Subgroups — C₆₁×#25592 #29529025

Order 15616.

1134 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3140	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3040	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2858	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2005	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2004	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2003	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2002	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1967	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1888	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#1887	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₁×#1886	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1885	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#104×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#559	—	2	2	no	—
19	C₆₁×#556	—	2	2	no	—
20	C₆₁×#551	—	2	4	no	—
21	C₆₁×#542	—	2	2	no	—
22	C₆₁×#536	C₂²	1	1	yes	characteristic
23	C₆₁×#535	C₂²	1	1	yes	characteristic
24	C₆₁×#533	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#529	—	2	2	no	—
28	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#519	C₂²	1	1	yes	characteristic
33	C₆₁×#518	C₂²	1	1	yes	characteristic
34	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#489	—	2	2	no	—
36	C₆₁×#488	—	2	2	no	—
37	C₆₁×#464	—	2	2	no	—
38	C₆₁×#463	—	2	2	no	—
39	C₆₁×#423	C₂²	1	1	yes	characteristic
40	C₂×#104×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#104×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#419	C₂²	1	1	yes	characteristic
43	C₆₁×#419	C₂²	1	1	yes	characteristic
44	C₆₁×#419	C₂²	1	1	yes	characteristic
45	C₆₁×#418	C₂²	1	1	yes	characteristic
46	C₆₁×#418	C₂²	1	1	yes	characteristic
47	C₆₁×#418	C₂²	1	1	yes	characteristic
48	C₂×#101×C₆₁	C₂²	1	1	yes	characteristic
49	C₂×#101×C₆₁	C₂²	1	1	yes	characteristic
50	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
51	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
52	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
53	C₆₁×#390	—	2	4	no	—
54	C₄³×C₆₁	C₂²	1	1	yes	characteristic
55	C₆₁×#339	C₂²	1	1	yes	characteristic
56	C₆₁×#339	C₂²	1	1	yes	characteristic
57	C₆₁×#339	C₂²	1	1	yes	characteristic
58	C₆₁×#339	C₂²	1	1	yes	characteristic
59	C₆₁×#339	C₂²	1	1	yes	characteristic
60	C₆₁×#339	C₂²	1	1	yes	characteristic
61	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₆₁	—	2	2	no	—
65	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
67	#137×C₆₁	—	4	1	no	—
68	#137×C₆₁	—	4	1	no	—
69	AGL(1,ℤ/8)×C₆₁	—	4	1	no	—
70	AGL(1,ℤ/8)×C₆₁	—	4	1	no	—
71	C₂×Q₁₆×C₆₁	—	2	1	no	—
72	C₂×SD₁₆×C₆₁	—	2	1	no	—
73	C₂×SD₁₆×C₆₁	—	2	1	no	—
74	C₂×D₈×C₆₁	—	2	1	no	—
75	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
77	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
79	C₄⋊Q₈×C₆₁	—	2	1	no	—
80	C₄⋊Q₈×C₆₁	—	2	2	no	—
81	C₄⋊Q₈×C₆₁	—	4	1	no	—
82	C₄⋊Q₈×C₆₁	D₄	1	2	yes	complemented
83	#127×C₆₁	—	2	1	no	—
84	#127×C₆₁	—	2	2	no	—
85	#127×C₆₁	—	4	1	no	—
86	#126×C₆₁	—	2	4	no	—
87	#126×C₆₁	—	2	4	no	—
88	#125×C₆₁	D₄	1	2	yes	complemented
89	#124×C₆₁	—	2	4	no	—
90	#124×C₆₁	—	2	1	no	—
91	#124×C₆₁	—	2	1	no	—
92	#124×C₆₁	—	2	1	no	—
93	#124×C₆₁	—	2	2	no	—
94	#124×C₆₁	—	2	1	no	—
95	#124×C₆₁	—	2	2	no	—
96	#124×C₆₁	—	2	1	no	—
97	#124×C₆₁	—	2	1	no	—
98	#124×C₆₁	—	2	2	no	—
99	#124×C₆₁	—	2	2	no	—
100	#124×C₆₁	—	4	1	no	—
101	#124×C₆₁	—	4	1	no	—
102	#123×C₆₁	—	2	4	no	—
103	#123×C₆₁	—	2	4	no	—
104	#123×C₆₁	—	2	2	no	—
105	C₂²⋊Q₈×C₆₁	—	2	4	no	—
106	C₂²⋊Q₈×C₆₁	—	2	2	no	—
107	C₂²⋊Q₈×C₆₁	—	2	2	no	—
108	#121×C₆₁	—	2	2	no	—
109	#120×C₆₁	—	2	2	no	—
110	C₄×Q₈×C₆₁	—	2	2	no	—
111	C₄×Q₈×C₆₁	—	2	2	no	—
112	C₄×Q₈×C₆₁	—	2	2	no	—
113	C₄×Q₈×C₆₁	D₄	1	2	yes	complemented
114	C₄×D₄×C₆₁	—	2	4	no	—
115	C₄×D₄×C₆₁	—	2	2	no	—
116	C₄×D₄×C₆₁	—	2	2	no	—
117	C₄×D₄×C₆₁	—	2	2	no	—
118	C₄×D₄×C₆₁	D₄	1	2	yes	complemented
119	#117×C₆₁	—	2	4	no	—
120	#117×C₆₁	—	2	2	no	—
121	#117×C₆₁	—	2	1	no	—
122	#117×C₆₁	C₂³	1	1	yes	characteristic
123	#117×C₆₁	C₂³	1	1	yes	characteristic
124	#117×C₆₁	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
126	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
127	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
128	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
129	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
130	C₂×C₂²⋊C₄×C₆₁	D₄	1	4	yes	—
131	C₂×C₄²×C₆₁	—	2	1	no	—
132	C₂×C₄²×C₆₁	—	2	2	no	—
133	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
136	C₄⋊C₈×C₆₁	—	2	2	no	—
137	#104×C₆₁	—	2	2	no	—
138	#104×C₆₁	—	2	2	no	—
139	#103×C₆₁	—	2	2	no	—
140	#103×C₆₁	—	2	2	no	—
141	#103×C₆₁	—	2	2	no	—
142	#102×C₆₁	—	2	2	no	—
143	#102×C₆₁	—	2	2	no	—
144	#102×C₆₁	—	2	2	no	—
145	#101×C₆₁	—	4	1	no	—
146	#101×C₆₁	—	4	1	no	—
147	#100×C₆₁	—	4	1	no	—
148	#100×C₆₁	—	4	1	no	—
149	#97×C₆₁	—	2	1	no	—
150	C₄×C₈×C₆₁	—	2	1	no	—
151	C₂⁴×C₆₁	—	2	1	no	—
152	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
153	#41×C₆₁	—	2	1	no	—
154	#41×C₆₁	—	2	1	no	—
155	#41×C₆₁	—	2	1	no	—
156	#41×C₆₁	—	2	1	no	—
157	C₂×Q₈×C₆₁	—	2	2	no	—
158	C₂×Q₈×C₆₁	—	2	1	no	—
159	C₂×Q₈×C₆₁	—	2	1	no	—
160	C₂×Q₈×C₆₁	—	2	1	no	—
161	C₂×Q₈×C₆₁	—	2	1	no	—
162	C₂×Q₈×C₆₁	—	2	1	no	—
163	C₂×Q₈×C₆₁	—	2	1	no	—
164	C₂×Q₈×C₆₁	—	2	1	no	—
165	C₂×Q₈×C₆₁	—	2	1	no	—
166	C₂×Q₈×C₆₁	—	2	1	no	—
167	C₂×Q₈×C₆₁	—	2	1	no	—
168	C₂×Q₈×C₆₁	—	2	1	no	—
169	C₂×Q₈×C₆₁	—	2	1	no	—
170	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
171	C₂×D₄×C₆₁	—	2	1	no	—
172	C₂×D₄×C₆₁	—	2	4	no	—
173	C₂×D₄×C₆₁	—	2	1	no	—
174	C₂×D₄×C₆₁	—	2	1	no	—
175	C₂×D₄×C₆₁	—	2	1	no	—
176	C₂×D₄×C₆₁	—	2	1	no	—
177	C₂×D₄×C₆₁	—	2	1	no	—
178	C₂×D₄×C₆₁	—	2	1	no	—
179	C₂×D₄×C₆₁	—	2	1	no	—
180	C₂×D₄×C₆₁	—	2	1	no	—
181	C₂×D₄×C₆₁	—	2	1	no	—
182	C₂×D₄×C₆₁	—	2	1	no	—
183	C₂×D₄×C₆₁	—	2	1	no	—
184	C₂×D₄×C₆₁	—	2	1	no	—
185	C₂×D₄×C₆₁	—	4	4	no	—
186	C₂×D₄×C₆₁	—	4	8	no	—
187	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₆₁	—	2	1	no	—
189	C₂²×C₄×C₆₁	—	2	1	no	—
190	C₂²×C₄×C₆₁	—	2	4	no	—
191	C₂²×C₄×C₆₁	—	2	1	no	—
192	C₂²×C₄×C₆₁	—	2	2	no	—
193	C₂²×C₄×C₆₁	—	2	1	no	—
194	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
197	Q₁₆×C₆₁	—	4	2	no	—
198	SD₁₆×C₆₁	—	4	2	no	—
199	SD₁₆×C₆₁	—	4	2	no	—
200	D₈×C₆₁	—	4	2	no	—
201	M₁₆×C₆₁	—	2	1	no	—
202	M₁₆×C₆₁	—	2	1	no	—
203	M₁₆×C₆₁	—	2	1	no	—
204	M₁₆×C₆₁	—	2	1	no	—
205	M₁₆×C₆₁	—	2	1	no	—
206	M₁₆×C₆₁	—	2	1	no	—
207	M₁₆×C₆₁	—	2	1	no	—
208	M₁₆×C₆₁	—	2	1	no	—
209	C₂×C₈×C₆₁	—	2	1	no	—
210	C₂×C₈×C₆₁	—	2	1	no	—
211	C₂×C₈×C₆₁	—	2	1	no	—
212	C₂×C₈×C₆₁	—	2	1	no	—
213	C₄⋊C₄×C₆₁	—	2	4	no	—
214	C₄⋊C₄×C₆₁	—	2	4	no	—
215	C₄⋊C₄×C₆₁	—	2	2	no	—
216	C₄⋊C₄×C₆₁	—	2	2	no	—
217	C₄⋊C₄×C₆₁	—	2	2	no	—
218	C₄⋊C₄×C₆₁	—	2	2	no	—
219	C₄⋊C₄×C₆₁	—	4	4	no	—
220	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
221	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
222	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
223	C₂²⋊C₄×C₆₁	—	2	4	no	—
224	C₂²⋊C₄×C₆₁	—	2	4	no	—
225	C₂²⋊C₄×C₆₁	—	2	4	no	—
226	C₂²⋊C₄×C₆₁	—	2	1	no	—
227	C₂²⋊C₄×C₆₁	—	2	1	no	—
228	C₂²⋊C₄×C₆₁	—	2	1	no	—
229	C₂²⋊C₄×C₆₁	—	2	4	no	—
230	C₂²⋊C₄×C₆₁	—	2	4	no	—
231	C₂²⋊C₄×C₆₁	—	2	2	no	—
232	C₂²⋊C₄×C₆₁	—	4	4	no	—
233	C₂²⋊C₄×C₆₁	—	4	4	no	—
234	C₂²⋊C₄×C₆₁	—	4	2	no	—
235	C₂²⋊C₄×C₆₁	—	4	2	no	—
236	C₄²×C₆₁	—	2	4	no	—
237	C₄²×C₆₁	—	2	2	no	—
238	C₄²×C₆₁	—	2	1	no	—
239	C₄²×C₆₁	—	2	1	no	—
240	C₄²×C₆₁	—	2	1	no	—
241	C₄²×C₆₁	—	2	1	no	—
242	C₄²×C₆₁	—	2	1	no	—
243	C₄²×C₆₁	—	4	4	no	—
244	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
245	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
246	C₄²×C₆₁	C₂×D₄	1	2	yes	—
247	C₄²×C₆₁	#41	1	2	yes	—
248	C₄²×C₆₁	#41	1	2	yes	—
249	C₂³×C₆₁	—	2	2	no	—
250	C₂³×C₆₁	—	2	1	no	—
251	C₂³×C₆₁	—	2	2	no	—
252	C₂³×C₆₁	—	2	2	no	—
253	C₂³×C₆₁	—	4	4	no	—
254	C₂³×C₆₁	—	4	4	no	—
255	C₂³×C₆₁	—	4	1	no	—
256	C₂³×C₆₁	—	4	1	no	—
257	C₂³×C₆₁	—	4	1	no	—
258	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
259	Q₈×C₆₁	—	2	1	no	—
260	Q₈×C₆₁	—	2	1	no	—
261	Q₈×C₆₁	—	4	1	no	—
262	Q₈×C₆₁	—	4	1	no	—
263	Q₈×C₆₁	—	4	1	no	—
264	Q₈×C₆₁	—	4	1	no	—
265	Q₈×C₆₁	—	4	1	no	—
266	Q₈×C₆₁	—	4	1	no	—
267	Q₈×C₆₁	—	4	1	no	—
268	Q₈×C₆₁	—	4	1	no	—
269	Q₈×C₆₁	—	8	2	no	—
270	D₄×C₆₁	—	2	1	no	—
271	D₄×C₆₁	—	2	1	no	—
272	D₄×C₆₁	—	4	1	no	—
273	D₄×C₆₁	—	4	1	no	—
274	D₄×C₆₁	—	4	1	no	—
275	D₄×C₆₁	—	4	1	no	—
276	D₄×C₆₁	—	4	1	no	—
277	D₄×C₆₁	—	4	1	no	—
278	D₄×C₆₁	—	4	1	no	—
279	D₄×C₆₁	—	4	1	no	—
280	D₄×C₆₁	—	4	1	no	—
281	D₄×C₆₁	—	4	1	no	—
282	D₄×C₆₁	—	8	8	no	—
283	D₄×C₆₁	—	8	4	no	—
284	C₂×C₄×C₆₁	—	2	4	no	—
285	C₂×C₄×C₆₁	—	2	2	no	—
286	C₂×C₄×C₆₁	—	2	2	no	—
287	C₂×C₄×C₆₁	—	2	2	no	—
288	C₂×C₄×C₆₁	—	2	2	no	—
289	C₂×C₄×C₆₁	—	2	1	no	—
290	C₂×C₄×C₆₁	—	2	2	no	—
291	C₂×C₄×C₆₁	—	2	2	no	—
292	C₂×C₄×C₆₁	—	2	2	no	—
293	C₂×C₄×C₆₁	—	2	2	no	—
294	C₂×C₄×C₆₁	—	4	2	no	—
295	C₂×C₄×C₆₁	—	4	2	no	—
296	C₂×C₄×C₆₁	—	4	1	no	—
297	C₂×C₄×C₆₁	—	4	1	no	—
298	C₂×C₄×C₆₁	—	4	1	no	—
299	C₂×C₄×C₆₁	—	4	1	no	—
300	C₂×C₄×C₆₁	—	4	1	no	—
301	C₂×C₄×C₆₁	—	8	2	no	—
302	C₂×C₄×C₆₁	—	8	4	no	—
303	C₂×C₄×C₆₁	#120	1	2	yes	—
304	C₂×C₄×C₆₁	#120	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	C₂×#41	1	1	yes	characteristic
308	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
309	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
310	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
311	C₈×C₆₁	—	4	1	no	—
312	C₈×C₆₁	—	4	1	no	—
313	C₈×C₆₁	—	4	1	no	—
314	C₈×C₆₁	—	4	1	no	—
315	#25592	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂²×C₆₁	—	2	2	no	—
317	C₂²×C₆₁	—	4	1	no	—
318	C₂²×C₆₁	—	4	2	no	—
319	C₂²×C₆₁	—	4	2	no	—
320	C₂²×C₆₁	—	4	4	no	—
321	C₂²×C₆₁	—	8	2	no	—
322	C₂²×C₆₁	—	8	2	no	—
323	C₂²×C₆₁	—	8	2	no	—
324	C₂²×C₆₁	—	8	2	no	—
325	C₂²×C₆₁	—	8	1	no	—
326	C₂²×C₆₁	—	8	2	no	—
327	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
328	C₂²×C₆₁	#546	1	1	yes	characteristic
329	C₂²×C₆₁	#546	1	1	yes	characteristic
330	C₄×C₆₁	—	2	4	no	—
331	C₄×C₆₁	—	2	2	no	—
332	C₄×C₆₁	—	2	2	no	—
333	C₄×C₆₁	—	4	2	no	—
334	C₄×C₆₁	—	4	2	no	—
335	C₄×C₆₁	—	4	1	no	—
336	C₄×C₆₁	—	4	2	no	—
337	C₄×C₆₁	—	4	2	no	—
338	C₄×C₆₁	—	4	2	no	—
339	C₄×C₆₁	—	8	1	no	—
340	C₄×C₆₁	#532	1	1	yes	characteristic
341	C₄×C₆₁	#535	1	1	yes	characteristic
342	C₄×C₆₁	#550	1	1	yes	characteristic
343	C₄×C₆₁	#550	1	1	yes	characteristic
344	#3510	C₂×C₆₁	1	1	yes	characteristic, complemented
345	#3140	C₂×C₆₁	1	1	yes	characteristic, complemented
346	#3040	C₂×C₆₁	1	1	yes	characteristic, complemented
347	#2858	C₂×C₆₁	1	1	yes	characteristic, complemented
348	#2005	C₂×C₆₁	1	1	yes	characteristic, complemented
349	#2004	C₂×C₆₁	1	1	yes	characteristic, complemented
350	#2003	C₂×C₆₁	1	1	yes	characteristic, complemented
351	#2002	C₂×C₆₁	1	1	yes	characteristic
352	#1994	C₂×C₆₁	1	1	yes	characteristic, complemented
353	#1967	C₂×C₆₁	1	1	yes	characteristic, complemented
354	#1888	C₂×C₆₁	1	1	yes	characteristic, complemented
355	#1887	C₂×C₆₁	1	1	yes	characteristic, complemented
356	#1886	C₂×C₆₁	1	1	yes	characteristic, complemented
357	#1885	C₂×C₆₁	1	1	yes	characteristic, complemented
358	C₄×#104	C₂×C₆₁	1	1	yes	characteristic, complemented
359	C₂×C₆₁	—	2	1	no	—
360	C₂×C₆₁	—	2	1	no	—
361	C₂×C₆₁	—	8	1	no	—
362	C₂×C₆₁	—	8	2	no	—
363	C₂×C₆₁	—	16	1	no	—
364	C₂×C₆₁	#2431	1	1	yes	characteristic
365	C₂×C₆₁	#3018	1	2	yes	—
366	C₂×#137	C₂²×C₆₁	1	1	yes	characteristic
367	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	1	yes	characteristic
368	#559	—	2	2	no	—
369	#556	—	2	2	no	—
370	#551	—	2	4	no	—
371	#542	—	2	2	no	—
372	#536	C₂²×C₆₁	1	1	yes	characteristic
373	#535	C₂²×C₆₁	1	1	yes	characteristic
374	#533	C₂²×C₆₁	1	1	yes	characteristic
375	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
376	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
377	#529	—	2	2	no	—
378	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
379	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
380	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
381	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
382	#519	C₂²×C₆₁	1	1	yes	characteristic
383	#518	C₂²×C₆₁	1	1	yes	characteristic
384	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
385	#489	—	2	2	no	—
386	#488	—	2	2	no	—
387	#464	—	2	2	no	—
388	#463	—	2	2	no	—
389	#423	C₂²×C₆₁	1	1	yes	characteristic
390	C₂×#104	C₂²×C₆₁	1	1	yes	characteristic
391	C₂×#104	C₂²×C₆₁	1	1	yes	characteristic
392	#419	C₂²×C₆₁	1	1	yes	characteristic
393	#419	C₂²×C₆₁	1	1	yes	characteristic
394	#419	C₂²×C₆₁	1	1	yes	characteristic
395	#418	C₂²×C₆₁	1	1	yes	characteristic
396	#418	C₂²×C₆₁	1	1	yes	characteristic
397	#418	C₂²×C₆₁	1	1	yes	characteristic
398	C₂×#101	C₂²×C₆₁	1	1	yes	characteristic
399	C₂×#101	C₂²×C₆₁	1	1	yes	characteristic
400	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
401	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
402	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
403	#390	—	2	4	no	—
404	C₄³	C₂²×C₆₁	1	1	yes	characteristic
405	#339	C₂²×C₆₁	1	1	yes	characteristic
406	#339	C₂²×C₆₁	1	1	yes	characteristic
407	#339	C₂²×C₆₁	1	1	yes	characteristic
408	#339	C₂²×C₆₁	1	1	yes	characteristic
409	#339	C₂²×C₆₁	1	1	yes	characteristic
410	#339	C₂²×C₆₁	1	1	yes	characteristic
411	C₆₁	#25592	1	1	yes	characteristic, direct product factor, Sylow
412	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
413	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
414	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
415	C₂²×D₄	—	2	2	no	—
416	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
417	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
418	#137	—	4	1	no	—
419	#137	—	4	1	no	—
420	AGL(1,ℤ/8)	—	4	1	no	—
421	AGL(1,ℤ/8)	—	4	1	no	—
422	C₂×Q₁₆	—	2	1	no	—
423	C₂×SD₁₆	—	2	1	no	—
424	C₂×SD₁₆	—	2	1	no	—
425	C₂×D₈	—	2	1	no	—
426	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
427	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
428	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
429	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
430	C₄⋊Q₈	—	2	1	no	—
431	C₄⋊Q₈	—	2	2	no	—
432	C₄⋊Q₈	—	4	1	no	—
433	C₄⋊Q₈	D₄×C₆₁	1	2	yes	complemented
434	#127	—	2	1	no	—
435	#127	—	2	2	no	—
436	#127	—	4	1	no	—
437	#126	—	2	4	no	—
438	#126	—	2	4	no	—
439	#125	D₄×C₆₁	1	2	yes	complemented
440	#124	—	2	4	no	—
441	#124	—	2	1	no	—
442	#124	—	2	1	no	—
443	#124	—	2	1	no	—
444	#124	—	2	1	no	—
445	#124	—	2	2	no	—
446	#124	—	2	2	no	—
447	#124	—	2	1	no	—
448	#124	—	2	1	no	—
449	#124	—	2	2	no	—
450	#124	—	2	2	no	—
451	#124	—	4	1	no	—
452	#124	—	4	1	no	—
453	#123	—	2	4	no	—
454	#123	—	2	4	no	—
455	#123	—	2	2	no	—
456	C₂²⋊Q₈	—	2	4	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	2	no	—
459	#121	—	2	2	no	—
460	#120	—	2	2	no	—
461	C₄×Q₈	—	2	2	no	—
462	C₄×Q₈	—	2	2	no	—
463	C₄×Q₈	—	2	2	no	—
464	C₄×Q₈	D₄×C₆₁	1	2	yes	complemented
465	C₄×D₄	—	2	4	no	—
466	C₄×D₄	—	2	2	no	—
467	C₄×D₄	—	2	2	no	—
468	C₄×D₄	—	2	2	no	—
469	C₄×D₄	D₄×C₆₁	1	2	yes	complemented
470	#117	—	2	4	no	—
471	#117	—	2	2	no	—
472	#117	—	2	1	no	—
473	#117	C₂³×C₆₁	1	1	yes	characteristic
474	#117	C₂³×C₆₁	1	1	yes	characteristic
475	#117	C₂³×C₆₁	1	1	yes	characteristic
476	C₂×C₄⋊C₄	—	2	2	no	—
477	C₂×C₂²⋊C₄	—	2	2	no	—
478	C₂×C₂²⋊C₄	—	2	2	no	—
479	C₂×C₂²⋊C₄	—	2	2	no	—
480	C₂×C₂²⋊C₄	—	2	2	no	—
481	C₂×C₂²⋊C₄	D₄×C₆₁	1	4	yes	—
482	C₂×C₄²	—	2	1	no	—
483	C₂×C₄²	—	2	2	no	—
484	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
486	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
487	C₄⋊C₈	—	2	2	no	—
488	#104	—	2	2	no	—
489	#104	—	2	2	no	—
490	#103	—	2	2	no	—
491	#103	—	2	2	no	—
492	#103	—	2	2	no	—
493	#102	—	2	2	no	—
494	#102	—	2	2	no	—
495	#102	—	2	2	no	—
496	#101	—	4	1	no	—
497	#101	—	4	1	no	—
498	#100	—	4	1	no	—
499	#100	—	4	1	no	—
500	#97	—	2	1	no	—
501	C₄×C₈	—	2	1	no	—
502	C₂⁴	—	2	1	no	—
503	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
504	#41	—	2	1	no	—
505	#41	—	2	1	no	—
506	#41	—	2	1	no	—
507	#41	—	2	1	no	—
508	C₂×Q₈	—	2	2	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	—	2	1	no	—
511	C₂×Q₈	—	2	1	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	2	1	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	2	1	no	—
518	C₂×Q₈	—	2	1	no	—
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	—	2	1	no	—
521	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	—	2	4	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	—	2	1	no	—
527	C₂×D₄	—	2	1	no	—
528	C₂×D₄	—	2	1	no	—
529	C₂×D₄	—	2	1	no	—
530	C₂×D₄	—	2	1	no	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	1	no	—
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	—	4	4	no	—
537	C₂×D₄	—	4	8	no	—
538	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	4	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	2	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
546	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
547	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
548	Q₁₆	—	4	2	no	—
549	SD₁₆	—	4	2	no	—
550	SD₁₆	—	4	2	no	—
551	D₈	—	4	2	no	—
552	M₁₆	—	2	1	no	—
553	M₁₆	—	2	1	no	—
554	M₁₆	—	2	1	no	—
555	M₁₆	—	2	1	no	—
556	M₁₆	—	2	1	no	—
557	M₁₆	—	2	1	no	—
558	M₁₆	—	2	1	no	—
559	M₁₆	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	4	4	no	—
571	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
572	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
573	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
574	C₂²⋊C₄	—	2	4	no	—
575	C₂²⋊C₄	—	2	4	no	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	1	no	—
578	C₂²⋊C₄	—	2	1	no	—
579	C₂²⋊C₄	—	2	1	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	4	no	—
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	4	4	no	—
584	C₂²⋊C₄	—	4	4	no	—
585	C₂²⋊C₄	—	4	2	no	—
586	C₂²⋊C₄	—	4	2	no	—
587	C₄²	—	2	4	no	—
588	C₄²	—	2	2	no	—
589	C₄²	—	2	1	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	—	2	1	no	—
593	C₄²	—	2	1	no	—
594	C₄²	—	4	4	no	—
595	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
596	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
597	C₄²	C₂×D₄×C₆₁	1	2	yes	—
598	C₄²	#41×C₆₁	1	2	yes	—
599	C₄²	#41×C₆₁	1	2	yes	—
600	C₂³	—	2	2	no	—
601	C₂³	—	2	1	no	—
602	C₂³	—	2	2	no	—
603	C₂³	—	2	2	no	—
604	C₂³	—	4	4	no	—
605	C₂³	—	4	4	no	—
606	C₂³	—	4	1	no	—
607	C₂³	—	4	1	no	—
608	C₂³	—	4	1	no	—
609	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
610	Q₈	—	2	1	no	—
611	Q₈	—	2	1	no	—
612	Q₈	—	4	1	no	—
613	Q₈	—	4	1	no	—
614	Q₈	—	4	1	no	—
615	Q₈	—	4	1	no	—
616	Q₈	—	4	1	no	—
617	Q₈	—	4	1	no	—
618	Q₈	—	4	1	no	—
619	Q₈	—	4	1	no	—
620	Q₈	—	8	2	no	—
621	D₄	—	2	1	no	—
622	D₄	—	2	1	no	—
623	D₄	—	4	1	no	—
624	D₄	—	4	1	no	—
625	D₄	—	4	1	no	—
626	D₄	—	4	1	no	—
627	D₄	—	4	1	no	—
628	D₄	—	4	1	no	—
629	D₄	—	4	1	no	—
630	D₄	—	4	1	no	—
631	D₄	—	4	1	no	—
632	D₄	—	4	1	no	—
633	D₄	—	8	8	no	—
634	D₄	—	8	4	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	4	2	no	—
646	C₂×C₄	—	4	2	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	8	2	no	—
653	C₂×C₄	—	8	4	no	—
654	C₂×C₄	#120×C₆₁	1	2	yes	—
655	C₂×C₄	#120×C₆₁	1	2	yes	—
656	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
657	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
658	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
659	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
660	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
661	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
662	C₈	—	4	1	no	—
663	C₈	—	4	1	no	—
664	C₈	—	4	1	no	—
665	C₈	—	4	1	no	—
666	C₂²	—	2	2	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	2	no	—
669	C₂²	—	4	2	no	—
670	C₂²	—	4	4	no	—
671	C₂²	—	8	2	no	—
672	C₂²	—	8	2	no	—
673	C₂²	—	8	2	no	—
674	C₂²	—	8	2	no	—
675	C₂²	—	8	1	no	—
676	C₂²	—	8	2	no	—
677	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
678	C₂²	C₆₁×#546	1	1	yes	characteristic
679	C₂²	C₆₁×#546	1	1	yes	characteristic
680	C₄	—	2	4	no	—
681	C₄	—	2	2	no	—
682	C₄	—	2	2	no	—
683	C₄	—	4	2	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	1	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	2	no	—
689	C₄	—	8	1	no	—
690	C₄	C₆₁×#532	1	1	yes	characteristic
691	C₄	C₆₁×#535	1	1	yes	characteristic
692	C₄	C₆₁×#550	1	1	yes	characteristic
693	C₄	C₆₁×#550	1	1	yes	characteristic
694	C₂	—	2	1	no	—
695	C₂	—	2	1	no	—
696	C₂	—	8	2	no	—
697	C₂	—	8	1	no	—
698	C₂	—	16	1	no	—
699	C₂	C₆₁×#2431	1	1	yes	characteristic
700	C₂	C₆₁×#3018	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#2002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3510	C₂	semidirect
2. C₆₁×#3140	C₂	semidirect
3. C₆₁×#3040	C₂	semidirect
4. C₆₁×#2858	C₂	semidirect
5. C₆₁×#2005	C₂	semidirect
6. C₆₁×#2004	C₂	semidirect
7. C₆₁×#2003	C₂	semidirect
9. C₆₁×#1994	C₂	semidirect
10. C₆₁×#1967	C₂	semidirect
11. C₆₁×#1888	C₂	semidirect
12. C₆₁×#1887	C₂	semidirect
13. C₆₁×#1886	C₂	semidirect
14. C₆₁×#1885	C₂	semidirect
15. C₄×#104×C₆₁	C₂	semidirect
82. C₄⋊Q₈×C₆₁	D₄	semidirect
88. #125×C₆₁	D₄	semidirect
113. C₄×Q₈×C₆₁	D₄	semidirect
118. C₄×D₄×C₆₁	D₄	semidirect
315. #25592	C₆₁	direct (N × H)
344. #3510	C₂×C₆₁	semidirect
345. #3140	C₂×C₆₁	semidirect
346. #3040	C₂×C₆₁	semidirect
347. #2858	C₂×C₆₁	semidirect
348. #2005	C₂×C₆₁	semidirect
349. #2004	C₂×C₆₁	semidirect
350. #2003	C₂×C₆₁	semidirect
352. #1994	C₂×C₆₁	semidirect
353. #1967	C₂×C₆₁	semidirect
354. #1888	C₂×C₆₁	semidirect
355. #1887	C₂×C₆₁	semidirect
356. #1886	C₂×C₆₁	semidirect
357. #1885	C₂×C₆₁	semidirect
358. C₄×#104	C₂×C₆₁	semidirect
411. C₆₁	#25592	direct (N × H)
433. C₄⋊Q₈	D₄×C₆₁	semidirect
439. #125	D₄×C₆₁	semidirect
464. C₄×Q₈	D₄×C₆₁	semidirect
469. C₄×D₄	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3510	2
C₆₁×#3140	2
C₆₁×#3040	2
C₆₁×#2858	2
C₆₁×#2005	2
C₆₁×#2004	2
C₆₁×#2003	2
C₆₁×#2002	2
C₆₁×#1994	2
C₆₁×#1967	2
C₆₁×#1888	2
C₆₁×#1887	2
C₆₁×#1886	2
C₆₁×#1885	2
C₄×#104×C₆₁	2
#25592	61

Subgroups — C61×#25592 #29529025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25592 #29529025