Subgroups — C₆₁×#25880 #29529313

Order 15616.

1242 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3500	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2640	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2038	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2037	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1967	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1963	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1793	C₂	1	2	yes	maximal, complemented
10	C₆₁×#1792	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1759	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#548	—	2	1	no	—
15	C₆₁×#548	—	2	1	no	—
16	C₆₁×#546	—	2	1	no	—
17	C₆₁×#546	—	2	1	no	—
18	C₆₁×#546	—	2	2	no	—
19	C₆₁×#546	—	2	2	no	—
20	C₆₁×#536	—	2	1	no	—
21	C₆₁×#536	C₂²	1	1	yes	characteristic
22	C₆₁×#535	—	2	1	no	—
23	C₆₁×#535	C₂²	1	1	yes	characteristic
24	C₆₁×#532	—	2	1	no	—
25	C₆₁×#532	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
27	C₂×#124×C₆₁	—	2	1	no	—
28	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
29	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#489	—	2	1	no	—
31	C₆₁×#489	—	2	1	no	—
32	C₆₁×#487	—	2	1	no	—
33	C₆₁×#486	—	2	1	no	—
34	C₆₁×#477	—	2	1	no	—
35	C₆₁×#476	—	2	1	no	—
36	C₆₁×#475	—	2	1	no	—
37	C₆₁×#474	—	2	1	no	—
38	C₆₁×#428	C₂²	1	2	yes	complemented
39	C₆₁×#423	C₂²	1	1	yes	characteristic, complemented
40	C₂×#104×C₆₁	C₂²	1	2	yes	—
41	C₂×#104×C₆₁	C₂²	1	2	yes	—
42	C₂×#104×C₆₁	C₂²	1	2	yes	complemented
43	C₆₁×#417	C₂²	1	2	yes	—
44	C₆₁×#417	C₂²	1	2	yes	—
45	C₆₁×#417	C₂²	1	2	yes	complemented
46	C₂×#101×C₆₁	C₂²	1	2	yes	complemented
47	C₂×#100×C₆₁	C₂²	1	2	yes	—
48	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
49	C₆₁×#397	—	2	2	no	—
50	C₆₁×#393	—	2	2	no	—
51	C₆₁×#391	C₂²	1	2	yes	complemented
52	C₆₁×#390	C₂²	1	2	yes	—
53	C₆₁×#376	C₂²	1	1	yes	characteristic, complemented
54	C₆₁×#339	C₂²	1	2	yes	complemented
55	2₋¹⁺⁴×C₆₁	—	2	4	no	—
56	2₋¹⁺⁴×C₆₁	—	4	2	no	—
57	H(5,𝔽₂)×C₆₁	—	2	4	no	—
58	H(5,𝔽₂)×C₆₁	—	4	2	no	—
59	C₂×#41×C₆₁	—	2	1	no	—
60	C₂×#41×C₆₁	—	2	1	no	—
61	C₂×#41×C₆₁	—	2	1	no	—
62	C₂×#41×C₆₁	—	2	1	no	—
63	C₂×#41×C₆₁	—	2	2	no	—
64	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
65	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
66	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₆₁	—	2	1	no	—
68	C₂²×Q₈×C₆₁	—	2	1	no	—
69	C₂²×Q₈×C₆₁	—	2	1	no	—
70	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₆₁	—	2	2	no	—
72	C₂²×D₄×C₆₁	—	2	1	no	—
73	C₂²×D₄×C₆₁	—	2	1	no	—
74	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
76	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
77	C₄⋊Q₈×C₆₁	—	2	1	no	—
78	C₄⋊Q₈×C₆₁	—	2	1	no	—
79	C₄⋊Q₈×C₆₁	—	2	1	no	—
80	C₄⋊Q₈×C₆₁	—	2	1	no	—
81	C₄⋊Q₈×C₆₁	—	2	1	no	—
82	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
83	#127×C₆₁	—	2	1	no	—
84	#127×C₆₁	D₄	1	1	yes	characteristic
85	#124×C₆₁	—	2	2	no	—
86	#124×C₆₁	—	2	2	no	—
87	#124×C₆₁	—	2	1	no	—
88	#124×C₆₁	—	2	1	no	—
89	#124×C₆₁	—	2	1	no	—
90	#124×C₆₁	—	2	1	no	—
91	#124×C₆₁	—	2	1	no	—
92	#124×C₆₁	—	2	1	no	—
93	#124×C₆₁	D₄	1	1	yes	characteristic
94	#124×C₆₁	D₄	1	1	yes	characteristic
95	#123×C₆₁	—	2	2	no	—
96	#123×C₆₁	—	2	2	no	—
97	#123×C₆₁	—	2	2	no	—
98	C₂²⋊Q₈×C₆₁	—	2	2	no	—
99	C₂²⋊Q₈×C₆₁	—	2	2	no	—
100	C₂²⋊Q₈×C₆₁	—	2	1	no	—
101	C₂²⋊Q₈×C₆₁	—	2	1	no	—
102	#121×C₆₁	—	2	2	no	—
103	#121×C₆₁	—	2	2	no	—
104	#121×C₆₁	—	2	2	no	—
105	#121×C₆₁	—	2	2	no	—
106	#121×C₆₁	—	2	1	no	—
107	#121×C₆₁	—	2	1	no	—
108	#121×C₆₁	—	2	1	no	—
109	#121×C₆₁	—	2	1	no	—
110	#120×C₆₁	—	2	2	no	—
111	#120×C₆₁	—	2	2	no	—
112	#120×C₆₁	—	2	2	no	—
113	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
114	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
115	C₄×D₄×C₆₁	—	2	2	no	—
116	C₄×D₄×C₆₁	—	2	2	no	—
117	C₄×D₄×C₆₁	—	2	1	no	—
118	C₄×D₄×C₆₁	—	2	1	no	—
119	C₄×D₄×C₆₁	—	2	1	no	—
120	C₄×D₄×C₆₁	—	2	1	no	—
121	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
122	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
123	#117×C₆₁	—	2	1	no	—
124	#117×C₆₁	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
126	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
128	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
129	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
130	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
131	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
132	C₂×C₄²×C₆₁	—	2	1	no	—
133	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
134	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
135	#107×C₆₁	—	2	2	no	—
136	#106×C₆₁	—	2	2	no	—
137	C₄⋊C₈×C₆₁	—	2	1	no	—
138	C₄⋊C₈×C₆₁	—	2	1	no	—
139	#104×C₆₁	—	2	4	no	—
140	#104×C₆₁	—	4	2	no	—
141	#104×C₆₁	—	4	2	no	—
142	#103×C₆₁	—	2	2	no	—
143	#103×C₆₁	—	2	2	no	—
144	#103×C₆₁	—	2	2	no	—
145	#102×C₆₁	—	2	2	no	—
146	#102×C₆₁	—	2	2	no	—
147	#102×C₆₁	—	2	2	no	—
148	#101×C₆₁	—	4	2	no	—
149	#100×C₆₁	—	4	2	no	—
150	#97×C₆₁	—	2	1	no	—
151	C₄×C₈×C₆₁	—	2	1	no	—
152	#95×C₆₁	—	2	1	no	—
153	#95×C₆₁	—	2	1	no	—
154	#95×C₆₁	—	2	4	no	—
155	C₂⁴×C₆₁	—	2	2	no	—
156	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
157	#41×C₆₁	—	2	2	no	—
158	#41×C₆₁	—	2	2	no	—
159	#41×C₆₁	—	2	2	no	—
160	#41×C₆₁	—	2	2	no	—
161	#41×C₆₁	—	2	2	no	—
162	#41×C₆₁	—	4	4	no	—
163	#41×C₆₁	—	4	8	no	—
164	#41×C₆₁	—	4	4	no	—
165	#41×C₆₁	—	4	2	no	—
166	#41×C₆₁	—	4	2	no	—
167	#41×C₆₁	—	4	2	no	—
168	#41×C₆₁	—	4	2	no	—
169	#41×C₆₁	#41	1	4	yes	—
170	C₂×Q₈×C₆₁	—	2	2	no	—
171	C₂×Q₈×C₆₁	—	2	1	no	—
172	C₂×Q₈×C₆₁	—	2	1	no	—
173	C₂×Q₈×C₆₁	—	2	2	no	—
174	C₂×Q₈×C₆₁	—	2	2	no	—
175	C₂×Q₈×C₆₁	—	4	4	no	—
176	C₂×Q₈×C₆₁	—	4	2	no	—
177	C₂×Q₈×C₆₁	—	4	2	no	—
178	C₂×Q₈×C₆₁	—	4	1	no	—
179	C₂×Q₈×C₆₁	—	4	1	no	—
180	C₂×Q₈×C₆₁	—	8	2	no	—
181	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
182	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
183	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
184	C₂×D₄×C₆₁	—	2	1	no	—
185	C₂×D₄×C₆₁	—	2	2	no	—
186	C₂×D₄×C₆₁	—	2	2	no	—
187	C₂×D₄×C₆₁	—	2	2	no	—
188	C₂×D₄×C₆₁	—	2	1	no	—
189	C₂×D₄×C₆₁	—	2	1	no	—
190	C₂×D₄×C₆₁	—	2	2	no	—
191	C₂×D₄×C₆₁	—	2	2	no	—
192	C₂×D₄×C₆₁	—	2	1	no	—
193	C₂×D₄×C₆₁	—	2	1	no	—
194	C₂×D₄×C₆₁	—	2	1	no	—
195	C₂×D₄×C₆₁	—	2	1	no	—
196	C₂×D₄×C₆₁	—	2	2	no	—
197	C₂×D₄×C₆₁	—	2	2	no	—
198	C₂×D₄×C₆₁	—	4	4	no	—
199	C₂×D₄×C₆₁	—	4	8	no	—
200	C₂×D₄×C₆₁	—	4	2	no	—
201	C₂×D₄×C₆₁	—	4	2	no	—
202	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
203	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
204	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₆₁	—	2	1	no	—
206	C₂²×C₄×C₆₁	—	2	2	no	—
207	C₂²×C₄×C₆₁	—	2	2	no	—
208	C₂²×C₄×C₆₁	—	2	2	no	—
209	C₂²×C₄×C₆₁	—	2	1	no	—
210	C₂²×C₄×C₆₁	—	2	1	no	—
211	C₂²×C₄×C₆₁	—	2	1	no	—
212	C₂²×C₄×C₆₁	—	2	1	no	—
213	C₂²×C₄×C₆₁	—	2	2	no	—
214	C₂²×C₄×C₆₁	—	2	2	no	—
215	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
216	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
217	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
218	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
219	M₁₆×C₆₁	—	2	2	no	—
220	M₁₆×C₆₁	—	2	2	no	—
221	M₁₆×C₆₁	—	2	2	no	—
222	M₁₆×C₆₁	—	2	2	no	—
223	C₂×C₈×C₆₁	—	2	2	no	—
224	C₂×C₈×C₆₁	—	2	2	no	—
225	C₄⋊C₄×C₆₁	—	2	2	no	—
226	C₄⋊C₄×C₆₁	—	2	2	no	—
227	C₄⋊C₄×C₆₁	—	2	1	no	—
228	C₄⋊C₄×C₆₁	—	2	1	no	—
229	C₄⋊C₄×C₆₁	—	2	1	no	—
230	C₄⋊C₄×C₆₁	—	2	1	no	—
231	C₄⋊C₄×C₆₁	—	2	2	no	—
232	C₄⋊C₄×C₆₁	—	2	2	no	—
233	C₄⋊C₄×C₆₁	—	4	1	no	—
234	C₄⋊C₄×C₆₁	—	4	1	no	—
235	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
236	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
237	C₂²⋊C₄×C₆₁	—	2	2	no	—
238	C₂²⋊C₄×C₆₁	—	2	2	no	—
239	C₂²⋊C₄×C₆₁	—	2	2	no	—
240	C₂²⋊C₄×C₆₁	—	2	2	no	—
241	C₂²⋊C₄×C₆₁	—	2	1	no	—
242	C₂²⋊C₄×C₆₁	—	2	1	no	—
243	C₂²⋊C₄×C₆₁	—	2	1	no	—
244	C₂²⋊C₄×C₆₁	—	2	2	no	—
245	C₂²⋊C₄×C₆₁	—	2	2	no	—
246	C₂²⋊C₄×C₆₁	—	2	2	no	—
247	C₂²⋊C₄×C₆₁	—	2	2	no	—
248	C₂²⋊C₄×C₆₁	—	2	1	no	—
249	C₂²⋊C₄×C₆₁	—	2	1	no	—
250	C₂²⋊C₄×C₆₁	—	2	1	no	—
251	C₂²⋊C₄×C₆₁	—	2	1	no	—
252	C₂²⋊C₄×C₆₁	—	8	4	no	—
253	C₄²×C₆₁	—	2	1	no	—
254	C₄²×C₆₁	—	2	1	no	—
255	C₄²×C₆₁	—	2	1	no	—
256	C₄²×C₆₁	—	2	2	no	—
257	C₄²×C₆₁	—	2	2	no	—
258	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
259	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
261	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
262	C₂³×C₆₁	—	2	1	no	—
263	C₂³×C₆₁	—	2	2	no	—
264	C₂³×C₆₁	—	2	1	no	—
265	C₂³×C₆₁	—	2	1	no	—
266	C₂³×C₆₁	—	2	4	no	—
267	C₂³×C₆₁	—	4	4	no	—
268	C₂³×C₆₁	—	4	2	no	—
269	C₂³×C₆₁	—	4	2	no	—
270	C₂³×C₆₁	—	8	2	no	—
271	C₂³×C₆₁	—	8	2	no	—
272	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
273	Q₈×C₆₁	—	2	1	no	—
274	Q₈×C₆₁	—	2	2	no	—
275	Q₈×C₆₁	—	2	2	no	—
276	Q₈×C₆₁	—	4	4	no	—
277	Q₈×C₆₁	—	4	1	no	—
278	Q₈×C₆₁	—	4	1	no	—
279	Q₈×C₆₁	—	4	2	no	—
280	Q₈×C₆₁	—	8	2	no	—
281	Q₈×C₆₁	—	8	2	no	—
282	Q₈×C₆₁	#124	1	2	yes	—
283	D₄×C₆₁	—	2	4	no	—
284	D₄×C₆₁	—	2	1	no	—
285	D₄×C₆₁	—	2	2	no	—
286	D₄×C₆₁	—	2	2	no	—
287	D₄×C₆₁	—	4	4	no	—
288	D₄×C₆₁	—	4	4	no	—
289	D₄×C₆₁	—	4	4	no	—
290	D₄×C₆₁	—	4	2	no	—
291	D₄×C₆₁	—	4	2	no	—
292	D₄×C₆₁	—	4	1	no	—
293	D₄×C₆₁	—	4	1	no	—
294	D₄×C₆₁	—	4	2	no	—
295	D₄×C₆₁	—	8	1	no	—
296	D₄×C₆₁	—	8	1	no	—
297	D₄×C₆₁	—	8	2	no	—
298	D₄×C₆₁	—	8	2	no	—
299	D₄×C₆₁	#124	1	2	yes	—
300	C₂×C₄×C₆₁	—	2	1	no	—
301	C₂×C₄×C₆₁	—	2	4	no	—
302	C₂×C₄×C₆₁	—	2	1	no	—
303	C₂×C₄×C₆₁	—	2	1	no	—
304	C₂×C₄×C₆₁	—	2	1	no	—
305	C₂×C₄×C₆₁	—	2	2	no	—
306	C₂×C₄×C₆₁	—	2	1	no	—
307	C₂×C₄×C₆₁	—	2	4	no	—
308	C₂×C₄×C₆₁	—	4	1	no	—
309	C₂×C₄×C₆₁	—	4	1	no	—
310	C₂×C₄×C₆₁	—	4	4	no	—
311	C₂×C₄×C₆₁	—	4	4	no	—
312	C₂×C₄×C₆₁	—	4	4	no	—
313	C₂×C₄×C₆₁	—	4	1	no	—
314	C₂×C₄×C₆₁	—	4	1	no	—
315	C₂×C₄×C₆₁	—	4	1	no	—
316	C₂×C₄×C₆₁	—	4	1	no	—
317	C₂×C₄×C₆₁	—	4	2	no	—
318	C₂×C₄×C₆₁	—	4	4	no	—
319	C₂×C₄×C₆₁	—	4	2	no	—
320	C₂×C₄×C₆₁	—	4	2	no	—
321	C₂×C₄×C₆₁	—	4	2	no	—
322	C₂×C₄×C₆₁	—	8	1	no	—
323	C₂×C₄×C₆₁	—	8	1	no	—
324	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
325	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
326	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
327	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
328	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
329	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
330	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
331	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
332	C₈×C₆₁	—	4	2	no	—
333	C₈×C₆₁	—	4	2	no	—
334	#25880	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
335	C₂²×C₆₁	—	2	1	no	—
336	C₂²×C₆₁	—	2	1	no	—
337	C₂²×C₆₁	—	2	2	no	—
338	C₂²×C₆₁	—	4	1	no	—
339	C₂²×C₆₁	—	4	1	no	—
340	C₂²×C₆₁	—	4	1	no	—
341	C₂²×C₆₁	—	4	1	no	—
342	C₂²×C₆₁	—	4	2	no	—
343	C₂²×C₆₁	—	4	1	no	—
344	C₂²×C₆₁	—	4	2	no	—
345	C₂²×C₆₁	—	4	1	no	—
346	C₂²×C₆₁	—	4	1	no	—
347	C₂²×C₆₁	—	4	2	no	—
348	C₂²×C₆₁	—	4	1	no	—
349	C₂²×C₆₁	—	8	4	no	—
350	C₂²×C₆₁	—	16	4	no	—
351	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
352	C₂²×C₆₁	#546	1	2	yes	—
353	C₄×C₆₁	—	2	2	no	—
354	C₄×C₆₁	—	2	1	no	—
355	C₄×C₆₁	—	2	1	no	—
356	C₄×C₆₁	—	2	1	no	—
357	C₄×C₆₁	—	2	1	no	—
358	C₄×C₆₁	—	4	2	no	—
359	C₄×C₆₁	—	4	1	no	—
360	C₄×C₆₁	—	4	1	no	—
361	C₄×C₆₁	—	4	1	no	—
362	C₄×C₆₁	—	4	1	no	—
363	C₄×C₆₁	—	8	2	no	—
364	C₄×C₆₁	—	8	2	no	—
365	C₄×C₆₁	C₂×#124	1	1	yes	characteristic
366	C₄×C₆₁	#535	1	1	yes	characteristic
367	C₄×C₆₁	#546	1	2	yes	—
368	#3500	C₂×C₆₁	1	1	yes	characteristic
369	#3141	C₂×C₆₁	1	1	yes	characteristic, complemented
370	#3079	C₂×C₆₁	1	1	yes	characteristic, complemented
371	#2640	C₂×C₆₁	1	1	yes	characteristic, complemented
372	#2038	C₂×C₆₁	1	2	yes	complemented
373	#2037	C₂×C₆₁	1	2	yes	complemented
374	#1967	C₂×C₆₁	1	1	yes	characteristic, complemented
375	#1963	C₂×C₆₁	1	1	yes	characteristic, complemented
376	#1793	C₂×C₆₁	1	2	yes	complemented
377	#1792	C₂×C₆₁	1	2	yes	complemented
378	#1759	C₂×C₆₁	1	1	yes	characteristic, complemented
379	C₂×C₆₁	—	2	2	no	—
380	C₂×C₆₁	—	4	2	no	—
381	C₂×C₆₁	—	8	1	no	—
382	C₂×C₆₁	—	8	2	no	—
383	C₂×C₆₁	—	8	1	no	—
384	C₂×C₆₁	#2435	1	1	yes	characteristic
385	C₂×C₆₁	#3021	1	1	yes	characteristic
386	C₂×C₆₁	#3022	1	1	yes	characteristic
387	C₂×2₋¹⁺⁴	C₂²×C₆₁	1	1	yes	characteristic
388	C₂×H(5,𝔽₂)	C₂²×C₆₁	1	1	yes	characteristic
389	#548	—	2	1	no	—
390	#548	—	2	1	no	—
391	#546	—	2	1	no	—
392	#546	—	2	1	no	—
393	#546	—	2	2	no	—
394	#546	—	2	2	no	—
395	#536	—	2	1	no	—
396	#536	C₂²×C₆₁	1	1	yes	characteristic
397	#535	—	2	1	no	—
398	#535	C₂²×C₆₁	1	1	yes	characteristic
399	#532	—	2	1	no	—
400	#532	C₂²×C₆₁	1	1	yes	characteristic
401	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
402	C₂×#124	—	2	1	no	—
403	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
404	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
405	#489	—	2	1	no	—
406	#489	—	2	1	no	—
407	#487	—	2	1	no	—
408	#486	—	2	1	no	—
409	#477	—	2	1	no	—
410	#476	—	2	1	no	—
411	#475	—	2	1	no	—
412	#474	—	2	1	no	—
413	#428	C₂²×C₆₁	1	2	yes	complemented
414	#423	C₂²×C₆₁	1	1	yes	characteristic, complemented
415	C₂×#104	C₂²×C₆₁	1	2	yes	—
416	C₂×#104	C₂²×C₆₁	1	2	yes	—
417	C₂×#104	C₂²×C₆₁	1	2	yes	complemented
418	#417	C₂²×C₆₁	1	2	yes	—
419	#417	C₂²×C₆₁	1	2	yes	—
420	#417	C₂²×C₆₁	1	2	yes	complemented
421	C₂×#101	C₂²×C₆₁	1	2	yes	complemented
422	C₂×#100	C₂²×C₆₁	1	2	yes	—
423	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic, complemented
424	#397	—	2	2	no	—
425	#393	—	2	2	no	—
426	#391	C₂²×C₆₁	1	2	yes	complemented
427	#390	C₂²×C₆₁	1	2	yes	—
428	#376	C₂²×C₆₁	1	1	yes	characteristic, complemented
429	#339	C₂²×C₆₁	1	2	yes	complemented
430	C₆₁	#25880	1	1	yes	characteristic, direct product factor, Sylow
431	2₋¹⁺⁴	—	2	4	no	—
432	2₋¹⁺⁴	—	4	2	no	—
433	H(5,𝔽₂)	—	2	4	no	—
434	H(5,𝔽₂)	—	4	2	no	—
435	C₂×#41	—	2	1	no	—
436	C₂×#41	—	2	1	no	—
437	C₂×#41	—	2	1	no	—
438	C₂×#41	—	2	1	no	—
439	C₂×#41	—	2	2	no	—
440	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
441	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
442	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
443	C₂²×Q₈	—	2	1	no	—
444	C₂²×Q₈	—	2	1	no	—
445	C₂²×Q₈	—	2	1	no	—
446	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
447	C₂²×D₄	—	2	2	no	—
448	C₂²×D₄	—	2	1	no	—
449	C₂²×D₄	—	2	1	no	—
450	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
451	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
452	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
453	C₄⋊Q₈	—	2	1	no	—
454	C₄⋊Q₈	—	2	1	no	—
455	C₄⋊Q₈	—	2	1	no	—
456	C₄⋊Q₈	—	2	1	no	—
457	C₄⋊Q₈	—	2	1	no	—
458	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
459	#127	—	2	1	no	—
460	#127	D₄×C₆₁	1	1	yes	characteristic
461	#124	—	2	2	no	—
462	#124	—	2	2	no	—
463	#124	—	2	1	no	—
464	#124	—	2	1	no	—
465	#124	—	2	1	no	—
466	#124	—	2	1	no	—
467	#124	—	2	1	no	—
468	#124	—	2	1	no	—
469	#124	D₄×C₆₁	1	1	yes	characteristic
470	#124	D₄×C₆₁	1	1	yes	characteristic
471	#123	—	2	2	no	—
472	#123	—	2	2	no	—
473	#123	—	2	2	no	—
474	C₂²⋊Q₈	—	2	2	no	—
475	C₂²⋊Q₈	—	2	2	no	—
476	C₂²⋊Q₈	—	2	1	no	—
477	C₂²⋊Q₈	—	2	1	no	—
478	#121	—	2	2	no	—
479	#121	—	2	2	no	—
480	#121	—	2	2	no	—
481	#121	—	2	2	no	—
482	#121	—	2	1	no	—
483	#121	—	2	1	no	—
484	#121	—	2	1	no	—
485	#121	—	2	1	no	—
486	#120	—	2	2	no	—
487	#120	—	2	2	no	—
488	#120	—	2	2	no	—
489	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
490	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
491	C₄×D₄	—	2	2	no	—
492	C₄×D₄	—	2	2	no	—
493	C₄×D₄	—	2	1	no	—
494	C₄×D₄	—	2	1	no	—
495	C₄×D₄	—	2	1	no	—
496	C₄×D₄	—	2	1	no	—
497	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
498	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
499	#117	—	2	1	no	—
500	#117	C₂³×C₆₁	1	1	yes	characteristic
501	C₂×C₄⋊C₄	—	2	1	no	—
502	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
503	C₂×C₂²⋊C₄	—	2	2	no	—
504	C₂×C₂²⋊C₄	—	2	2	no	—
505	C₂×C₂²⋊C₄	—	2	4	no	—
506	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
507	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
508	C₂×C₄²	—	2	1	no	—
509	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
510	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
511	#107	—	2	2	no	—
512	#106	—	2	2	no	—
513	C₄⋊C₈	—	2	1	no	—
514	C₄⋊C₈	—	2	1	no	—
515	#104	—	2	4	no	—
516	#104	—	4	2	no	—
517	#104	—	4	2	no	—
518	#103	—	2	2	no	—
519	#103	—	2	2	no	—
520	#103	—	2	2	no	—
521	#102	—	2	2	no	—
522	#102	—	2	2	no	—
523	#102	—	2	2	no	—
524	#101	—	4	2	no	—
525	#100	—	4	2	no	—
526	#97	—	2	1	no	—
527	C₄×C₈	—	2	1	no	—
528	#95	—	2	1	no	—
529	#95	—	2	1	no	—
530	#95	—	2	4	no	—
531	C₂⁴	—	2	2	no	—
532	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
533	#41	—	2	2	no	—
534	#41	—	2	2	no	—
535	#41	—	2	2	no	—
536	#41	—	2	2	no	—
537	#41	—	2	2	no	—
538	#41	—	4	4	no	—
539	#41	—	4	8	no	—
540	#41	—	4	4	no	—
541	#41	—	4	2	no	—
542	#41	—	4	2	no	—
543	#41	—	4	2	no	—
544	#41	—	4	2	no	—
545	#41	#41×C₆₁	1	4	yes	—
546	C₂×Q₈	—	2	2	no	—
547	C₂×Q₈	—	2	1	no	—
548	C₂×Q₈	—	2	1	no	—
549	C₂×Q₈	—	2	2	no	—
550	C₂×Q₈	—	2	2	no	—
551	C₂×Q₈	—	4	4	no	—
552	C₂×Q₈	—	4	2	no	—
553	C₂×Q₈	—	4	2	no	—
554	C₂×Q₈	—	4	1	no	—
555	C₂×Q₈	—	4	1	no	—
556	C₂×Q₈	—	8	2	no	—
557	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
558	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
559	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
560	C₂×D₄	—	2	1	no	—
561	C₂×D₄	—	2	2	no	—
562	C₂×D₄	—	2	2	no	—
563	C₂×D₄	—	2	2	no	—
564	C₂×D₄	—	2	1	no	—
565	C₂×D₄	—	2	1	no	—
566	C₂×D₄	—	2	2	no	—
567	C₂×D₄	—	2	2	no	—
568	C₂×D₄	—	2	1	no	—
569	C₂×D₄	—	2	1	no	—
570	C₂×D₄	—	2	1	no	—
571	C₂×D₄	—	2	1	no	—
572	C₂×D₄	—	2	2	no	—
573	C₂×D₄	—	2	2	no	—
574	C₂×D₄	—	4	4	no	—
575	C₂×D₄	—	4	8	no	—
576	C₂×D₄	—	4	2	no	—
577	C₂×D₄	—	4	2	no	—
578	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
579	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
580	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
581	C₂²×C₄	—	2	1	no	—
582	C₂²×C₄	—	2	2	no	—
583	C₂²×C₄	—	2	2	no	—
584	C₂²×C₄	—	2	2	no	—
585	C₂²×C₄	—	2	1	no	—
586	C₂²×C₄	—	2	1	no	—
587	C₂²×C₄	—	2	1	no	—
588	C₂²×C₄	—	2	1	no	—
589	C₂²×C₄	—	2	2	no	—
590	C₂²×C₄	—	2	2	no	—
591	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
592	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
593	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
594	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
595	M₁₆	—	2	2	no	—
596	M₁₆	—	2	2	no	—
597	M₁₆	—	2	2	no	—
598	M₁₆	—	2	2	no	—
599	C₂×C₈	—	2	2	no	—
600	C₂×C₈	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	1	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	2	no	—
608	C₄⋊C₄	—	2	2	no	—
609	C₄⋊C₄	—	4	1	no	—
610	C₄⋊C₄	—	4	1	no	—
611	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
612	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
613	C₂²⋊C₄	—	2	2	no	—
614	C₂²⋊C₄	—	2	2	no	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	2	no	—
617	C₂²⋊C₄	—	2	1	no	—
618	C₂²⋊C₄	—	2	1	no	—
619	C₂²⋊C₄	—	2	1	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	—	2	2	no	—
622	C₂²⋊C₄	—	2	2	no	—
623	C₂²⋊C₄	—	2	2	no	—
624	C₂²⋊C₄	—	2	1	no	—
625	C₂²⋊C₄	—	2	1	no	—
626	C₂²⋊C₄	—	2	1	no	—
627	C₂²⋊C₄	—	2	1	no	—
628	C₂²⋊C₄	—	8	4	no	—
629	C₄²	—	2	1	no	—
630	C₄²	—	2	1	no	—
631	C₄²	—	2	1	no	—
632	C₄²	—	2	2	no	—
633	C₄²	—	2	2	no	—
634	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
635	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
636	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
637	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
638	C₂³	—	2	1	no	—
639	C₂³	—	2	2	no	—
640	C₂³	—	2	1	no	—
641	C₂³	—	2	1	no	—
642	C₂³	—	2	4	no	—
643	C₂³	—	4	4	no	—
644	C₂³	—	4	2	no	—
645	C₂³	—	4	2	no	—
646	C₂³	—	8	2	no	—
647	C₂³	—	8	2	no	—
648	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
649	Q₈	—	2	1	no	—
650	Q₈	—	2	2	no	—
651	Q₈	—	2	2	no	—
652	Q₈	—	4	4	no	—
653	Q₈	—	4	1	no	—
654	Q₈	—	4	2	no	—
655	Q₈	—	4	1	no	—
656	Q₈	—	8	2	no	—
657	Q₈	—	8	2	no	—
658	Q₈	#124×C₆₁	1	2	yes	—
659	D₄	—	2	4	no	—
660	D₄	—	2	1	no	—
661	D₄	—	2	2	no	—
662	D₄	—	2	2	no	—
663	D₄	—	4	4	no	—
664	D₄	—	4	4	no	—
665	D₄	—	4	4	no	—
666	D₄	—	4	2	no	—
667	D₄	—	4	2	no	—
668	D₄	—	4	1	no	—
669	D₄	—	4	2	no	—
670	D₄	—	4	1	no	—
671	D₄	—	8	1	no	—
672	D₄	—	8	1	no	—
673	D₄	—	8	2	no	—
674	D₄	—	8	2	no	—
675	D₄	#124×C₆₁	1	2	yes	—
676	C₂×C₄	—	2	1	no	—
677	C₂×C₄	—	2	4	no	—
678	C₂×C₄	—	2	4	no	—
679	C₂×C₄	—	2	1	no	—
680	C₂×C₄	—	2	1	no	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	2	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	4	1	no	—
685	C₂×C₄	—	4	1	no	—
686	C₂×C₄	—	4	4	no	—
687	C₂×C₄	—	4	4	no	—
688	C₂×C₄	—	4	4	no	—
689	C₂×C₄	—	4	1	no	—
690	C₂×C₄	—	4	1	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	4	1	no	—
693	C₂×C₄	—	4	2	no	—
694	C₂×C₄	—	4	4	no	—
695	C₂×C₄	—	4	2	no	—
696	C₂×C₄	—	4	2	no	—
697	C₂×C₄	—	4	2	no	—
698	C₂×C₄	—	8	1	no	—
699	C₂×C₄	—	8	1	no	—
700	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
701	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
702	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
703	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
704	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
705	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
706	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
707	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
708	C₈	—	4	2	no	—
709	C₈	—	4	2	no	—
710	C₂²	—	2	1	no	—
711	C₂²	—	2	1	no	—
712	C₂²	—	2	2	no	—
713	C₂²	—	4	1	no	—
714	C₂²	—	4	1	no	—
715	C₂²	—	4	1	no	—
716	C₂²	—	4	1	no	—
717	C₂²	—	4	2	no	—
718	C₂²	—	4	1	no	—
719	C₂²	—	4	2	no	—
720	C₂²	—	4	1	no	—
721	C₂²	—	4	1	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	1	no	—
724	C₂²	—	8	4	no	—
725	C₂²	—	16	4	no	—
726	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
727	C₂²	C₆₁×#546	1	2	yes	—
728	C₄	—	2	2	no	—
729	C₄	—	2	1	no	—
730	C₄	—	2	1	no	—
731	C₄	—	2	1	no	—
732	C₄	—	2	1	no	—
733	C₄	—	4	2	no	—
734	C₄	—	4	1	no	—
735	C₄	—	4	1	no	—
736	C₄	—	4	1	no	—
737	C₄	—	4	1	no	—
738	C₄	—	8	2	no	—
739	C₄	—	8	2	no	—
740	C₄	C₂×#124×C₆₁	1	1	yes	characteristic
741	C₄	C₆₁×#535	1	1	yes	characteristic
742	C₄	C₆₁×#546	1	2	yes	—
743	C₂	—	2	2	no	—
744	C₂	—	4	2	no	—
745	C₂	—	8	2	no	—
746	C₂	—	8	1	no	—
747	C₂	—	8	1	no	—
748	C₂	C₆₁×#2435	1	1	yes	characteristic
749	C₂	C₆₁×#3021	1	1	yes	characteristic
750	C₂	C₆₁×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#3500	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
2. C₆₁×#3141	C₂	semidirect
3. C₆₁×#3079	C₂	semidirect
4. C₆₁×#2640	C₂	semidirect
5. C₆₁×#2038	C₂	semidirect
6. C₆₁×#2037	C₂	semidirect
7. C₆₁×#1967	C₂	semidirect
8. C₆₁×#1963	C₂	semidirect
9. C₆₁×#1793	C₂	semidirect
10. C₆₁×#1792	C₂	semidirect
11. C₆₁×#1759	C₂	semidirect
26. C₂×C₄⋊Q₈×C₆₁	C₂²	semidirect
38. C₆₁×#428	C₂²	semidirect
39. C₆₁×#423	C₂²	semidirect
42. C₂×#104×C₆₁	C₂²	semidirect
45. C₆₁×#417	C₂²	semidirect
46. C₂×#101×C₆₁	C₂²	semidirect
48. C₄×M₁₆×C₆₁	C₂²	semidirect
51. C₆₁×#391	C₂²	semidirect
53. C₆₁×#376	C₂²	semidirect
54. C₆₁×#339	C₂²	semidirect
334. #25880	C₆₁	direct (N × H)
369. #3141	C₂×C₆₁	semidirect
370. #3079	C₂×C₆₁	semidirect
371. #2640	C₂×C₆₁	semidirect
372. #2038	C₂×C₆₁	semidirect
373. #2037	C₂×C₆₁	semidirect
374. #1967	C₂×C₆₁	semidirect
375. #1963	C₂×C₆₁	semidirect
376. #1793	C₂×C₆₁	semidirect
377. #1792	C₂×C₆₁	semidirect
378. #1759	C₂×C₆₁	semidirect
401. C₂×C₄⋊Q₈	C₂²×C₆₁	semidirect
413. #428	C₂²×C₆₁	semidirect
414. #423	C₂²×C₆₁	semidirect
417. C₂×#104	C₂²×C₆₁	semidirect
420. #417	C₂²×C₆₁	semidirect
421. C₂×#101	C₂²×C₆₁	semidirect
423. C₄×M₁₆	C₂²×C₆₁	semidirect
426. #391	C₂²×C₆₁	semidirect
428. #376	C₂²×C₆₁	semidirect
429. #339	C₂²×C₆₁	semidirect
430. C₆₁	#25880	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×#3500	2
C₆₁×#3141	2
C₆₁×#3079	2
C₆₁×#2640	2
C₆₁×#2038	2
C₆₁×#2037	2
C₆₁×#1967	2
C₆₁×#1963	2
C₆₁×#1793	2
C₆₁×#1792	2
C₆₁×#1759	2
#25880	61

Subgroups — C61×#25880 #29529313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25880 #29529313