Subgroups — C₆₁×#22972 #29526405

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#478	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2370	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2349	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1991	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1949	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1944	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1919	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1864	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1862	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₁×#1762	C₂	1	1	yes	maximal, characteristic
14	C₆₁×#1762	C₂	1	1	yes	maximal, characteristic
15	C₄×#102×C₆₁	C₂	1	1	yes	maximal, characteristic
16	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#478	C₄	1	8	yes	complemented
21	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#102×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×#97×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₆₁×#389	—	2	1	no	—
34	C₆₁×#389	C₂²	1	1	yes	characteristic
35	C₆₁×#389	C₂²	1	1	yes	characteristic
36	C₆₁×#389	C₂²	1	1	yes	characteristic
37	C₆₁×#389	C₂²	1	1	yes	characteristic
38	C₆₁×#387	—	2	1	no	—
39	C₆₁×#387	—	2	1	no	—
40	C₆₁×#385	—	2	1	no	—
41	C₆₁×#385	—	2	1	no	—
42	C₆₁×#382	C₂²	1	1	yes	characteristic
43	C₆₁×#380	—	2	1	no	—
44	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
45	C₄×C₄⋊C₄×C₆₁	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	C₂×#95×C₆₁	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₁	—	2	1	no	—
62	C₂³×C₄×C₆₁	—	2	1	no	—
63	C₂³×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	#125×C₆₁	—	2	2	no	—
69	#125×C₆₁	C₂×C₄	1	4	yes	complemented
70	C₄×D₄×C₆₁	—	2	2	no	—
71	C₄×D₄×C₆₁	—	2	2	no	—
72	C₄×D₄×C₆₁	—	2	2	no	—
73	C₄×D₄×C₆₁	—	4	1	no	—
74	C₄×D₄×C₆₁	—	4	1	no	—
75	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
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×C₂²⋊C₄×C₆₁	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
94	C₂×C₄²×C₆₁	—	2	1	no	—
95	C₂×C₄²×C₆₁	—	2	1	no	—
96	C₂×C₄²×C₆₁	—	2	1	no	—
97	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
100	#107×C₆₁	—	4	1	no	—
101	#107×C₆₁	C₂×C₄	1	4	yes	complemented
102	#106×C₆₁	—	4	1	no	—
103	#106×C₆₁	C₂×C₄	1	4	yes	complemented
104	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	complemented
105	#102×C₆₁	—	2	2	no	—
106	#102×C₆₁	C₂×C₄	1	4	yes	—
107	#102×C₆₁	C₂×C₄	1	4	yes	—
108	#102×C₆₁	Q₈	1	4	yes	—
109	#97×C₆₁	—	4	1	no	—
110	#97×C₆₁	—	4	1	no	—
111	C₄×C₈×C₆₁	—	2	2	no	—
112	#95×C₆₁	—	2	1	no	—
113	#95×C₆₁	—	2	1	no	—
114	#95×C₆₁	—	2	1	no	—
115	#95×C₆₁	—	2	1	no	—
116	#95×C₆₁	—	2	1	no	—
117	#95×C₆₁	—	2	1	no	—
118	#95×C₆₁	—	2	1	no	—
119	#95×C₆₁	—	2	1	no	—
120	#95×C₆₁	—	2	1	no	—
121	#95×C₆₁	D₄	1	1	yes	characteristic
122	#95×C₆₁	D₄	1	1	yes	characteristic
123	C₂⁴×C₆₁	—	2	1	no	—
124	C₂×D₄×C₆₁	—	2	1	no	—
125	C₂×D₄×C₆₁	—	2	1	no	—
126	C₂×D₄×C₆₁	—	2	1	no	—
127	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
128	C₂×D₄×C₆₁	C₂×Q₈	1	2	yes	—
129	C₂×D₄×C₆₁	#41	1	2	yes	—
130	C₂²×C₄×C₆₁	—	2	1	no	—
131	C₂²×C₄×C₆₁	—	2	1	no	—
132	C₂²×C₄×C₆₁	—	2	1	no	—
133	C₂²×C₄×C₆₁	—	2	1	no	—
134	C₂²×C₄×C₆₁	—	2	4	no	—
135	C₂²×C₄×C₆₁	—	2	1	no	—
136	C₂²×C₄×C₆₁	—	2	1	no	—
137	C₂²×C₄×C₆₁	—	2	1	no	—
138	C₂²×C₄×C₆₁	—	4	2	no	—
139	C₂²×C₄×C₆₁	—	4	2	no	—
140	C₂²×C₄×C₆₁	—	4	2	no	—
141	C₂²×C₄×C₆₁	—	4	2	no	—
142	C₂²×C₄×C₆₁	—	8	1	no	—
143	C₂²×C₄×C₆₁	—	8	1	no	—
144	C₂²×C₄×C₆₁	—	8	1	no	—
145	C₂²×C₄×C₆₁	—	8	1	no	—
146	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
153	C₂×C₈×C₆₁	—	2	1	no	—
154	C₂×C₈×C₆₁	—	2	1	no	—
155	C₂×C₈×C₆₁	—	2	1	no	—
156	C₂×C₈×C₆₁	—	2	1	no	—
157	C₂×C₈×C₆₁	—	2	1	no	—
158	C₂×C₈×C₆₁	—	2	1	no	—
159	C₂×C₈×C₆₁	—	2	1	no	—
160	C₂×C₈×C₆₁	—	2	1	no	—
161	C₂×C₈×C₆₁	—	2	1	no	—
162	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
163	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
164	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
165	C₄⋊C₄×C₆₁	—	2	4	no	—
166	C₄⋊C₄×C₆₁	—	2	4	no	—
167	C₄⋊C₄×C₆₁	—	2	1	no	—
168	C₄⋊C₄×C₆₁	—	2	1	no	—
169	C₄⋊C₄×C₆₁	—	2	1	no	—
170	C₄⋊C₄×C₆₁	—	2	1	no	—
171	C₄⋊C₄×C₆₁	—	2	1	no	—
172	C₄⋊C₄×C₆₁	—	2	1	no	—
173	C₄⋊C₄×C₆₁	—	2	1	no	—
174	C₄⋊C₄×C₆₁	—	2	1	no	—
175	C₄⋊C₄×C₆₁	—	2	1	no	—
176	C₄⋊C₄×C₆₁	—	2	1	no	—
177	C₄⋊C₄×C₆₁	—	4	1	no	—
178	C₄⋊C₄×C₆₁	—	4	1	no	—
179	C₄⋊C₄×C₆₁	—	4	2	no	—
180	C₄⋊C₄×C₆₁	—	8	1	no	—
181	C₄⋊C₄×C₆₁	—	8	1	no	—
182	C₄⋊C₄×C₆₁	—	8	1	no	—
183	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
184	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
185	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
186	C₄⋊C₄×C₆₁	C₂×Q₈	1	2	yes	—
187	C₂²⋊C₄×C₆₁	—	2	4	no	—
188	C₂²⋊C₄×C₆₁	—	4	2	no	—
189	C₂²⋊C₄×C₆₁	—	4	2	no	—
190	C₄²×C₆₁	—	2	1	no	—
191	C₄²×C₆₁	—	2	1	no	—
192	C₄²×C₆₁	—	2	1	no	—
193	C₄²×C₆₁	—	2	1	no	—
194	C₄²×C₆₁	—	4	2	no	—
195	C₄²×C₆₁	—	8	1	no	—
196	C₄²×C₆₁	—	8	1	no	—
197	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
198	C₄²×C₆₁	#41	1	2	yes	—
199	C₂³×C₆₁	—	2	2	no	—
200	C₂³×C₆₁	—	2	2	no	—
201	C₂³×C₆₁	—	2	2	no	—
202	C₂³×C₆₁	—	4	1	no	—
203	C₂³×C₆₁	—	4	1	no	—
204	C₂³×C₆₁	—	4	1	no	—
205	C₂³×C₆₁	—	4	1	no	—
206	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
207	D₄×C₆₁	—	2	2	no	—
208	D₄×C₆₁	—	2	2	no	—
209	D₄×C₆₁	—	2	2	no	—
210	D₄×C₆₁	C₄×Q₈	1	4	yes	—
211	C₂×C₄×C₆₁	—	2	1	no	—
212	C₂×C₄×C₆₁	—	2	1	no	—
213	C₂×C₄×C₆₁	—	2	1	no	—
214	C₂×C₄×C₆₁	—	2	1	no	—
215	C₂×C₄×C₆₁	—	2	2	no	—
216	C₂×C₄×C₆₁	—	2	2	no	—
217	C₂×C₄×C₆₁	—	2	2	no	—
218	C₂×C₄×C₆₁	—	2	1	no	—
219	C₂×C₄×C₆₁	—	2	1	no	—
220	C₂×C₄×C₆₁	—	2	1	no	—
221	C₂×C₄×C₆₁	—	2	1	no	—
222	C₂×C₄×C₆₁	—	2	2	no	—
223	C₂×C₄×C₆₁	—	2	1	no	—
224	C₂×C₄×C₆₁	—	2	1	no	—
225	C₂×C₄×C₆₁	—	2	1	no	—
226	C₂×C₄×C₆₁	—	2	1	no	—
227	C₂×C₄×C₆₁	—	2	1	no	—
228	C₂×C₄×C₆₁	—	2	1	no	—
229	C₂×C₄×C₆₁	—	2	1	no	—
230	C₂×C₄×C₆₁	—	2	1	no	—
231	C₂×C₄×C₆₁	—	4	1	no	—
232	C₂×C₄×C₆₁	—	4	1	no	—
233	C₂×C₄×C₆₁	—	4	1	no	—
234	C₂×C₄×C₆₁	—	4	1	no	—
235	C₂×C₄×C₆₁	—	4	1	no	—
236	C₂×C₄×C₆₁	—	4	1	no	—
237	C₂×C₄×C₆₁	—	4	1	no	—
238	C₂×C₄×C₆₁	—	4	1	no	—
239	C₂×C₄×C₆₁	—	4	1	no	—
240	C₂×C₄×C₆₁	—	4	1	no	—
241	C₂×C₄×C₆₁	—	4	1	no	—
242	C₂×C₄×C₆₁	—	4	1	no	—
243	C₂×C₄×C₆₁	—	4	1	no	—
244	C₂×C₄×C₆₁	—	4	1	no	—
245	C₂×C₄×C₆₁	—	4	1	no	—
246	C₂×C₄×C₆₁	—	4	1	no	—
247	C₂×C₄×C₆₁	—	4	1	no	—
248	C₂×C₄×C₆₁	—	8	4	no	—
249	C₂×C₄×C₆₁	—	8	4	no	—
250	C₂×C₄×C₆₁	—	8	4	no	—
251	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
252	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
253	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
255	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
257	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
259	C₈×C₆₁	—	2	2	no	—
260	C₈×C₆₁	—	2	2	no	—
261	C₈×C₆₁	—	2	2	no	—
262	C₈×C₆₁	—	4	1	no	—
263	#22972	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂²×C₆₁	—	2	4	no	—
265	C₂²×C₆₁	—	4	2	no	—
266	C₂²×C₆₁	—	4	2	no	—
267	C₂²×C₆₁	—	4	2	no	—
268	C₂²×C₆₁	—	4	2	no	—
269	C₂²×C₆₁	—	4	2	no	—
270	C₂²×C₆₁	—	4	2	no	—
271	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
272	C₂²×C₆₁	#546	1	1	yes	characteristic
273	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
274	C₂²×C₆₁	#575	1	1	yes	characteristic
275	C₂²×C₆₁	#575	1	1	yes	characteristic
276	C₂²×C₆₁	#576	1	1	yes	characteristic
277	C₂²×C₆₁	#577	1	1	yes	characteristic
278	C₄×C₆₁	—	2	2	no	—
279	C₄×C₆₁	—	2	2	no	—
280	C₄×C₆₁	—	4	2	no	—
281	C₄×C₆₁	—	4	2	no	—
282	C₄×C₆₁	—	4	2	no	—
283	C₄×C₆₁	—	4	2	no	—
284	C₄×C₆₁	—	4	1	no	—
285	C₄×C₆₁	—	4	1	no	—
286	C₄×C₆₁	—	4	1	no	—
287	C₄×C₆₁	—	4	1	no	—
288	C₄×C₆₁	—	8	1	no	—
289	C₄×C₆₁	—	8	1	no	—
290	C₄×C₆₁	—	8	1	no	—
291	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
292	C₄×C₆₁	#520	1	1	yes	characteristic
293	C₄×C₆₁	#552	1	1	yes	characteristic
294	C₄×C₆₁	#554	1	1	yes	characteristic
295	C₂×#478	C₂×C₆₁	1	1	yes	characteristic
296	#2370	C₂×C₆₁	1	1	yes	characteristic, complemented
297	#2349	C₂×C₆₁	1	1	yes	characteristic
298	#1991	C₂×C₆₁	1	1	yes	characteristic, complemented
299	#1949	C₂×C₆₁	1	1	yes	characteristic, complemented
300	#1944	C₂×C₆₁	1	1	yes	characteristic, complemented
301	#1943	C₂×C₆₁	1	1	yes	characteristic, complemented
302	#1919	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#1918	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#1864	C₂×C₆₁	1	1	yes	characteristic
305	#1862	C₂×C₆₁	1	1	yes	characteristic
306	#1770	C₂×C₆₁	1	1	yes	characteristic, complemented
307	#1762	C₂×C₆₁	1	1	yes	characteristic
308	#1762	C₂×C₆₁	1	1	yes	characteristic
309	C₄×#102	C₂×C₆₁	1	1	yes	characteristic
310	C₂×C₆₁	—	4	4	no	—
311	C₂×C₆₁	#2360	1	1	yes	characteristic
312	C₂×C₆₁	#2943	1	1	yes	characteristic
313	C₂×C₆₁	#2946	1	1	yes	characteristic
314	C₂×C₆₁	#3322	1	1	yes	characteristic
315	C₂×C₆₁	#3323	1	1	yes	characteristic
316	C₂×C₆₁	#3388	1	1	yes	characteristic
317	C₂×C₆₁	#3389	1	1	yes	characteristic
318	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
319	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
320	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
321	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
322	#478	C₄×C₆₁	1	8	yes	complemented
323	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
324	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
325	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
326	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
327	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
328	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
329	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
330	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
331	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
332	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
333	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
334	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
335	#389	—	2	1	no	—
336	#389	C₂²×C₆₁	1	1	yes	characteristic
337	#389	C₂²×C₆₁	1	1	yes	characteristic
338	#389	C₂²×C₆₁	1	1	yes	characteristic
339	#389	C₂²×C₆₁	1	1	yes	characteristic
340	#387	—	2	1	no	—
341	#387	—	2	1	no	—
342	#385	—	2	1	no	—
343	#385	—	2	1	no	—
344	#382	C₂²×C₆₁	1	1	yes	characteristic
345	#380	—	2	1	no	—
346	C₄×C₄⋊C₄	—	2	1	no	—
347	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
348	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
350	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
351	C₂×#95	—	2	1	no	—
352	#340	C₂²×C₆₁	1	1	yes	characteristic
353	#340	C₂²×C₆₁	1	1	yes	characteristic
354	#340	C₂²×C₆₁	1	1	yes	characteristic
355	#340	C₂²×C₆₁	1	1	yes	characteristic
356	#340	C₂²×C₆₁	1	1	yes	characteristic
357	#340	C₂²×C₆₁	1	1	yes	characteristic
358	#340	C₂²×C₆₁	1	1	yes	characteristic
359	#340	C₂²×C₆₁	1	1	yes	characteristic
360	#336	C₂²×C₆₁	1	1	yes	characteristic
361	#336	C₂²×C₆₁	1	1	yes	characteristic
362	C₆₁	#22972	1	1	yes	characteristic, direct product factor, Sylow
363	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
364	C₂³×C₄	—	2	1	no	—
365	C₂³×C₄	—	2	1	no	—
366	C₂³×C₄	—	2	1	no	—
367	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
370	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
371	#125	—	2	2	no	—
372	#125	C₂×C₄×C₆₁	1	4	yes	complemented
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	4	1	no	—
377	C₄×D₄	—	4	1	no	—
378	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
386	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	2	1	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	—	2	1	no	—
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
403	#107	—	4	1	no	—
404	#107	C₂×C₄×C₆₁	1	4	yes	complemented
405	#106	—	4	1	no	—
406	#106	C₂×C₄×C₆₁	1	4	yes	complemented
407	C₄⋊C₈	C₂×C₄×C₆₁	1	4	yes	complemented
408	#102	—	2	2	no	—
409	#102	C₂×C₄×C₆₁	1	4	yes	—
410	#102	C₂×C₄×C₆₁	1	4	yes	—
411	#102	Q₈×C₆₁	1	4	yes	—
412	#97	—	4	1	no	—
413	#97	—	4	1	no	—
414	C₄×C₈	—	2	2	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	—	2	1	no	—
418	#95	—	2	1	no	—
419	#95	—	2	1	no	—
420	#95	—	2	1	no	—
421	#95	—	2	1	no	—
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	D₄×C₆₁	1	1	yes	characteristic
425	#95	D₄×C₆₁	1	1	yes	characteristic
426	C₂⁴	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
431	C₂×D₄	C₂×Q₈×C₆₁	1	2	yes	—
432	C₂×D₄	#41×C₆₁	1	2	yes	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	8	1	no	—
446	C₂²×C₄	—	8	1	no	—
447	C₂²×C₄	—	8	1	no	—
448	C₂²×C₄	—	8	1	no	—
449	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
466	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
467	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	—	4	2	no	—
483	C₄⋊C₄	—	8	1	no	—
484	C₄⋊C₄	—	8	1	no	—
485	C₄⋊C₄	—	8	1	no	—
486	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
487	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
488	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
489	C₄⋊C₄	C₂×Q₈×C₆₁	1	2	yes	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	4	2	no	—
498	C₄²	—	8	1	no	—
499	C₄²	—	8	1	no	—
500	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
501	C₄²	#41×C₆₁	1	2	yes	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	2	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	4	1	no	—
506	C₂³	—	4	1	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
510	D₄	—	2	2	no	—
511	D₄	—	2	2	no	—
512	D₄	—	2	2	no	—
513	D₄	C₄×Q₈×C₆₁	1	4	yes	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	8	4	no	—
552	C₂×C₄	—	8	4	no	—
553	C₂×C₄	—	8	4	no	—
554	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
555	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
556	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
557	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
558	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
559	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
560	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
561	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
562	C₈	—	2	2	no	—
563	C₈	—	2	2	no	—
564	C₈	—	2	2	no	—
565	C₈	—	4	1	no	—
566	C₂²	—	2	4	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	2	no	—
570	C₂²	—	4	2	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	2	no	—
573	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
574	C₂²	C₆₁×#546	1	1	yes	characteristic
575	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
576	C₂²	C₆₁×#575	1	1	yes	characteristic
577	C₂²	C₆₁×#575	1	1	yes	characteristic
578	C₂²	C₆₁×#576	1	1	yes	characteristic
579	C₂²	C₆₁×#577	1	1	yes	characteristic
580	C₄	—	2	2	no	—
581	C₄	—	2	2	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	—	4	1	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	2	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	2	no	—
589	C₄	—	4	2	no	—
590	C₄	—	8	1	no	—
591	C₄	—	8	1	no	—
592	C₄	—	8	1	no	—
593	C₄	C₂×C₄×Q₈×C₆₁	1	1	yes	characteristic
594	C₄	C₆₁×#520	1	1	yes	characteristic
595	C₄	C₆₁×#552	1	1	yes	characteristic
596	C₄	C₆₁×#554	1	1	yes	characteristic
597	C₂	—	4	4	no	—
598	C₂	C₆₁×#2360	1	1	yes	characteristic
599	C₂	C₆₁×#2943	1	1	yes	characteristic
600	C₂	C₆₁×#2946	1	1	yes	characteristic
601	C₂	C₆₁×#3322	1	1	yes	characteristic
602	C₂	C₆₁×#3323	1	1	yes	characteristic
603	C₂	C₆₁×#3388	1	1	yes	characteristic
604	C₂	C₆₁×#3389	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	C₂²×D₄×C₆₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄×C₄⋊C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2370	C₂	semidirect
4. C₆₁×#1991	C₂	semidirect
5. C₆₁×#1949	C₂	semidirect
6. C₆₁×#1944	C₂	semidirect
7. C₆₁×#1943	C₂	semidirect
8. C₆₁×#1919	C₂	semidirect
9. C₆₁×#1918	C₂	semidirect
12. C₆₁×#1770	C₂	semidirect
20. C₆₁×#478	C₄	semidirect
69. #125×C₆₁	C₂×C₄	semidirect
101. #107×C₆₁	C₂×C₄	semidirect
103. #106×C₆₁	C₂×C₄	semidirect
104. C₄⋊C₈×C₆₁	C₂×C₄	semidirect
263. #22972	C₆₁	direct (N × H)
296. #2370	C₂×C₆₁	semidirect
298. #1991	C₂×C₆₁	semidirect
299. #1949	C₂×C₆₁	semidirect
300. #1944	C₂×C₆₁	semidirect
301. #1943	C₂×C₆₁	semidirect
302. #1919	C₂×C₆₁	semidirect
303. #1918	C₂×C₆₁	semidirect
306. #1770	C₂×C₆₁	semidirect
322. #478	C₄×C₆₁	semidirect
362. C₆₁	#22972	direct (N × H)
372. #125	C₂×C₄×C₆₁	semidirect
404. #107	C₂×C₄×C₆₁	semidirect
406. #106	C₂×C₄×C₆₁	semidirect
407. C₄⋊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×#478	2
C₆₁×#2370	2
C₆₁×#2349	2
C₆₁×#1991	2
C₆₁×#1949	2
C₆₁×#1944	2
C₆₁×#1943	2
C₆₁×#1919	2
C₆₁×#1918	2
C₆₁×#1864	2
C₆₁×#1862	2
C₆₁×#1770	2
C₆₁×#1762	2
C₄×#102×C₆₁	2
#22972	61

Subgroups — C61×#22972 #29526405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22972 #29526405