Subgroups — C₆₁×#22916 #29526349

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#480	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2361	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2344	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1934	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1925	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1893	C₂	1	1	yes	maximal, characteristic
10	C₆₁×#1862	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1852	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#107×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#102×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₆₁×#340	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#121×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₁	—	2	1	no	—
19	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₁	—	2	1	no	—
21	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#480	C₄	1	8	yes	complemented
24	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#390	C₂²	1	1	yes	characteristic
36	C₆₁×#389	C₂²	1	1	yes	characteristic
37	C₆₁×#389	C₂²	1	1	yes	characteristic
38	C₆₁×#389	C₂²	1	1	yes	characteristic
39	C₆₁×#388	—	2	1	no	—
40	C₆₁×#387	—	2	1	no	—
41	C₆₁×#387	—	2	1	no	—
42	C₆₁×#386	—	2	1	no	—
43	C₆₁×#385	—	2	1	no	—
44	C₆₁×#381	—	2	1	no	—
45	C₆₁×#380	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
49	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
50	C₆₁×#340	C₂²	1	1	yes	characteristic
51	C₆₁×#340	C₂²	1	1	yes	characteristic
52	C₆₁×#340	C₂²	1	1	yes	characteristic
53	C₆₁×#340	C₂²	1	1	yes	characteristic
54	C₆₁×#340	C₂²	1	1	yes	characteristic
55	C₆₁×#340	C₂²	1	1	yes	characteristic
56	C₆₁×#340	C₂²	1	1	yes	characteristic
57	C₆₁×#340	C₂²	1	1	yes	characteristic
58	C₆₁×#336	C₂²	1	1	yes	characteristic
59	C₆₁×#336	C₂²	1	1	yes	characteristic
60	C₂²×D₄×C₆₁	—	2	1	no	—
61	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₁	—	2	1	no	—
63	C₂³×C₄×C₆₁	—	2	1	no	—
64	C₂³×C₄×C₆₁	—	2	1	no	—
65	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₁	—	2	2	no	—
67	C₂²×C₈×C₆₁	—	2	2	no	—
68	C₂²×C₈×C₆₁	—	4	1	no	—
69	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
73	#121×C₆₁	—	2	2	no	—
74	#121×C₆₁	C₂×C₄	1	4	yes	—
75	C₄×D₄×C₆₁	—	2	2	no	—
76	C₄×D₄×C₆₁	—	4	1	no	—
77	C₄×D₄×C₆₁	—	4	4	no	—
78	#117×C₆₁	—	4	4	no	—
79	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
82	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
87	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	complemented
88	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₆₁	—	2	1	no	—
103	C₂×C₄²×C₆₁	—	2	1	no	—
104	C₂×C₄²×C₆₁	—	2	1	no	—
105	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
107	#107×C₆₁	—	2	2	no	—
108	#107×C₆₁	—	4	1	no	—
109	#107×C₆₁	C₂×C₄	1	4	yes	complemented
110	#107×C₆₁	C₂×C₄	1	4	yes	complemented
111	#102×C₆₁	—	2	2	no	—
112	#102×C₆₁	—	4	1	no	—
113	#102×C₆₁	C₂×C₄	1	4	yes	complemented
114	#102×C₆₁	C₂×C₄	1	4	yes	complemented
115	C₂²⋊C₈×C₆₁	C₂×C₄	1	4	yes	complemented
116	C₄×C₈×C₆₁	—	2	2	no	—
117	C₄×C₈×C₆₁	—	4	1	no	—
118	#95×C₆₁	—	2	1	no	—
119	#95×C₆₁	—	2	1	no	—
120	#95×C₆₁	—	2	1	no	—
121	#95×C₆₁	—	2	1	no	—
122	#95×C₆₁	—	2	1	no	—
123	#95×C₆₁	—	2	1	no	—
124	#95×C₆₁	—	2	1	no	—
125	#95×C₆₁	D₄	1	1	yes	characteristic
126	#95×C₆₁	D₄	1	1	yes	characteristic
127	C₂⁴×C₆₁	—	2	1	no	—
128	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₆₁	—	2	4	no	—
130	C₂×D₄×C₆₁	—	2	1	no	—
131	C₂×D₄×C₆₁	—	2	1	no	—
132	C₂×D₄×C₆₁	—	2	1	no	—
133	C₂×D₄×C₆₁	—	2	1	no	—
134	C₂×D₄×C₆₁	—	2	1	no	—
135	C₂×D₄×C₆₁	—	4	2	no	—
136	C₂×D₄×C₆₁	—	4	2	no	—
137	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₆₁	—	2	1	no	—
139	C₂²×C₄×C₆₁	—	2	1	no	—
140	C₂²×C₄×C₆₁	—	2	1	no	—
141	C₂²×C₄×C₆₁	—	2	1	no	—
142	C₂²×C₄×C₆₁	—	2	4	no	—
143	C₂²×C₄×C₆₁	—	2	1	no	—
144	C₂²×C₄×C₆₁	—	2	1	no	—
145	C₂²×C₄×C₆₁	—	2	1	no	—
146	C₂²×C₄×C₆₁	—	2	1	no	—
147	C₂²×C₄×C₆₁	—	2	1	no	—
148	C₂²×C₄×C₆₁	—	2	1	no	—
149	C₂²×C₄×C₆₁	—	2	1	no	—
150	C₂²×C₄×C₆₁	—	4	2	no	—
151	C₂²×C₄×C₆₁	—	4	2	no	—
152	C₂²×C₄×C₆₁	—	4	2	no	—
153	C₂²×C₄×C₆₁	—	4	2	no	—
154	C₂²×C₄×C₆₁	—	4	2	no	—
155	C₂²×C₄×C₆₁	—	4	2	no	—
156	C₂²×C₄×C₆₁	—	8	1	no	—
157	C₂²×C₄×C₆₁	—	8	1	no	—
158	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
159	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	#41	1	2	yes	—
165	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
166	C₂×C₈×C₆₁	—	2	1	no	—
167	C₂×C₈×C₆₁	—	2	1	no	—
168	C₂×C₈×C₆₁	—	2	1	no	—
169	C₂×C₈×C₆₁	—	2	1	no	—
170	C₂×C₈×C₆₁	—	2	1	no	—
171	C₂×C₈×C₆₁	—	2	1	no	—
172	C₂×C₈×C₆₁	—	2	1	no	—
173	C₂×C₈×C₆₁	—	2	1	no	—
174	C₂×C₈×C₆₁	—	2	1	no	—
175	C₂×C₈×C₆₁	—	4	4	no	—
176	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
177	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
178	C₂×C₈×C₆₁	#41	1	2	yes	—
179	C₄⋊C₄×C₆₁	—	2	4	no	—
180	C₄⋊C₄×C₆₁	—	2	1	no	—
181	C₄⋊C₄×C₆₁	—	2	1	no	—
182	C₄⋊C₄×C₆₁	—	2	1	no	—
183	C₄⋊C₄×C₆₁	—	2	1	no	—
184	C₄⋊C₄×C₆₁	—	2	1	no	—
185	C₄⋊C₄×C₆₁	—	2	1	no	—
186	C₄⋊C₄×C₆₁	—	2	1	no	—
187	C₄⋊C₄×C₆₁	—	2	1	no	—
188	C₄⋊C₄×C₆₁	—	2	1	no	—
189	C₄⋊C₄×C₆₁	—	2	1	no	—
190	C₄⋊C₄×C₆₁	—	4	2	no	—
191	C₄⋊C₄×C₆₁	—	4	2	no	—
192	C₄⋊C₄×C₆₁	—	4	2	no	—
193	C₄⋊C₄×C₆₁	—	4	2	no	—
194	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
195	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
196	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
197	C₄⋊C₄×C₆₁	#41	1	2	yes	—
198	C₂²⋊C₄×C₆₁	—	2	2	no	—
199	C₂²⋊C₄×C₆₁	—	2	4	no	—
200	C₂²⋊C₄×C₆₁	—	4	2	no	—
201	C₂²⋊C₄×C₆₁	—	4	2	no	—
202	C₂²⋊C₄×C₆₁	—	4	2	no	—
203	C₂²⋊C₄×C₆₁	—	8	1	no	—
204	C₂²⋊C₄×C₆₁	D₈	1	4	yes	—
205	C₄²×C₆₁	—	2	1	no	—
206	C₄²×C₆₁	—	2	1	no	—
207	C₄²×C₆₁	—	2	1	no	—
208	C₄²×C₆₁	—	2	1	no	—
209	C₄²×C₆₁	—	4	2	no	—
210	C₄²×C₆₁	—	4	2	no	—
211	C₄²×C₆₁	—	4	2	no	—
212	C₂³×C₆₁	—	2	1	no	—
213	C₂³×C₆₁	—	2	1	no	—
214	C₂³×C₆₁	—	2	1	no	—
215	C₂³×C₆₁	—	2	1	no	—
216	C₂³×C₆₁	—	2	2	no	—
217	C₂³×C₆₁	—	4	1	no	—
218	C₂³×C₆₁	—	4	1	no	—
219	C₂³×C₆₁	—	4	1	no	—
220	C₂³×C₆₁	—	4	1	no	—
221	C₂³×C₆₁	—	4	1	no	—
222	C₂³×C₆₁	—	4	1	no	—
223	C₂³×C₆₁	C₄×D₄	1	2	yes	—
224	C₂³×C₆₁	C₂×D₈	1	2	yes	—
225	C₂³×C₆₁	#135	1	2	yes	—
226	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
227	D₄×C₆₁	—	2	2	no	—
228	D₄×C₆₁	—	2	2	no	—
229	D₄×C₆₁	—	4	8	no	—
230	D₄×C₆₁	—	4	1	no	—
231	D₄×C₆₁	—	4	1	no	—
232	C₂×C₄×C₆₁	—	2	1	no	—
233	C₂×C₄×C₆₁	—	2	1	no	—
234	C₂×C₄×C₆₁	—	2	1	no	—
235	C₂×C₄×C₆₁	—	2	1	no	—
236	C₂×C₄×C₆₁	—	2	1	no	—
237	C₂×C₄×C₆₁	—	2	1	no	—
238	C₂×C₄×C₆₁	—	2	2	no	—
239	C₂×C₄×C₆₁	—	2	2	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	1	no	—
246	C₂×C₄×C₆₁	—	2	2	no	—
247	C₂×C₄×C₆₁	—	2	1	no	—
248	C₂×C₄×C₆₁	—	2	1	no	—
249	C₂×C₄×C₆₁	—	2	1	no	—
250	C₂×C₄×C₆₁	—	2	1	no	—
251	C₂×C₄×C₆₁	—	2	1	no	—
252	C₂×C₄×C₆₁	—	4	1	no	—
253	C₂×C₄×C₆₁	—	4	1	no	—
254	C₂×C₄×C₆₁	—	4	1	no	—
255	C₂×C₄×C₆₁	—	4	1	no	—
256	C₂×C₄×C₆₁	—	4	1	no	—
257	C₂×C₄×C₆₁	—	4	1	no	—
258	C₂×C₄×C₆₁	—	4	1	no	—
259	C₂×C₄×C₆₁	—	4	1	no	—
260	C₂×C₄×C₆₁	—	4	8	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	—	8	4	no	—
271	C₂×C₄×C₆₁	—	8	4	no	—
272	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
273	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
274	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
275	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
276	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
281	C₈×C₆₁	—	2	2	no	—
282	C₈×C₆₁	—	2	2	no	—
283	C₈×C₆₁	—	2	2	no	—
284	C₈×C₆₁	—	4	1	no	—
285	#22916	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
286	C₂²×C₆₁	—	2	2	no	—
287	C₂²×C₆₁	—	2	2	no	—
288	C₂²×C₆₁	—	2	2	no	—
289	C₂²×C₆₁	—	2	2	no	—
290	C₂²×C₆₁	—	2	2	no	—
291	C₂²×C₆₁	—	2	2	no	—
292	C₂²×C₆₁	—	4	2	no	—
293	C₂²×C₆₁	—	4	2	no	—
294	C₂²×C₆₁	—	4	2	no	—
295	C₂²×C₆₁	—	8	1	no	—
296	C₂²×C₆₁	—	8	1	no	—
297	C₂²×C₆₁	—	8	1	no	—
298	C₂²×C₆₁	—	8	1	no	—
299	C₂²×C₆₁	C₄×D₈	1	4	yes	—
300	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
301	C₂²×C₆₁	#546	1	1	yes	characteristic
302	C₂²×C₆₁	#547	1	1	yes	characteristic
303	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
304	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
305	C₂²×C₆₁	#577	1	1	yes	characteristic
306	C₂²×C₆₁	#578	1	1	yes	characteristic
307	C₄×C₆₁	—	2	2	no	—
308	C₄×C₆₁	—	2	2	no	—
309	C₄×C₆₁	—	2	2	no	—
310	C₄×C₆₁	—	4	1	no	—
311	C₄×C₆₁	—	4	2	no	—
312	C₄×C₆₁	—	4	2	no	—
313	C₄×C₆₁	—	4	2	no	—
314	C₄×C₆₁	—	4	2	no	—
315	C₄×C₆₁	—	4	1	no	—
316	C₄×C₆₁	—	8	1	no	—
317	C₄×C₆₁	—	8	1	no	—
318	C₄×C₆₁	—	8	1	no	—
319	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
320	C₄×C₆₁	#520	1	1	yes	characteristic
321	C₄×C₆₁	#551	1	1	yes	characteristic
322	C₄×C₆₁	#552	1	1	yes	characteristic
323	C₂×#480	C₂×C₆₁	1	1	yes	characteristic
324	#2361	C₂×C₆₁	1	1	yes	characteristic
325	#2344	C₂×C₆₁	1	1	yes	characteristic, complemented
326	#1942	C₂×C₆₁	1	1	yes	characteristic, complemented
327	#1934	C₂×C₆₁	1	1	yes	characteristic, complemented
328	#1925	C₂×C₆₁	1	1	yes	characteristic, complemented
329	#1918	C₂×C₆₁	1	1	yes	characteristic, complemented
330	#1918	C₂×C₆₁	1	1	yes	characteristic, complemented
331	#1893	C₂×C₆₁	1	1	yes	characteristic
332	#1862	C₂×C₆₁	1	1	yes	characteristic, complemented
333	#1852	C₂×C₆₁	1	1	yes	characteristic, complemented
334	C₄×#107	C₂×C₆₁	1	1	yes	characteristic, complemented
335	C₄×#102	C₂×C₆₁	1	1	yes	characteristic, complemented
336	C₈×C₂²⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
337	C₂×#340	C₂×C₆₁	1	1	yes	characteristic, complemented
338	C₂×C₆₁	—	2	4	no	—
339	C₂×C₆₁	—	8	2	no	—
340	C₂×C₆₁	#2359	1	1	yes	characteristic
341	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
342	C₂×C₆₁	#2947	1	1	yes	characteristic
343	C₂×C₆₁	#3294	1	1	yes	characteristic
344	C₂×C₆₁	#3300	1	1	yes	characteristic
345	C₂×C₆₁	#3334	1	1	yes	characteristic
346	C₂×C₆₁	#3337	1	1	yes	characteristic
347	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
348	C₂×#121	C₂²×C₆₁	1	1	yes	characteristic
349	C₂×C₄×D₄	—	2	1	no	—
350	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
351	C₂×#117	—	2	1	no	—
352	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
353	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
354	#480	C₄×C₆₁	1	8	yes	complemented
355	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
356	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
357	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
358	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
359	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
360	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
361	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
362	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
363	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
364	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
365	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
366	#390	C₂²×C₆₁	1	1	yes	characteristic
367	#389	C₂²×C₆₁	1	1	yes	characteristic
368	#389	C₂²×C₆₁	1	1	yes	characteristic
369	#389	C₂²×C₆₁	1	1	yes	characteristic
370	#388	—	2	1	no	—
371	#387	—	2	1	no	—
372	#387	—	2	1	no	—
373	#386	—	2	1	no	—
374	#385	—	2	1	no	—
375	#381	—	2	1	no	—
376	#380	C₂²×C₆₁	1	1	yes	characteristic
377	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
378	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
379	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
380	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
381	#340	C₂²×C₆₁	1	1	yes	characteristic
382	#340	C₂²×C₆₁	1	1	yes	characteristic
383	#340	C₂²×C₆₁	1	1	yes	characteristic
384	#340	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	#22916	1	1	yes	characteristic, direct product factor, Sylow
392	C₂²×D₄	—	2	1	no	—
393	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
394	C₂³×C₄	—	2	1	no	—
395	C₂³×C₄	—	2	1	no	—
396	C₂³×C₄	—	2	1	no	—
397	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
398	C₂²×C₈	—	2	2	no	—
399	C₂²×C₈	—	2	2	no	—
400	C₂²×C₈	—	4	1	no	—
401	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
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	#121	—	2	2	no	—
406	#121	C₂×C₄×C₆₁	1	4	yes	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	4	1	no	—
409	C₄×D₄	—	4	4	no	—
410	#117	—	4	4	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	2	no	—
413	C₂×C₄⋊C₄	—	2	2	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	4	1	no	—
419	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	complemented
420	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₂²⋊C₄	—	2	1	no	—
432	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
434	C₂×C₄²	—	2	1	no	—
435	C₂×C₄²	—	2	1	no	—
436	C₂×C₄²	—	2	1	no	—
437	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
439	#107	—	2	2	no	—
440	#107	—	4	1	no	—
441	#107	C₂×C₄×C₆₁	1	4	yes	complemented
442	#107	C₂×C₄×C₆₁	1	4	yes	complemented
443	#102	—	2	2	no	—
444	#102	—	4	1	no	—
445	#102	C₂×C₄×C₆₁	1	4	yes	complemented
446	#102	C₂×C₄×C₆₁	1	4	yes	complemented
447	C₂²⋊C₈	C₂×C₄×C₆₁	1	4	yes	complemented
448	C₄×C₈	—	2	2	no	—
449	C₄×C₈	—	4	1	no	—
450	#95	—	2	1	no	—
451	#95	—	2	1	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	#95	D₄×C₆₁	1	1	yes	characteristic
458	#95	D₄×C₆₁	1	1	yes	characteristic
459	C₂⁴	—	2	1	no	—
460	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	4	2	no	—
468	C₂×D₄	—	4	2	no	—
469	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	4	2	no	—
483	C₂²×C₄	—	4	2	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	—	4	2	no	—
488	C₂²×C₄	—	8	1	no	—
489	C₂²×C₄	—	8	1	no	—
490	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
491	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
496	C₂²×C₄	#41×C₆₁	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	4	4	no	—
508	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
509	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
510	C₂×C₈	#41×C₆₁	1	2	yes	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	4	2	no	—
523	C₄⋊C₄	—	4	2	no	—
524	C₄⋊C₄	—	4	2	no	—
525	C₄⋊C₄	—	4	2	no	—
526	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
527	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
528	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
529	C₄⋊C₄	#41×C₆₁	1	2	yes	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	4	2	no	—
533	C₂²⋊C₄	—	4	2	no	—
534	C₂²⋊C₄	—	4	2	no	—
535	C₂²⋊C₄	—	8	1	no	—
536	C₂²⋊C₄	D₈×C₆₁	1	4	yes	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	4	2	no	—
542	C₄²	—	4	2	no	—
543	C₄²	—	4	2	no	—
544	C₂³	—	2	1	no	—
545	C₂³	—	2	1	no	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	4	1	no	—
550	C₂³	—	4	1	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	4	1	no	—
553	C₂³	—	4	1	no	—
554	C₂³	—	4	1	no	—
555	C₂³	C₄×D₄×C₆₁	1	2	yes	—
556	C₂³	C₂×D₈×C₆₁	1	2	yes	—
557	C₂³	#135×C₆₁	1	2	yes	—
558	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
559	D₄	—	2	2	no	—
560	D₄	—	2	2	no	—
561	D₄	—	4	8	no	—
562	D₄	—	4	1	no	—
563	D₄	—	4	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	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	1	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	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	8	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	—	8	4	no	—
603	C₂×C₄	—	8	4	no	—
604	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
605	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
606	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
607	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
608	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
609	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
610	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
611	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
612	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
613	C₈	—	2	2	no	—
614	C₈	—	2	2	no	—
615	C₈	—	2	2	no	—
616	C₈	—	4	1	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	2	2	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	2	no	—
626	C₂²	—	8	1	no	—
627	C₂²	—	8	1	no	—
628	C₂²	—	8	1	no	—
629	C₂²	—	8	1	no	—
630	C₂²	C₄×D₈×C₆₁	1	4	yes	—
631	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
632	C₂²	C₆₁×#546	1	1	yes	characteristic
633	C₂²	C₆₁×#547	1	1	yes	characteristic
634	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
635	C₂²	C₂×#135×C₆₁	1	1	yes	characteristic
636	C₂²	C₆₁×#577	1	1	yes	characteristic
637	C₂²	C₆₁×#578	1	1	yes	characteristic
638	C₄	—	2	2	no	—
639	C₄	—	2	2	no	—
640	C₄	—	2	2	no	—
641	C₄	—	4	2	no	—
642	C₄	—	4	1	no	—
643	C₄	—	4	1	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	8	1	no	—
648	C₄	—	8	1	no	—
649	C₄	—	8	1	no	—
650	C₄	C₄×#41×C₆₁	1	1	yes	characteristic
651	C₄	C₆₁×#520	1	1	yes	characteristic
652	C₄	C₆₁×#551	1	1	yes	characteristic
653	C₄	C₆₁×#552	1	1	yes	characteristic
654	C₂	—	2	4	no	—
655	C₂	—	8	2	no	—
656	C₂	C₆₁×#2359	1	1	yes	characteristic
657	C₂	C₂×C₄×D₈×C₆₁	1	1	yes	characteristic
658	C₂	C₆₁×#2947	1	1	yes	characteristic
659	C₂	C₆₁×#3294	1	1	yes	characteristic
660	C₂	C₆₁×#3300	1	1	yes	characteristic
661	C₂	C₆₁×#3334	1	1	yes	characteristic
662	C₂	C₆₁×#3337	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	C₂×#121×C₆₁	C₂²
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
3. C₆₁×#2344	C₂	semidirect
4. C₆₁×#1942	C₂	semidirect
5. C₆₁×#1934	C₂	semidirect
6. C₆₁×#1925	C₂	semidirect
7. C₆₁×#1918	C₂	semidirect
8. C₆₁×#1918	C₂	semidirect
10. C₆₁×#1862	C₂	semidirect
11. C₆₁×#1852	C₂	semidirect
12. C₄×#107×C₆₁	C₂	semidirect
13. C₄×#102×C₆₁	C₂	semidirect
14. C₈×C₂²⋊C₄×C₆₁	C₂	semidirect
15. C₂×C₆₁×#340	C₂	semidirect
23. C₆₁×#480	C₄	semidirect
87. C₂×C₄⋊C₄×C₆₁	C₂×C₄	semidirect
109. #107×C₆₁	C₂×C₄	semidirect
110. #107×C₆₁	C₂×C₄	semidirect
113. #102×C₆₁	C₂×C₄	semidirect
114. #102×C₆₁	C₂×C₄	semidirect
115. C₂²⋊C₈×C₆₁	C₂×C₄	semidirect
285. #22916	C₆₁	direct (N × H)
325. #2344	C₂×C₆₁	semidirect
326. #1942	C₂×C₆₁	semidirect
327. #1934	C₂×C₆₁	semidirect
328. #1925	C₂×C₆₁	semidirect
329. #1918	C₂×C₆₁	semidirect
330. #1918	C₂×C₆₁	semidirect
332. #1862	C₂×C₆₁	semidirect
333. #1852	C₂×C₆₁	semidirect
334. C₄×#107	C₂×C₆₁	semidirect
335. C₄×#102	C₂×C₆₁	semidirect
336. C₈×C₂²⋊C₄	C₂×C₆₁	semidirect
337. C₂×#340	C₂×C₆₁	semidirect
354. #480	C₄×C₆₁	semidirect
391. C₆₁	#22916	direct (N × H)
419. C₂×C₄⋊C₄	C₂×C₄×C₆₁	semidirect
441. #107	C₂×C₄×C₆₁	semidirect
442. #107	C₂×C₄×C₆₁	semidirect
445. #102	C₂×C₄×C₆₁	semidirect
446. #102	C₂×C₄×C₆₁	semidirect
447. C₂²⋊C₈	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×#480	2
C₆₁×#2361	2
C₆₁×#2344	2
C₆₁×#1942	2
C₆₁×#1934	2
C₆₁×#1925	2
C₆₁×#1918	2
C₆₁×#1893	2
C₆₁×#1862	2
C₆₁×#1852	2
C₄×#107×C₆₁	2
C₄×#102×C₆₁	2
C₈×C₂²⋊C₄×C₆₁	2
C₂×C₆₁×#340	2
#22916	61

Subgroups — C61×#22916 #29526349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22916 #29526349