Subgroups — C₆₁×#23036 #29526469

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#428	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2362	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2360	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1928	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1927	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1873	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1872	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1865	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₁×#1863	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1776	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#1753	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#470	C₄	1	8	yes	complemented
25	C₄×Q₁₆×C₆₁	—	2	4	no	—
26	C₆₁×#428	—	2	4	no	—
27	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₆₁×#391	C₂²	1	1	yes	characteristic
42	C₆₁×#391	C₂²	1	1	yes	characteristic
43	C₆₁×#389	—	2	1	no	—
44	C₆₁×#389	C₂²	1	1	yes	characteristic
45	C₆₁×#389	C₂²	1	1	yes	characteristic
46	C₆₁×#388	—	2	1	no	—
47	C₆₁×#388	—	2	1	no	—
48	C₆₁×#387	—	2	1	no	—
49	C₆₁×#387	—	2	1	no	—
50	C₆₁×#385	—	2	1	no	—
51	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
52	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
53	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
54	C₄×C₂²⋊C₄×C₆₁	—	2	1	no	—
55	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
56	C₆₁×#340	C₂²	1	1	yes	characteristic
57	C₆₁×#340	C₂²	1	1	yes	characteristic
58	C₆₁×#340	C₂²	1	1	yes	characteristic
59	C₆₁×#340	C₂²	1	1	yes	characteristic
60	C₆₁×#336	C₂²	1	1	yes	characteristic
61	C₆₁×#336	C₂²	1	1	yes	characteristic
62	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₆₁	—	2	1	no	—
65	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₆₁	—	2	2	no	—
67	C₂×Q₁₆×C₆₁	—	2	2	no	—
68	C₂×Q₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
69	C₂×M₁₆×C₆₁	—	2	2	no	—
70	C₂×M₁₆×C₆₁	—	2	2	no	—
71	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
72	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
76	C₂²⋊Q₈×C₆₁	—	2	2	no	—
77	C₂²⋊Q₈×C₆₁	—	2	2	no	—
78	C₂²⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
79	C₂²⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
80	C₄×Q₈×C₆₁	—	2	2	no	—
81	C₄×Q₈×C₆₁	—	2	2	no	—
82	C₄×Q₈×C₆₁	—	4	1	no	—
83	C₄×Q₈×C₆₁	—	4	1	no	—
84	#117×C₆₁	—	2	2	no	—
85	#117×C₆₁	—	4	1	no	—
86	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
89	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
95	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₆₁	—	2	1	no	—
107	C₂×C₄²×C₆₁	—	2	1	no	—
108	C₂×C₄²×C₆₁	—	2	1	no	—
109	C₂×C₄²×C₆₁	—	2	1	no	—
110	C₂×C₄²×C₆₁	—	2	1	no	—
111	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
113	#107×C₆₁	—	2	2	no	—
114	#107×C₆₁	D₄	1	4	yes	—
115	#106×C₆₁	—	2	2	no	—
116	#106×C₆₁	—	2	2	no	—
117	#106×C₆₁	C₂×C₄	1	4	yes	complemented
118	#103×C₆₁	—	2	2	no	—
119	#103×C₆₁	—	2	2	no	—
120	#103×C₆₁	—	4	1	no	—
121	#103×C₆₁	—	4	1	no	—
122	#103×C₆₁	C₂×C₄	1	4	yes	complemented
123	#103×C₆₁	C₂×C₄	1	4	yes	complemented
124	C₂²⋊C₈×C₆₁	—	2	2	no	—
125	#97×C₆₁	—	2	2	no	—
126	C₄×C₈×C₆₁	—	2	2	no	—
127	#95×C₆₁	—	2	1	no	—
128	#95×C₆₁	—	2	1	no	—
129	#95×C₆₁	—	2	1	no	—
130	#95×C₆₁	—	2	1	no	—
131	#95×C₆₁	—	2	1	no	—
132	#95×C₆₁	—	2	1	no	—
133	#95×C₆₁	—	2	1	no	—
134	#95×C₆₁	—	2	1	no	—
135	#95×C₆₁	D₄	1	1	yes	characteristic
136	#95×C₆₁	D₄	1	1	yes	characteristic
137	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
138	C₂×Q₈×C₆₁	—	2	1	no	—
139	C₂×Q₈×C₆₁	—	2	1	no	—
140	C₂×Q₈×C₆₁	—	2	1	no	—
141	C₂×Q₈×C₆₁	—	2	1	no	—
142	C₂×Q₈×C₆₁	—	2	1	no	—
143	C₂×Q₈×C₆₁	—	2	1	no	—
144	C₂×Q₈×C₆₁	—	2	1	no	—
145	C₂×Q₈×C₆₁	—	2	1	no	—
146	C₂×Q₈×C₆₁	—	2	1	no	—
147	C₂×Q₈×C₆₁	—	2	1	no	—
148	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
149	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₆₁	—	2	1	no	—
151	C₂²×C₄×C₆₁	—	2	1	no	—
152	C₂²×C₄×C₆₁	—	2	1	no	—
153	C₂²×C₄×C₆₁	—	2	1	no	—
154	C₂²×C₄×C₆₁	—	2	1	no	—
155	C₂²×C₄×C₆₁	—	2	1	no	—
156	C₂²×C₄×C₆₁	—	2	1	no	—
157	C₂²×C₄×C₆₁	—	2	1	no	—
158	C₂²×C₄×C₆₁	—	2	1	no	—
159	C₂²×C₄×C₆₁	—	2	1	no	—
160	C₂²×C₄×C₆₁	—	2	1	no	—
161	C₂²×C₄×C₆₁	—	2	1	no	—
162	C₂²×C₄×C₆₁	—	4	2	no	—
163	C₂²×C₄×C₆₁	—	4	2	no	—
164	C₂²×C₄×C₆₁	—	8	1	no	—
165	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
166	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₁	#41	1	2	yes	—
172	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
173	Q₁₆×C₆₁	—	2	8	no	—
174	M₁₆×C₆₁	—	2	8	no	—
175	C₂×C₈×C₆₁	—	2	1	no	—
176	C₂×C₈×C₆₁	—	2	1	no	—
177	C₂×C₈×C₆₁	—	2	1	no	—
178	C₂×C₈×C₆₁	—	2	1	no	—
179	C₂×C₈×C₆₁	—	2	1	no	—
180	C₂×C₈×C₆₁	—	2	1	no	—
181	C₂×C₈×C₆₁	—	2	1	no	—
182	C₂×C₈×C₆₁	—	2	1	no	—
183	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
184	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
185	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
186	C₂×C₈×C₆₁	#41	1	2	yes	—
187	C₄⋊C₄×C₆₁	—	2	4	no	—
188	C₄⋊C₄×C₆₁	—	2	4	no	—
189	C₄⋊C₄×C₆₁	—	2	1	no	—
190	C₄⋊C₄×C₆₁	—	2	1	no	—
191	C₄⋊C₄×C₆₁	—	2	1	no	—
192	C₄⋊C₄×C₆₁	—	2	1	no	—
193	C₄⋊C₄×C₆₁	—	2	1	no	—
194	C₄⋊C₄×C₆₁	—	2	1	no	—
195	C₄⋊C₄×C₆₁	—	2	1	no	—
196	C₄⋊C₄×C₆₁	—	2	1	no	—
197	C₄⋊C₄×C₆₁	—	4	2	no	—
198	C₄⋊C₄×C₆₁	—	4	2	no	—
199	C₄⋊C₄×C₆₁	—	4	2	no	—
200	C₄⋊C₄×C₆₁	—	4	2	no	—
201	C₄⋊C₄×C₆₁	—	8	1	no	—
202	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
203	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
204	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
205	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
206	C₂²⋊C₄×C₆₁	—	2	4	no	—
207	C₂²⋊C₄×C₆₁	—	2	4	no	—
208	C₂²⋊C₄×C₆₁	—	2	2	no	—
209	C₂²⋊C₄×C₆₁	—	4	1	no	—
210	C₂²⋊C₄×C₆₁	—	4	2	no	—
211	C₄²×C₆₁	—	2	1	no	—
212	C₄²×C₆₁	—	2	1	no	—
213	C₄²×C₆₁	—	2	1	no	—
214	C₄²×C₆₁	—	2	1	no	—
215	C₄²×C₆₁	—	4	2	no	—
216	C₄²×C₆₁	—	4	2	no	—
217	C₄²×C₆₁	—	4	2	no	—
218	C₄²×C₆₁	—	4	2	no	—
219	C₄²×C₆₁	—	8	1	no	—
220	C₂³×C₆₁	—	2	1	no	—
221	C₂³×C₆₁	—	2	1	no	—
222	C₂³×C₆₁	—	2	1	no	—
223	C₂³×C₆₁	—	2	1	no	—
224	C₂³×C₆₁	—	2	1	no	—
225	C₂³×C₆₁	—	2	1	no	—
226	C₂³×C₆₁	C₄×D₄	1	2	yes	—
227	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₆₁	—	2	2	no	—
229	Q₈×C₆₁	—	2	2	no	—
230	Q₈×C₆₁	—	2	2	no	—
231	Q₈×C₆₁	—	2	2	no	—
232	Q₈×C₆₁	—	4	1	no	—
233	Q₈×C₆₁	—	4	1	no	—
234	Q₈×C₆₁	—	4	1	no	—
235	Q₈×C₆₁	—	4	1	no	—
236	C₂×C₄×C₆₁	—	2	1	no	—
237	C₂×C₄×C₆₁	—	2	1	no	—
238	C₂×C₄×C₆₁	—	2	1	no	—
239	C₂×C₄×C₆₁	—	2	1	no	—
240	C₂×C₄×C₆₁	—	2	2	no	—
241	C₂×C₄×C₆₁	—	2	2	no	—
242	C₂×C₄×C₆₁	—	2	2	no	—
243	C₂×C₄×C₆₁	—	2	2	no	—
244	C₂×C₄×C₆₁	—	2	2	no	—
245	C₂×C₄×C₆₁	—	2	2	no	—
246	C₂×C₄×C₆₁	—	2	2	no	—
247	C₂×C₄×C₆₁	—	2	2	no	—
248	C₂×C₄×C₆₁	—	2	1	no	—
249	C₂×C₄×C₆₁	—	2	2	no	—
250	C₂×C₄×C₆₁	—	2	2	no	—
251	C₂×C₄×C₆₁	—	2	1	no	—
252	C₂×C₄×C₆₁	—	2	1	no	—
253	C₂×C₄×C₆₁	—	2	1	no	—
254	C₂×C₄×C₆₁	—	2	1	no	—
255	C₂×C₄×C₆₁	—	2	1	no	—
256	C₂×C₄×C₆₁	—	2	1	no	—
257	C₂×C₄×C₆₁	—	2	1	no	—
258	C₂×C₄×C₆₁	—	2	1	no	—
259	C₂×C₄×C₆₁	—	4	1	no	—
260	C₂×C₄×C₆₁	—	4	1	no	—
261	C₂×C₄×C₆₁	—	4	1	no	—
262	C₂×C₄×C₆₁	—	4	1	no	—
263	C₂×C₄×C₆₁	—	4	1	no	—
264	C₂×C₄×C₆₁	—	4	1	no	—
265	C₂×C₄×C₆₁	—	4	1	no	—
266	C₂×C₄×C₆₁	—	4	1	no	—
267	C₂×C₄×C₆₁	—	4	1	no	—
268	C₂×C₄×C₆₁	—	4	1	no	—
269	C₂×C₄×C₆₁	—	4	1	no	—
270	C₂×C₄×C₆₁	—	4	1	no	—
271	C₂×C₄×C₆₁	—	4	1	no	—
272	C₂×C₄×C₆₁	—	4	1	no	—
273	C₂×C₄×C₆₁	—	4	1	no	—
274	C₂×C₄×C₆₁	—	4	1	no	—
275	C₂×C₄×C₆₁	—	4	1	no	—
276	C₂×C₄×C₆₁	—	4	1	no	—
277	C₂×C₄×C₆₁	—	8	4	no	—
278	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
279	C₂×C₄×C₆₁	#137	1	2	yes	—
280	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
281	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
285	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
286	C₈×C₆₁	—	2	2	no	—
287	C₈×C₆₁	—	4	1	no	—
288	C₈×C₆₁	—	4	1	no	—
289	C₈×C₆₁	C₄×D₄	1	4	yes	—
290	#23036	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂²×C₆₁	—	2	2	no	—
292	C₂²×C₆₁	—	2	2	no	—
293	C₂²×C₆₁	—	2	2	no	—
294	C₂²×C₆₁	—	4	1	no	—
295	C₂²×C₆₁	—	4	1	no	—
296	C₂²×C₆₁	—	4	1	no	—
297	C₂²×C₆₁	—	4	1	no	—
298	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
299	C₂²×C₆₁	D₄²	1	1	yes	characteristic
300	C₂²×C₆₁	#547	1	1	yes	characteristic
301	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
302	C₂²×C₆₁	#575	1	1	yes	characteristic
303	C₂²×C₆₁	#577	1	1	yes	characteristic
304	C₂²×C₆₁	#578	1	1	yes	characteristic
305	C₄×C₆₁	—	2	2	no	—
306	C₄×C₆₁	—	2	2	no	—
307	C₄×C₆₁	—	4	1	no	—
308	C₄×C₆₁	—	4	1	no	—
309	C₄×C₆₁	—	4	2	no	—
310	C₄×C₆₁	—	4	2	no	—
311	C₄×C₆₁	—	4	2	no	—
312	C₄×C₆₁	—	4	2	no	—
313	C₄×C₆₁	—	4	1	no	—
314	C₄×C₆₁	—	4	2	no	—
315	C₄×C₆₁	—	4	2	no	—
316	C₄×C₆₁	—	8	1	no	—
317	C₄×C₆₁	—	8	1	no	—
318	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
319	C₄×C₆₁	#520	1	1	yes	characteristic
320	C₄×C₆₁	#548	1	1	yes	characteristic
321	C₄×C₆₁	#553	1	1	yes	characteristic
322	C₂×#470	C₂×C₆₁	1	1	yes	characteristic
323	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
324	C₂×#428	C₂×C₆₁	1	1	yes	characteristic
325	#2362	C₂×C₆₁	1	1	yes	characteristic
326	#2360	C₂×C₆₁	1	1	yes	characteristic
327	#1971	C₂×C₆₁	1	1	yes	characteristic, complemented
328	#1941	C₂×C₆₁	1	1	yes	characteristic, complemented
329	#1928	C₂×C₆₁	1	1	yes	characteristic, complemented
330	#1927	C₂×C₆₁	1	1	yes	characteristic, complemented
331	#1873	C₂×C₆₁	1	1	yes	characteristic
332	#1872	C₂×C₆₁	1	1	yes	characteristic
333	#1865	C₂×C₆₁	1	1	yes	characteristic, complemented
334	#1863	C₂×C₆₁	1	1	yes	characteristic, complemented
335	#1776	C₂×C₆₁	1	1	yes	characteristic, complemented
336	#1753	C₂×C₆₁	1	1	yes	characteristic
337	C₂×C₆₁	—	4	2	no	—
338	C₂×C₆₁	#2361	1	1	yes	characteristic
339	C₂×C₆₁	C₄×#137	1	1	yes	characteristic
340	C₂×C₆₁	#2947	1	1	yes	characteristic
341	C₂×C₆₁	#3267	1	1	yes	characteristic
342	C₂×C₆₁	#3270	1	1	yes	characteristic
343	C₂×C₆₁	#3352	1	1	yes	characteristic
344	C₂×C₆₁	#3356	1	1	yes	characteristic
345	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
346	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
347	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
348	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
349	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
350	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
351	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
352	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
353	#470	C₄×C₆₁	1	8	yes	complemented
354	C₄×Q₁₆	—	2	4	no	—
355	#428	—	2	4	no	—
356	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
357	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
358	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
359	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
360	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
361	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
362	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
363	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
364	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
365	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
366	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
367	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
368	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
369	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
370	#391	C₂²×C₆₁	1	1	yes	characteristic
371	#391	C₂²×C₆₁	1	1	yes	characteristic
372	#389	—	2	1	no	—
373	#389	C₂²×C₆₁	1	1	yes	characteristic
374	#389	C₂²×C₆₁	1	1	yes	characteristic
375	#388	—	2	1	no	—
376	#388	—	2	1	no	—
377	#387	—	2	1	no	—
378	#387	—	2	1	no	—
379	#385	—	2	1	no	—
380	C₄×C₄⋊C₄	—	2	1	no	—
381	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
382	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
383	C₄×C₂²⋊C₄	—	2	1	no	—
384	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
385	#340	C₂²×C₆₁	1	1	yes	characteristic
386	#340	C₂²×C₆₁	1	1	yes	characteristic
387	#340	C₂²×C₆₁	1	1	yes	characteristic
388	#340	C₂²×C₆₁	1	1	yes	characteristic
389	#336	C₂²×C₆₁	1	1	yes	characteristic
390	#336	C₂²×C₆₁	1	1	yes	characteristic
391	C₆₁	#23036	1	1	yes	characteristic, direct product factor, Sylow
392	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
393	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
394	C₂³×C₄	—	2	1	no	—
395	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
396	C₂×Q₁₆	—	2	2	no	—
397	C₂×Q₁₆	—	2	2	no	—
398	C₂×Q₁₆	C₂×C₄×C₆₁	1	4	yes	complemented
399	C₂×M₁₆	—	2	2	no	—
400	C₂×M₁₆	—	2	2	no	—
401	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
402	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
403	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
404	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
405	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	C₂×C₄×C₆₁	1	4	yes	—
409	C₂²⋊Q₈	C₂×C₄×C₆₁	1	4	yes	—
410	C₄×Q₈	—	2	2	no	—
411	C₄×Q₈	—	2	2	no	—
412	C₄×Q₈	—	4	1	no	—
413	C₄×Q₈	—	4	1	no	—
414	#117	—	2	2	no	—
415	#117	—	4	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	2	2	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	4	1	no	—
425	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	—	2	1	no	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
436	C₂×C₄²	—	2	1	no	—
437	C₂×C₄²	—	2	1	no	—
438	C₂×C₄²	—	2	1	no	—
439	C₂×C₄²	—	2	1	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
442	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
443	#107	—	2	2	no	—
444	#107	D₄×C₆₁	1	4	yes	—
445	#106	—	2	2	no	—
446	#106	—	2	2	no	—
447	#106	C₂×C₄×C₆₁	1	4	yes	complemented
448	#103	—	2	2	no	—
449	#103	—	2	2	no	—
450	#103	—	4	1	no	—
451	#103	—	4	1	no	—
452	#103	C₂×C₄×C₆₁	1	4	yes	complemented
453	#103	C₂×C₄×C₆₁	1	4	yes	complemented
454	C₂²⋊C₈	—	2	2	no	—
455	#97	—	2	2	no	—
456	C₄×C₈	—	2	2	no	—
457	#95	—	2	1	no	—
458	#95	—	2	1	no	—
459	#95	—	2	1	no	—
460	#95	—	2	1	no	—
461	#95	—	2	1	no	—
462	#95	—	2	1	no	—
463	#95	—	2	1	no	—
464	#95	—	2	1	no	—
465	#95	D₄×C₆₁	1	1	yes	characteristic
466	#95	D₄×C₆₁	1	1	yes	characteristic
467	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
479	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	4	2	no	—
493	C₂²×C₄	—	4	2	no	—
494	C₂²×C₄	—	8	1	no	—
495	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
496	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
497	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
501	C₂²×C₄	#41×C₆₁	1	2	yes	—
502	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
503	Q₁₆	—	2	8	no	—
504	M₁₆	—	2	8	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
514	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
515	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
516	C₂×C₈	#41×C₆₁	1	2	yes	—
517	C₄⋊C₄	—	2	4	no	—
518	C₄⋊C₄	—	2	4	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	4	2	no	—
528	C₄⋊C₄	—	4	2	no	—
529	C₄⋊C₄	—	4	2	no	—
530	C₄⋊C₄	—	4	2	no	—
531	C₄⋊C₄	—	8	1	no	—
532	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
533	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
534	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
535	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	4	1	no	—
540	C₂²⋊C₄	—	4	2	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	4	2	no	—
546	C₄²	—	4	2	no	—
547	C₄²	—	4	2	no	—
548	C₄²	—	4	2	no	—
549	C₄²	—	8	1	no	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	1	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	2	1	no	—
556	C₂³	C₄×D₄×C₆₁	1	2	yes	—
557	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
558	Q₈	—	2	2	no	—
559	Q₈	—	2	2	no	—
560	Q₈	—	2	2	no	—
561	Q₈	—	2	2	no	—
562	Q₈	—	4	1	no	—
563	Q₈	—	4	1	no	—
564	Q₈	—	4	1	no	—
565	Q₈	—	4	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	8	4	no	—
608	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
609	C₂×C₄	#137×C₆₁	1	2	yes	—
610	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
611	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
612	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
613	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
614	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
615	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
616	C₈	—	2	2	no	—
617	C₈	—	4	1	no	—
618	C₈	—	4	1	no	—
619	C₈	C₄×D₄×C₆₁	1	4	yes	—
620	C₂²	—	2	2	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	4	1	no	—
624	C₂²	—	4	1	no	—
625	C₂²	—	4	1	no	—
626	C₂²	—	4	1	no	—
627	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
628	C₂²	D₄²×C₆₁	1	1	yes	characteristic
629	C₂²	C₆₁×#547	1	1	yes	characteristic
630	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
631	C₂²	C₆₁×#575	1	1	yes	characteristic
632	C₂²	C₆₁×#577	1	1	yes	characteristic
633	C₂²	C₆₁×#578	1	1	yes	characteristic
634	C₄	—	2	2	no	—
635	C₄	—	2	2	no	—
636	C₄	—	4	1	no	—
637	C₄	—	4	1	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	2	no	—
641	C₄	—	4	2	no	—
642	C₄	—	4	1	no	—
643	C₄	—	4	2	no	—
644	C₄	—	4	2	no	—
645	C₄	—	8	1	no	—
646	C₄	—	8	1	no	—
647	C₄	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
648	C₄	C₆₁×#520	1	1	yes	characteristic
649	C₄	C₆₁×#548	1	1	yes	characteristic
650	C₄	C₆₁×#553	1	1	yes	characteristic
651	C₂	—	4	2	no	—
652	C₂	C₆₁×#2361	1	1	yes	characteristic
653	C₂	C₄×#137×C₆₁	1	1	yes	characteristic
654	C₂	C₆₁×#2947	1	1	yes	characteristic
655	C₂	C₆₁×#3267	1	1	yes	characteristic
656	C₂	C₆₁×#3270	1	1	yes	characteristic
657	C₂	C₆₁×#3352	1	1	yes	characteristic
658	C₂	C₆₁×#3356	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄×C₂²⋊C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₆₁	C₂	semidirect
6. C₆₁×#1971	C₂	semidirect
7. C₆₁×#1941	C₂	semidirect
8. C₆₁×#1928	C₂	semidirect
9. C₆₁×#1927	C₂	semidirect
12. C₆₁×#1865	C₂	semidirect
13. C₆₁×#1863	C₂	semidirect
14. C₆₁×#1776	C₂	semidirect
24. C₆₁×#470	C₄	semidirect
68. C₂×Q₁₆×C₆₁	C₂×C₄	semidirect
117. #106×C₆₁	C₂×C₄	semidirect
122. #103×C₆₁	C₂×C₄	semidirect
123. #103×C₆₁	C₂×C₄	semidirect
290. #23036	C₆₁	direct (N × H)
323. C₂×C₄×Q₁₆	C₂×C₆₁	semidirect
327. #1971	C₂×C₆₁	semidirect
328. #1941	C₂×C₆₁	semidirect
329. #1928	C₂×C₆₁	semidirect
330. #1927	C₂×C₆₁	semidirect
333. #1865	C₂×C₆₁	semidirect
334. #1863	C₂×C₆₁	semidirect
335. #1776	C₂×C₆₁	semidirect
353. #470	C₄×C₆₁	semidirect
391. C₆₁	#23036	direct (N × H)
398. C₂×Q₁₆	C₂×C₄×C₆₁	semidirect
447. #106	C₂×C₄×C₆₁	semidirect
452. #103	C₂×C₄×C₆₁	semidirect
453. #103	C₂×C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#470	2
C₂×C₄×Q₁₆×C₆₁	2
C₂×C₆₁×#428	2
C₆₁×#2362	2
C₆₁×#2360	2
C₆₁×#1971	2
C₆₁×#1941	2
C₆₁×#1928	2
C₆₁×#1927	2
C₆₁×#1873	2
C₆₁×#1872	2
C₆₁×#1865	2
C₆₁×#1863	2
C₆₁×#1776	2
C₆₁×#1753	2
#23036	61

Subgroups — C61×#23036 #29526469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23036 #29526469