Subgroups — C₆₁×#61806 #29565239

Order 15616.

1668 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3550	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3507	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3356	C₂	1	2	yes	maximal, complemented
4	C₆₁×#3349	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3342	C₂	1	2	yes	maximal, complemented
6	C₆₁×#3328	C₂	1	2	yes	maximal, complemented
7	C₆₁×#3312	C₂	1	4	yes	maximal, complemented
8	C₆₁×#3299	C₂	1	2	yes	maximal, complemented
9	C₆₁×#3237	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#3231	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#3202	C₂	1	4	yes	maximal, complemented
12	C₆₁×#3113	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#3097	C₂	1	4	yes	maximal, complemented
15	C₆₁×#2981	C₂	1	1	yes	maximal, characteristic
16	C₆₁×#563	—	2	2	no	—
17	C₆₁×#559	—	2	1	no	—
18	C₆₁×#559	—	2	1	no	—
19	Q₈²×C₆₁	—	2	1	no	—
20	C₆₁×#557	—	2	1	no	—
21	C₆₁×#556	—	2	2	no	—
22	C₆₁×#556	C₂²	1	2	yes	—
23	C₆₁×#556	C₂²	1	2	yes	—
24	C₆₁×#556	C₂²	1	2	yes	—
25	C₆₁×#554	C₂²	1	2	yes	—
26	C₆₁×#552	C₂²	1	4	yes	—
27	D₄×Q₈×C₆₁	C₂²	1	2	yes	—
28	D₄×Q₈×C₆₁	C₂²	1	2	yes	—
29	C₆₁×#542	—	2	2	no	—
30	C₆₁×#542	—	2	2	no	—
31	C₆₁×#539	—	2	2	no	—
32	C₆₁×#536	C₂²	1	4	yes	—
33	C₆₁×#536	C₂²	1	2	yes	—
34	C₆₁×#533	C₂²	1	1	yes	characteristic
35	C₆₁×#533	C₂²	1	1	yes	characteristic
36	C₆₁×#533	C₂²	1	1	yes	characteristic
37	C₆₁×#532	C₂²	1	1	yes	characteristic
38	C₆₁×#529	—	2	2	no	—
39	C₆₁×#520	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₆₁×#501	C₂²	1	2	yes	—
42	C₆₁×#500	C₂²	1	1	yes	characteristic
43	C₆₁×#498	C₂²	1	1	yes	characteristic
44	C₆₁×#491	C₂²	1	2	yes	—
45	C₆₁×#487	C₂²	1	1	yes	characteristic
46	C₆₁×#487	C₂²	1	1	yes	characteristic
47	C₆₁×#485	C₂²	1	4	yes	—
48	C₆₁×#485	C₂²	1	4	yes	—
49	C₆₁×#485	C₂²	1	4	yes	—
50	C₆₁×#485	C₂²	1	4	yes	—
51	C₆₁×#485	C₂²	1	4	yes	—
52	C₆₁×#485	C₂²	1	4	yes	—
53	C₆₁×#484	C₂²	1	4	yes	—
54	C₆₁×#484	C₂²	1	2	yes	—
55	C₆₁×#483	C₂²	1	4	yes	—
56	C₆₁×#483	C₂²	1	2	yes	—
57	C₆₁×#479	C₂²	1	4	yes	—
58	C₆₁×#477	C₂²	1	2	yes	—
59	C₆₁×#477	C₂²	1	2	yes	—
60	C₆₁×#475	C₂²	1	2	yes	—
61	C₆₁×#475	C₂²	1	2	yes	—
62	C₆₁×#470	C₂²	1	4	yes	—
63	C₆₁×#470	C₂²	1	4	yes	—
64	C₆₁×#467	C₂²	1	4	yes	—
65	C₆₁×#467	C₂²	1	4	yes	—
66	C₆₁×#462	C₂²	1	2	yes	—
67	C₆₁×#462	C₂²	1	2	yes	—
68	C₆₁×#451	C₂²	1	4	yes	—
69	C₆₁×#451	C₂²	1	4	yes	—
70	C₆₁×#451	C₂²	1	2	yes	—
71	C₆₁×#451	C₂²	1	2	yes	—
72	C₆₁×#441	C₂²	1	4	yes	—
73	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
74	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
75	C₆₁×#436	C₂²	1	2	yes	—
76	C₆₁×#435	C₂²	1	4	yes	—
77	C₈×D₄×C₆₁	C₂²	1	2	yes	—
78	C₆₁×#432	C₂²	1	1	yes	characteristic
79	C₆₁×#431	C₂²	1	1	yes	characteristic
80	C₆₁×#428	C₂²	1	4	yes	—
81	C₆₁×#427	C₂²	1	4	yes	—
82	C₆₁×#419	C₂²	1	4	yes	—
83	C₂×#103×C₆₁	C₂²	1	4	yes	—
84	C₆₁×#408	C₂²	1	4	yes	—
85	C₂×#41×C₆₁	C₂³	1	2	yes	—
86	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
87	C₂²×Q₈×C₆₁	C₂³	1	4	yes	—
88	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
89	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
90	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
91	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
92	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
93	C₄⋊Q₈×C₆₁	—	2	2	no	—
94	C₄⋊Q₈×C₆₁	—	2	4	no	—
95	C₄⋊Q₈×C₆₁	—	2	2	no	—
96	C₄⋊Q₈×C₆₁	—	2	2	no	—
97	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
98	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
99	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
100	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
101	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
102	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
103	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
104	#127×C₆₁	D₄	1	1	yes	characteristic
105	#126×C₆₁	—	2	4	no	—
106	#126×C₆₁	—	2	4	no	—
107	#126×C₆₁	—	2	4	no	—
108	#126×C₆₁	—	2	4	no	—
109	#125×C₆₁	—	2	4	no	—
110	#125×C₆₁	—	2	2	no	—
111	#125×C₆₁	—	2	1	no	—
112	#125×C₆₁	—	2	1	no	—
113	#125×C₆₁	C₂³	1	2	yes	—
114	#124×C₆₁	—	2	4	no	—
115	#124×C₆₁	—	2	2	no	—
116	#124×C₆₁	—	2	2	no	—
117	#124×C₆₁	—	2	2	no	—
118	#124×C₆₁	—	2	2	no	—
119	#124×C₆₁	D₄	1	2	yes	—
120	#123×C₆₁	—	2	4	no	—
121	#123×C₆₁	—	2	4	no	—
122	#123×C₆₁	—	2	4	no	—
123	#123×C₆₁	—	2	4	no	—
124	C₂²⋊Q₈×C₆₁	—	2	1	no	—
125	C₂²⋊Q₈×C₆₁	—	2	1	no	—
126	C₂²⋊Q₈×C₆₁	—	2	1	no	—
127	C₂²⋊Q₈×C₆₁	—	2	1	no	—
128	C₂²⋊Q₈×C₆₁	—	2	4	no	—
129	C₂²⋊Q₈×C₆₁	—	2	4	no	—
130	C₂²⋊Q₈×C₆₁	—	2	4	no	—
131	C₂²⋊Q₈×C₆₁	C₂³	1	2	yes	—
132	C₂²⋊Q₈×C₆₁	C₂³	1	2	yes	—
133	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
134	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
135	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
136	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
137	#121×C₆₁	D₄	1	2	yes	—
138	#121×C₆₁	D₄	1	2	yes	—
139	C₄×Q₈×C₆₁	—	2	2	no	—
140	C₄×Q₈×C₆₁	—	2	2	no	—
141	C₄×Q₈×C₆₁	—	2	2	no	—
142	C₄×Q₈×C₆₁	—	2	2	no	—
143	C₄×Q₈×C₆₁	—	2	1	no	—
144	C₄×Q₈×C₆₁	—	2	1	no	—
145	C₄×Q₈×C₆₁	—	2	1	no	—
146	C₄×Q₈×C₆₁	—	2	1	no	—
147	C₄×Q₈×C₆₁	—	2	1	no	—
148	C₄×Q₈×C₆₁	—	2	1	no	—
149	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
150	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
151	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
152	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
153	C₄×D₄×C₆₁	—	2	4	no	—
154	C₄×D₄×C₆₁	—	2	2	no	—
155	C₄×D₄×C₆₁	—	2	2	no	—
156	C₄×D₄×C₆₁	—	2	2	no	—
157	C₄×D₄×C₆₁	—	2	2	no	—
158	C₄×D₄×C₆₁	C₂³	1	2	yes	—
159	C₄×D₄×C₆₁	C₂³	1	2	yes	—
160	#117×C₆₁	—	2	1	no	—
161	#117×C₆₁	—	2	1	no	—
162	#117×C₆₁	—	2	4	no	—
163	#117×C₆₁	C₂³	1	2	yes	—
164	#117×C₆₁	C₂³	1	4	yes	—
165	#117×C₆₁	C₂³	1	4	yes	—
166	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
167	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
168	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
169	C₂×C₄²×C₆₁	—	2	1	no	—
170	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
171	#107×C₆₁	C₂³	1	2	yes	—
172	#107×C₆₁	C₂³	1	4	yes	—
173	#107×C₆₁	C₂³	1	2	yes	—
174	#107×C₆₁	C₂³	1	4	yes	—
175	#106×C₆₁	C₂³	1	4	yes	—
176	#106×C₆₁	C₂³	1	4	yes	—
177	#106×C₆₁	C₂³	1	4	yes	—
178	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
179	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
180	#103×C₆₁	C₂³	1	2	yes	—
181	#103×C₆₁	C₂³	1	4	yes	—
182	#103×C₆₁	C₂³	1	4	yes	—
183	#103×C₆₁	C₂³	1	4	yes	—
184	#103×C₆₁	C₂³	1	2	yes	—
185	#103×C₆₁	C₂³	1	4	yes	—
186	#103×C₆₁	C₂³	1	4	yes	—
187	#103×C₆₁	C₂³	1	2	yes	—
188	#103×C₆₁	C₂³	1	2	yes	—
189	#103×C₆₁	C₂³	1	4	yes	—
190	C₂²⋊C₈×C₆₁	C₂³	1	2	yes	—
191	C₂²⋊C₈×C₆₁	C₂³	1	2	yes	—
192	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
193	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
194	#97×C₆₁	C₂³	1	2	yes	—
195	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
196	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
197	#41×C₆₁	—	4	4	no	—
198	C₂×Q₈×C₆₁	—	2	1	no	—
199	C₂×Q₈×C₆₁	—	2	1	no	—
200	C₂×Q₈×C₆₁	—	2	1	no	—
201	C₂×Q₈×C₆₁	—	2	1	no	—
202	C₂×Q₈×C₆₁	—	2	1	no	—
203	C₂×Q₈×C₆₁	—	2	4	no	—
204	C₂×Q₈×C₆₁	—	4	2	no	—
205	C₂×Q₈×C₆₁	—	4	4	no	—
206	C₂×Q₈×C₆₁	—	4	4	no	—
207	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
208	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
209	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
210	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
211	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
212	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
213	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
214	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
215	C₂²×C₄×C₆₁	—	2	1	no	—
216	C₂²×C₄×C₆₁	—	2	1	no	—
217	C₂²×C₄×C₆₁	—	2	4	no	—
218	C₂²×C₄×C₆₁	—	2	4	no	—
219	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
220	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
221	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
222	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
223	Q₁₆×C₆₁	—	4	2	no	—
224	Q₁₆×C₆₁	—	4	4	no	—
225	Q₁₆×C₆₁	—	4	2	no	—
226	M₁₆×C₆₁	—	4	4	no	—
227	C₂×C₈×C₆₁	—	4	4	no	—
228	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
229	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
230	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
231	C₄⋊C₄×C₆₁	—	2	2	no	—
232	C₄⋊C₄×C₆₁	—	2	2	no	—
233	C₄⋊C₄×C₆₁	—	2	1	no	—
234	C₄⋊C₄×C₆₁	—	2	4	no	—
235	C₄⋊C₄×C₆₁	—	2	2	no	—
236	C₄⋊C₄×C₆₁	—	2	1	no	—
237	C₄⋊C₄×C₆₁	—	2	1	no	—
238	C₄⋊C₄×C₆₁	—	2	4	no	—
239	C₄⋊C₄×C₆₁	—	2	1	no	—
240	C₄⋊C₄×C₆₁	—	2	1	no	—
241	C₄⋊C₄×C₆₁	—	2	1	no	—
242	C₄⋊C₄×C₆₁	—	2	1	no	—
243	C₄⋊C₄×C₆₁	—	2	1	no	—
244	C₄⋊C₄×C₆₁	—	2	4	no	—
245	C₄⋊C₄×C₆₁	—	2	4	no	—
246	C₄⋊C₄×C₆₁	—	2	2	no	—
247	C₄⋊C₄×C₆₁	—	2	1	no	—
248	C₄⋊C₄×C₆₁	—	2	2	no	—
249	C₄⋊C₄×C₆₁	—	2	2	no	—
250	C₄⋊C₄×C₆₁	—	2	4	no	—
251	C₄⋊C₄×C₆₁	—	2	2	no	—
252	C₄⋊C₄×C₆₁	—	2	2	no	—
253	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
254	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
255	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
256	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
257	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
258	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
259	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
260	C₂²⋊C₄×C₆₁	—	2	2	no	—
261	C₂²⋊C₄×C₆₁	—	2	2	no	—
262	C₂²⋊C₄×C₆₁	—	2	4	no	—
263	C₂²⋊C₄×C₆₁	—	2	2	no	—
264	C₂²⋊C₄×C₆₁	—	2	4	no	—
265	C₂²⋊C₄×C₆₁	—	2	4	no	—
266	C₂²⋊C₄×C₆₁	—	2	4	no	—
267	C₂²⋊C₄×C₆₁	—	2	4	no	—
268	C₂²⋊C₄×C₆₁	—	2	4	no	—
269	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
270	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
271	C₄²×C₆₁	—	2	2	no	—
272	C₄²×C₆₁	—	2	2	no	—
273	C₄²×C₆₁	—	2	4	no	—
274	C₄²×C₆₁	—	2	2	no	—
275	C₄²×C₆₁	—	2	2	no	—
276	C₄²×C₆₁	—	2	2	no	—
277	C₄²×C₆₁	—	2	4	no	—
278	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
279	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
280	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
281	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
282	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
283	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
284	Q₈×C₆₁	—	2	2	no	—
285	Q₈×C₆₁	—	2	2	no	—
286	Q₈×C₆₁	—	2	2	no	—
287	Q₈×C₆₁	—	2	4	no	—
288	Q₈×C₆₁	—	2	2	no	—
289	Q₈×C₆₁	—	4	2	no	—
290	Q₈×C₆₁	—	4	2	no	—
291	Q₈×C₆₁	—	4	2	no	—
292	Q₈×C₆₁	—	4	2	no	—
293	Q₈×C₆₁	—	4	2	no	—
294	Q₈×C₆₁	—	8	1	no	—
295	Q₈×C₆₁	—	8	4	no	—
296	D₄×C₆₁	—	2	4	no	—
297	D₄×C₆₁	—	4	2	no	—
298	D₄×C₆₁	—	4	2	no	—
299	D₄×C₆₁	—	4	4	no	—
300	C₂×C₄×C₆₁	—	2	2	no	—
301	C₂×C₄×C₆₁	—	2	4	no	—
302	C₂×C₄×C₆₁	—	2	4	no	—
303	C₂×C₄×C₆₁	—	2	2	no	—
304	C₂×C₄×C₆₁	—	2	2	no	—
305	C₂×C₄×C₆₁	—	2	2	no	—
306	C₂×C₄×C₆₁	—	2	2	no	—
307	C₂×C₄×C₆₁	—	2	2	no	—
308	C₂×C₄×C₆₁	—	2	2	no	—
309	C₂×C₄×C₆₁	—	2	2	no	—
310	C₂×C₄×C₆₁	—	2	2	no	—
311	C₂×C₄×C₆₁	—	2	2	no	—
312	C₂×C₄×C₆₁	—	4	1	no	—
313	C₂×C₄×C₆₁	—	4	2	no	—
314	C₂×C₄×C₆₁	—	8	1	no	—
315	C₂×C₄×C₆₁	—	8	4	no	—
316	C₂×C₄×C₆₁	—	8	4	no	—
317	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
318	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
321	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
324	C₈×C₆₁	—	2	2	no	—
325	C₈×C₆₁	—	2	4	no	—
326	C₈×C₆₁	—	2	2	no	—
327	#61806	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
328	C₂²×C₆₁	—	2	1	no	—
329	C₂²×C₆₁	—	2	4	no	—
330	C₂²×C₆₁	—	2	1	no	—
331	C₂²×C₆₁	—	2	4	no	—
332	C₂²×C₆₁	—	2	1	no	—
333	C₂²×C₆₁	—	2	4	no	—
334	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
335	C₄×C₆₁	—	2	1	no	—
336	C₄×C₆₁	—	2	1	no	—
337	C₄×C₆₁	—	2	1	no	—
338	C₄×C₆₁	—	2	4	no	—
339	C₄×C₆₁	—	2	1	no	—
340	C₄×C₆₁	—	4	2	no	—
341	C₄×C₆₁	—	4	2	no	—
342	C₄×C₆₁	—	4	2	no	—
343	C₄×C₆₁	—	4	2	no	—
344	C₄×C₆₁	—	4	2	no	—
345	C₄×C₆₁	—	4	2	no	—
346	C₄×C₆₁	—	4	2	no	—
347	C₄×C₆₁	—	4	2	no	—
348	C₄×C₆₁	#578	1	2	yes	—
349	C₄×C₆₁	#585	1	2	yes	—
350	#3550	C₂×C₆₁	1	1	yes	characteristic
351	#3507	C₂×C₆₁	1	1	yes	characteristic
352	#3356	C₂×C₆₁	1	2	yes	complemented
353	#3349	C₂×C₆₁	1	4	yes	complemented
354	#3342	C₂×C₆₁	1	2	yes	complemented
355	#3328	C₂×C₆₁	1	2	yes	complemented
356	#3312	C₂×C₆₁	1	4	yes	complemented
357	#3299	C₂×C₆₁	1	2	yes	complemented
358	#3237	C₂×C₆₁	1	1	yes	characteristic, complemented
359	#3231	C₂×C₆₁	1	1	yes	characteristic, complemented
360	#3202	C₂×C₆₁	1	4	yes	complemented
361	#3113	C₂×C₆₁	1	1	yes	characteristic, complemented
362	#3104	C₂×C₆₁	1	1	yes	characteristic, complemented
363	#3097	C₂×C₆₁	1	4	yes	complemented
364	#2981	C₂×C₆₁	1	1	yes	characteristic
365	C₂×C₆₁	—	4	1	no	—
366	C₂×C₆₁	—	4	4	no	—
367	C₂×C₆₁	#3485	1	1	yes	characteristic
368	C₂×C₆₁	C₂×#578	1	1	yes	characteristic
369	C₂×C₆₁	#3586	1	1	yes	characteristic
370	#563	—	2	2	no	—
371	#559	—	2	1	no	—
372	#559	—	2	1	no	—
373	Q₈²	—	2	1	no	—
374	#557	—	2	1	no	—
375	#556	—	2	2	no	—
376	#556	C₂²×C₆₁	1	2	yes	—
377	#556	C₂²×C₆₁	1	2	yes	—
378	#556	C₂²×C₆₁	1	2	yes	—
379	#554	C₂²×C₆₁	1	2	yes	—
380	#552	C₂²×C₆₁	1	4	yes	—
381	D₄×Q₈	C₂²×C₆₁	1	2	yes	—
382	D₄×Q₈	C₂²×C₆₁	1	2	yes	—
383	#542	—	2	2	no	—
384	#542	—	2	2	no	—
385	#539	—	2	2	no	—
386	#536	C₂²×C₆₁	1	4	yes	—
387	#536	C₂²×C₆₁	1	2	yes	—
388	#533	C₂²×C₆₁	1	1	yes	characteristic
389	#533	C₂²×C₆₁	1	1	yes	characteristic
390	#533	C₂²×C₆₁	1	1	yes	characteristic
391	#532	C₂²×C₆₁	1	1	yes	characteristic
392	#529	—	2	2	no	—
393	#520	C₂²×C₆₁	1	2	yes	—
394	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
395	#501	C₂²×C₆₁	1	2	yes	—
396	#500	C₂²×C₆₁	1	1	yes	characteristic
397	#498	C₂²×C₆₁	1	1	yes	characteristic
398	#491	C₂²×C₆₁	1	2	yes	—
399	#487	C₂²×C₆₁	1	1	yes	characteristic
400	#487	C₂²×C₆₁	1	1	yes	characteristic
401	#485	C₂²×C₆₁	1	4	yes	—
402	#485	C₂²×C₆₁	1	4	yes	—
403	#485	C₂²×C₆₁	1	4	yes	—
404	#485	C₂²×C₆₁	1	4	yes	—
405	#485	C₂²×C₆₁	1	4	yes	—
406	#485	C₂²×C₆₁	1	4	yes	—
407	#484	C₂²×C₆₁	1	4	yes	—
408	#484	C₂²×C₆₁	1	2	yes	—
409	#483	C₂²×C₆₁	1	2	yes	—
410	#483	C₂²×C₆₁	1	4	yes	—
411	#479	C₂²×C₆₁	1	4	yes	—
412	#477	C₂²×C₆₁	1	2	yes	—
413	#477	C₂²×C₆₁	1	2	yes	—
414	#475	C₂²×C₆₁	1	2	yes	—
415	#475	C₂²×C₆₁	1	2	yes	—
416	#470	C₂²×C₆₁	1	4	yes	—
417	#470	C₂²×C₆₁	1	4	yes	—
418	#467	C₂²×C₆₁	1	4	yes	—
419	#467	C₂²×C₆₁	1	4	yes	—
420	#462	C₂²×C₆₁	1	2	yes	—
421	#462	C₂²×C₆₁	1	2	yes	—
422	#451	C₂²×C₆₁	1	4	yes	—
423	#451	C₂²×C₆₁	1	2	yes	—
424	#451	C₂²×C₆₁	1	2	yes	—
425	#451	C₂²×C₆₁	1	4	yes	—
426	#441	C₂²×C₆₁	1	4	yes	—
427	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
428	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
429	#436	C₂²×C₆₁	1	2	yes	—
430	#435	C₂²×C₆₁	1	4	yes	—
431	C₈×D₄	C₂²×C₆₁	1	2	yes	—
432	#432	C₂²×C₆₁	1	1	yes	characteristic
433	#431	C₂²×C₆₁	1	1	yes	characteristic
434	#428	C₂²×C₆₁	1	4	yes	—
435	#427	C₂²×C₆₁	1	4	yes	—
436	#419	C₂²×C₆₁	1	4	yes	—
437	C₂×#103	C₂²×C₆₁	1	4	yes	—
438	#408	C₂²×C₆₁	1	4	yes	—
439	C₆₁	#61806	1	1	yes	characteristic, direct product factor, Sylow
440	C₂×#41	C₂³×C₆₁	1	2	yes	—
441	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
442	C₂²×Q₈	C₂³×C₆₁	1	4	yes	—
443	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
444	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
445	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
446	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
447	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
448	C₄⋊Q₈	—	2	2	no	—
449	C₄⋊Q₈	—	2	4	no	—
450	C₄⋊Q₈	—	2	2	no	—
451	C₄⋊Q₈	—	2	2	no	—
452	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
453	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
454	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
455	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
456	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
457	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
458	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
459	#127	D₄×C₆₁	1	1	yes	characteristic
460	#126	—	2	4	no	—
461	#126	—	2	4	no	—
462	#126	—	2	4	no	—
463	#126	—	2	4	no	—
464	#125	—	2	2	no	—
465	#125	—	2	4	no	—
466	#125	—	2	1	no	—
467	#125	—	2	1	no	—
468	#125	C₂³×C₆₁	1	2	yes	—
469	#124	—	2	2	no	—
470	#124	—	2	4	no	—
471	#124	—	2	2	no	—
472	#124	—	2	2	no	—
473	#124	—	2	2	no	—
474	#124	D₄×C₆₁	1	2	yes	—
475	#123	—	2	4	no	—
476	#123	—	2	4	no	—
477	#123	—	2	4	no	—
478	#123	—	2	4	no	—
479	C₂²⋊Q₈	—	2	1	no	—
480	C₂²⋊Q₈	—	2	1	no	—
481	C₂²⋊Q₈	—	2	1	no	—
482	C₂²⋊Q₈	—	2	1	no	—
483	C₂²⋊Q₈	—	2	4	no	—
484	C₂²⋊Q₈	—	2	4	no	—
485	C₂²⋊Q₈	—	2	4	no	—
486	C₂²⋊Q₈	C₂³×C₆₁	1	2	yes	—
487	C₂²⋊Q₈	C₂³×C₆₁	1	2	yes	—
488	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
489	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
490	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
491	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
492	#121	D₄×C₆₁	1	2	yes	—
493	#121	D₄×C₆₁	1	2	yes	—
494	C₄×Q₈	—	2	2	no	—
495	C₄×Q₈	—	2	2	no	—
496	C₄×Q₈	—	2	2	no	—
497	C₄×Q₈	—	2	2	no	—
498	C₄×Q₈	—	2	1	no	—
499	C₄×Q₈	—	2	1	no	—
500	C₄×Q₈	—	2	1	no	—
501	C₄×Q₈	—	2	1	no	—
502	C₄×Q₈	—	2	1	no	—
503	C₄×Q₈	—	2	1	no	—
504	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
505	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
506	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
507	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
508	C₄×D₄	—	2	4	no	—
509	C₄×D₄	—	2	2	no	—
510	C₄×D₄	—	2	2	no	—
511	C₄×D₄	—	2	2	no	—
512	C₄×D₄	—	2	2	no	—
513	C₄×D₄	C₂³×C₆₁	1	2	yes	—
514	C₄×D₄	C₂³×C₆₁	1	2	yes	—
515	#117	—	2	1	no	—
516	#117	—	2	1	no	—
517	#117	—	2	4	no	—
518	#117	C₂³×C₆₁	1	2	yes	—
519	#117	C₂³×C₆₁	1	4	yes	—
520	#117	C₂³×C₆₁	1	4	yes	—
521	C₂×C₄⋊C₄	—	2	1	no	—
522	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
523	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
524	C₂×C₄²	—	2	1	no	—
525	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
526	#107	C₂³×C₆₁	1	2	yes	—
527	#107	C₂³×C₆₁	1	4	yes	—
528	#107	C₂³×C₆₁	1	2	yes	—
529	#107	C₂³×C₆₁	1	4	yes	—
530	#106	C₂³×C₆₁	1	4	yes	—
531	#106	C₂³×C₆₁	1	4	yes	—
532	#106	C₂³×C₆₁	1	4	yes	—
533	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
534	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
535	#103	C₂³×C₆₁	1	2	yes	—
536	#103	C₂³×C₆₁	1	4	yes	—
537	#103	C₂³×C₆₁	1	4	yes	—
538	#103	C₂³×C₆₁	1	4	yes	—
539	#103	C₂³×C₆₁	1	2	yes	—
540	#103	C₂³×C₆₁	1	4	yes	—
541	#103	C₂³×C₆₁	1	4	yes	—
542	#103	C₂³×C₆₁	1	2	yes	—
543	#103	C₂³×C₆₁	1	2	yes	—
544	#103	C₂³×C₆₁	1	4	yes	—
545	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
546	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
547	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
548	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
549	#97	C₂³×C₆₁	1	2	yes	—
550	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
551	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
552	#41	—	4	4	no	—
553	C₂×Q₈	—	2	1	no	—
554	C₂×Q₈	—	2	1	no	—
555	C₂×Q₈	—	2	1	no	—
556	C₂×Q₈	—	2	1	no	—
557	C₂×Q₈	—	2	1	no	—
558	C₂×Q₈	—	2	4	no	—
559	C₂×Q₈	—	4	2	no	—
560	C₂×Q₈	—	4	4	no	—
561	C₂×Q₈	—	4	4	no	—
562	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
563	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
564	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
565	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
566	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
567	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
568	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
569	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
570	C₂²×C₄	—	2	1	no	—
571	C₂²×C₄	—	2	1	no	—
572	C₂²×C₄	—	2	4	no	—
573	C₂²×C₄	—	2	4	no	—
574	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
575	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
576	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
577	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
578	Q₁₆	—	4	2	no	—
579	Q₁₆	—	4	4	no	—
580	Q₁₆	—	4	2	no	—
581	M₁₆	—	4	4	no	—
582	C₂×C₈	—	4	4	no	—
583	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
584	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
585	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	4	no	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	4	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	2	1	no	—
597	C₄⋊C₄	—	2	1	no	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	4	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	1	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	4	no	—
606	C₄⋊C₄	—	2	2	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
609	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
610	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
611	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
612	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
613	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
614	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	2	no	—
617	C₂²⋊C₄	—	2	4	no	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	4	no	—
620	C₂²⋊C₄	—	2	4	no	—
621	C₂²⋊C₄	—	2	4	no	—
622	C₂²⋊C₄	—	2	4	no	—
623	C₂²⋊C₄	—	2	4	no	—
624	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
625	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
626	C₄²	—	2	2	no	—
627	C₄²	—	2	2	no	—
628	C₄²	—	2	4	no	—
629	C₄²	—	2	2	no	—
630	C₄²	—	2	2	no	—
631	C₄²	—	2	2	no	—
632	C₄²	—	2	4	no	—
633	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
634	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
635	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
636	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
637	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
638	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
639	Q₈	—	2	2	no	—
640	Q₈	—	2	2	no	—
641	Q₈	—	2	2	no	—
642	Q₈	—	2	4	no	—
643	Q₈	—	2	2	no	—
644	Q₈	—	4	2	no	—
645	Q₈	—	4	2	no	—
646	Q₈	—	4	2	no	—
647	Q₈	—	4	2	no	—
648	Q₈	—	4	2	no	—
649	Q₈	—	8	1	no	—
650	Q₈	—	8	4	no	—
651	D₄	—	2	4	no	—
652	D₄	—	4	2	no	—
653	D₄	—	4	2	no	—
654	D₄	—	4	4	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	4	no	—
657	C₂×C₄	—	2	4	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	2	no	—
665	C₂×C₄	—	2	2	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	2	no	—
669	C₂×C₄	—	8	1	no	—
670	C₂×C₄	—	8	4	no	—
671	C₂×C₄	—	8	4	no	—
672	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
673	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
674	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
675	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
676	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
677	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
678	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
679	C₈	—	2	2	no	—
680	C₈	—	2	4	no	—
681	C₈	—	2	2	no	—
682	C₂²	—	2	1	no	—
683	C₂²	—	2	4	no	—
684	C₂²	—	2	1	no	—
685	C₂²	—	2	4	no	—
686	C₂²	—	2	1	no	—
687	C₂²	—	2	4	no	—
688	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
689	C₄	—	2	1	no	—
690	C₄	—	2	1	no	—
691	C₄	—	2	1	no	—
692	C₄	—	2	1	no	—
693	C₄	—	2	4	no	—
694	C₄	—	4	2	no	—
695	C₄	—	4	2	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	2	no	—
698	C₄	—	4	2	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	4	2	no	—
702	C₄	C₆₁×#578	1	2	yes	—
703	C₄	C₆₁×#585	1	2	yes	—
704	C₂	—	4	1	no	—
705	C₂	—	4	4	no	—
706	C₂	C₆₁×#3485	1	1	yes	characteristic
707	C₂	C₂×C₆₁×#578	1	1	yes	characteristic
708	C₂	C₆₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₆₁×#532 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#3356	C₂	semidirect
4. C₆₁×#3349	C₂	semidirect
5. C₆₁×#3342	C₂	semidirect
6. C₆₁×#3328	C₂	semidirect
7. C₆₁×#3312	C₂	semidirect
8. C₆₁×#3299	C₂	semidirect
9. C₆₁×#3237	C₂	semidirect
10. C₆₁×#3231	C₂	semidirect
11. C₆₁×#3202	C₂	semidirect
12. C₆₁×#3113	C₂	semidirect
13. C₆₁×#3104	C₂	semidirect
14. C₆₁×#3097	C₂	semidirect
327. #61806	C₆₁	direct (N × H)
352. #3356	C₂×C₆₁	semidirect
353. #3349	C₂×C₆₁	semidirect
354. #3342	C₂×C₆₁	semidirect
355. #3328	C₂×C₆₁	semidirect
356. #3312	C₂×C₆₁	semidirect
357. #3299	C₂×C₆₁	semidirect
358. #3237	C₂×C₆₁	semidirect
359. #3231	C₂×C₆₁	semidirect
360. #3202	C₂×C₆₁	semidirect
361. #3113	C₂×C₆₁	semidirect
362. #3104	C₂×C₆₁	semidirect
363. #3097	C₂×C₆₁	semidirect
439. C₆₁	#61806	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×#3550	2
C₆₁×#3507	2
C₆₁×#3356	2
C₆₁×#3349	2
C₆₁×#3342	2
C₆₁×#3328	2
C₆₁×#3312	2
C₆₁×#3299	2
C₆₁×#3237	2
C₆₁×#3231	2
C₆₁×#3202	2
C₆₁×#3113	2
C₆₁×#3104	2
C₆₁×#3097	2
C₆₁×#2981	2
#61806	61

Subgroups — C61×#61806 #29565239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#61806 #29565239