199	C₂×C₄×C₆₁	—	2	4	no	—
200	C₂×C₄×C₆₁	—	2	2	no	—
201	C₂×C₄×C₆₁	—	2	4	no	—
202	C₂×C₄×C₆₁	—	2	8	no	—
203	C₂×C₄×C₆₁	—	4	2	no	—
204	C₂×C₄×C₆₁	—	4	2	no	—
205	C₂×C₄×C₆₁	—	4	2	no	—
206	C₂×C₄×C₆₁	—	4	2	no	—
207	C₂×C₄×C₆₁	—	4	8	no	—
208	C₂×C₄×C₆₁	—	4	8	no	—
209	C₂×C₄×C₆₁	—	4	8	no	—
210	C₂×C₄×C₆₁	—	4	8	no	—
211	C₂×C₄×C₆₁	—	4	4	no	—
212	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₁	2₋¹⁺⁴	1	4	yes	—
219	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
220	C₈×C₆₁	—	2	8	no	—
221	#60861	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂²×C₆₁	—	2	8	no	—
223	C₂²×C₆₁	—	2	2	no	—
224	C₂²×C₆₁	—	2	2	no	—
225	C₂²×C₆₁	—	2	2	no	—
226	C₂²×C₆₁	—	4	2	no	—
227	C₂²×C₆₁	—	4	4	no	—
228	C₂²×C₆₁	#536	1	2	yes	—
229	C₂²×C₆₁	#575	1	4	yes	—
230	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
231	C₄×C₆₁	—	2	2	no	—
232	C₄×C₆₁	—	2	2	no	—
233	C₄×C₆₁	—	2	2	no	—
234	C₄×C₆₁	—	2	2	no	—
235	C₄×C₆₁	—	2	2	no	—
236	C₄×C₆₁	—	4	2	no	—
237	C₄×C₆₁	—	4	8	no	—
238	C₄×C₆₁	—	4	4	no	—
239	C₄×C₆₁	#536	1	2	yes	—
240	C₄×C₆₁	C₂×2₋¹⁺⁴	1	2	yes	—
241	#3444	C₂×C₆₁	1	1	yes	characteristic
242	C₂²×#125	C₂×C₆₁	1	1	yes	characteristic, complemented
243	#3236	C₂×C₆₁	1	4	yes	complemented
244	#3234	C₂×C₆₁	1	4	yes	complemented
245	#3232	C₂×C₆₁	1	8	yes	complemented
246	#3166	C₂×C₆₁	1	2	yes	complemented
247	#3142	C₂×C₆₁	1	2	yes	complemented
248	#3092	C₂×C₆₁	1	4	yes	complemented
249	#3080	C₂×C₆₁	1	4	yes	complemented
250	#2920	C₂×C₆₁	1	1	yes	characteristic, complemented
251	C₂×C₆₁	—	2	2	no	—
252	C₂×C₆₁	—	2	4	no	—
253	C₂×C₆₁	—	8	2	no	—
254	C₂×C₆₁	C₂×#536	1	1	yes	characteristic
255	C₂×C₆₁	C₂×#575	1	2	yes	—
256	#533	C₂²×C₆₁	1	2	yes	—
257	#533	C₂²×C₆₁	1	1	yes	characteristic
258	#532	C₂²×C₆₁	1	1	yes	characteristic
259	#532	C₂²×C₆₁	1	2	yes	—
260	#529	—	2	2	no	—
261	#529	—	2	2	no	—
262	#529	C₂²×C₆₁	1	4	yes	—
263	C₂×#125	—	2	2	no	—
264	C₂×#125	—	2	2	no	—
265	C₂×#125	—	2	2	no	—
266	C₂×#125	C₂²×C₆₁	1	2	yes	—
267	C₂×#125	C₂²×C₆₁	1	2	yes	—
268	C₂×#125	C₂²×C₆₁	1	4	yes	—
269	C₂×#125	C₂²×C₆₁	1	4	yes	—
270	#525	—	2	2	no	—
271	#525	C₂²×C₆₁	1	2	yes	—
272	C₄×#41	—	2	2	no	—
273	C₄×#41	C₂²×C₆₁	1	2	yes	—
274	C₂²×C₄⋊C₄	—	2	2	no	—
275	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
276	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
277	#501	C₂²×C₆₁	1	8	yes	complemented
278	#499	C₂²×C₆₁	1	8	yes	complemented
279	#491	C₂²×C₆₁	1	4	yes	complemented
280	#490	C₂²×C₆₁	1	4	yes	complemented
281	#488	C₂²×C₆₁	1	8	yes	complemented
282	#484	C₂²×C₆₁	1	8	yes	—
283	#483	C₂²×C₆₁	1	8	yes	—
284	#481	C₂²×C₆₁	1	8	yes	—
285	#480	C₂²×C₆₁	1	8	yes	—
286	#479	C₂²×C₆₁	1	16	yes	complemented
287	#478	C₂²×C₆₁	1	16	yes	complemented
288	#432	C₂²×C₆₁	1	4	yes	—
289	#431	C₂²×C₆₁	1	4	yes	—
290	#428	C₂²×C₆₁	1	8	yes	—
291	#423	C₂²×C₆₁	1	2	yes	—
292	#417	C₂²×C₆₁	1	8	yes	—
293	#407	C₂²×C₆₁	1	2	yes	—
294	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
295	C₆₁	#60861	1	1	yes	characteristic, direct product factor, Sylow
296	C₂×#41	—	2	2	no	—
297	C₂×#41	C₂³×C₆₁	1	2	yes	—
298	C₂³×C₄	—	2	2	no	—
299	C₂³×C₄	D₄×C₆₁	1	2	yes	—
300	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
301	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
302	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
303	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
304	#127	C₂³×C₆₁	1	1	yes	characteristic
305	#126	—	2	4	no	—
306	#125	—	2	2	no	—
307	#125	—	2	2	no	—
308	#125	—	2	4	no	—
309	#125	—	2	4	no	—
310	#125	—	2	4	no	—
311	#125	—	2	4	no	—
312	#125	—	2	4	no	—
313	#125	—	2	2	no	—
314	#125	C₂³×C₆₁	1	8	yes	—
315	#125	C₂³×C₆₁	1	4	yes	—
316	#125	C₂³×C₆₁	1	4	yes	—
317	#125	C₂³×C₆₁	1	2	yes	—
318	#124	—	2	2	no	—
319	#124	C₂³×C₆₁	1	2	yes	—
320	#123	—	2	4	no	—
321	#123	—	2	2	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	—	2	2	no	—
324	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
325	#121	—	2	2	no	—
326	#121	—	2	2	no	—
327	#121	C₂³×C₆₁	1	4	yes	—
328	#120	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
332	C₄×D₄	—	2	4	no	—
333	C₄×D₄	—	2	4	no	—
334	C₄×D₄	—	2	4	no	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	—	2	2	no	—
337	C₄×D₄	C₂³×C₆₁	1	4	yes	—
338	#117	—	2	2	no	—
339	#117	—	2	4	no	—
340	#117	—	2	2	no	—
341	#117	C₂³×C₆₁	1	2	yes	—
342	C₂×C₄⋊C₄	—	2	4	no	—
343	C₂×C₄⋊C₄	—	2	4	no	—
344	C₂×C₄⋊C₄	—	2	4	no	—
345	C₂×C₄⋊C₄	—	2	4	no	—
346	C₂×C₄⋊C₄	—	2	4	no	—
347	C₂×C₄⋊C₄	—	2	4	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
352	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
353	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
354	C₂×C₄²	—	2	2	no	—
355	C₂×C₄²	D₄×C₆₁	1	2	yes	—
356	C₂×C₄²	D₄×C₆₁	1	2	yes	—
357	C₂×C₄²	D₄×C₆₁	1	2	yes	—
358	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
359	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
360	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
362	#107	C₂³×C₆₁	1	16	yes	—
363	#106	C₂³×C₆₁	1	16	yes	—
364	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
365	#103	C₂³×C₆₁	1	16	yes	—
366	#102	C₂³×C₆₁	1	16	yes	—
367	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
368	#97	C₂³×C₆₁	1	4	yes	—
369	C₄×C₈	C₂³×C₆₁	1	4	yes	—
370	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
371	#41	—	2	8	no	—
372	#41	—	4	8	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
391	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
393	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
395	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
397	M₁₆	—	2	8	no	—
398	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
399	C₄⋊C₄	—	2	8	no	—
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	C₂⁴×C₆₁	1	8	yes	—
412	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	2	4	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
421	C₄²	C₂×D₄×C₆₁	1	2	yes	—
422	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
423	C₄²	C₂×D₄×C₆₁	1	2	yes	—
424	C₄²	C₂×D₄×C₆₁	1	2	yes	—
425	C₄²	C₂×D₄×C₆₁	1	2	yes	—
426	C₄²	C₂×D₄×C₆₁	1	2	yes	—
427	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
428	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
429	C₄²	C₂⁴×C₆₁	1	2	yes	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	2	2	no	—
433	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
434	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
435	Q₈	—	2	4	no	—
436	Q₈	—	4	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	4	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	8	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	8	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	—	4	8	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
464	C₂×C₄	2₋¹⁺⁴×C₆₁	1	4	yes	—
465	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
466	C₈	—	2	8	no	—
467	C₂²	—	2	8	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	2	no	—
473	C₂²	C₆₁×#536	1	2	yes	—
474	C₂²	C₆₁×#575	1	4	yes	—
475	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	4	8	no	—
482	C₄	—	4	4	no	—
483	C₄	—	4	2	no	—
484	C₄	C₆₁×#536	1	2	yes	—
485	C₄	C₂×2₋¹⁺⁴×C₆₁	1	2	yes	—
486	C₂	—	2	2	no	—
487	C₂	—	2	4	no	—
488	C₂	—	8	2	no	—
489	C₂	C₂×C₆₁×#536	1	1	yes	characteristic
490	C₂	C₂×C₆₁×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₆₁×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#125×C₆₁	C₂	semidirect
3. C₆₁×#3236	C₂	semidirect
4. C₆₁×#3234	C₂	semidirect
5. C₆₁×#3232	C₂	semidirect
6. C₆₁×#3166	C₂	semidirect
7. C₆₁×#3142	C₂	semidirect
8. C₆₁×#3092	C₂	semidirect
9. C₆₁×#3080	C₂	semidirect
10. C₆₁×#2920	C₂	semidirect
32. C₆₁×#501	C₂²	semidirect
33. C₆₁×#499	C₂²	semidirect
34. C₆₁×#491	C₂²	semidirect
35. C₆₁×#490	C₂²	semidirect
36. C₆₁×#488	C₂²	semidirect
41. C₆₁×#479	C₂²	semidirect
42. C₆₁×#478	C₂²	semidirect
221. #60861	C₆₁	direct (N × H)
242. C₂²×#125	C₂×C₆₁	semidirect
243. #3236	C₂×C₆₁	semidirect
244. #3234	C₂×C₆₁	semidirect
245. #3232	C₂×C₆₁	semidirect
246. #3166	C₂×C₆₁	semidirect
247. #3142	C₂×C₆₁	semidirect
248. #3092	C₂×C₆₁	semidirect
249. #3080	C₂×C₆₁	semidirect
250. #2920	C₂×C₆₁	semidirect
277. #501	C₂²×C₆₁	semidirect
278. #499	C₂²×C₆₁	semidirect
279. #491	C₂²×C₆₁	semidirect
280. #490	C₂²×C₆₁	semidirect
281. #488	C₂²×C₆₁	semidirect
286. #479	C₂²×C₆₁	semidirect
287. #478	C₂²×C₆₁	semidirect
295. C₆₁	#60861	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×#3444	2
C₂²×#125×C₆₁	2
C₆₁×#3236	2
C₆₁×#3234	2
C₆₁×#3232	2
C₆₁×#3166	2
C₆₁×#3142	2
C₆₁×#3092	2
C₆₁×#3080	2
C₆₁×#2920	2
#60861	61

Subgroups — C61×#60861 #29564294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60861 #29564294