Subgroups — C₆₁×#22292 #29525725

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2396	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1917	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1917	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1851	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1851	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1798	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1752	C₂	1	1	yes	maximal, characteristic
14	C₈×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
15	C₆₁×#1729	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#400	C₄	1	1	yes	characteristic
33	C₆₁×#399	C₄	1	1	yes	characteristic
34	C₆₁×#398	C₄	1	1	yes	characteristic
35	C₆₁×#398	C₄	1	1	yes	characteristic
36	C₆₁×#396	C₄	1	1	yes	characteristic
37	C₆₁×#396	C₄	1	1	yes	characteristic
38	C₆₁×#395	C₄	1	1	yes	characteristic
39	C₆₁×#393	C₄	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
45	C₆₁×#336	C₂²	1	1	yes	characteristic
46	C₆₁×#336	C₂²	1	1	yes	characteristic
47	C₆₁×#336	C₂²	1	1	yes	characteristic
48	C₆₁×#336	C₂²	1	1	yes	characteristic
49	C₆₁×#336	C₂²	1	1	yes	characteristic
50	C₆₁×#336	C₂²	1	1	yes	characteristic
51	C₆₁×#336	C₂²	1	1	yes	characteristic
52	C₆₁×#336	C₂²	1	1	yes	characteristic
53	C₆₁×#336	C₂²	1	1	yes	characteristic
54	C₆₁×#336	C₂²	1	1	yes	characteristic
55	C₆₁×#336	C₂²	1	1	yes	characteristic
56	C₆₁×#336	C₂²	1	1	yes	characteristic
57	C₆₁×#336	C₂²	1	1	yes	characteristic
58	C₆₁×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₆₁	—	2	2	no	—
62	C₂×M₁₆×C₆₁	—	4	1	no	—
63	C₂×M₁₆×C₆₁	—	4	1	no	—
64	C₂²×C₈×C₆₁	—	4	1	no	—
65	C₂²×C₈×C₆₁	—	4	1	no	—
66	C₂²×C₈×C₆₁	Q₈	1	4	yes	complemented
67	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
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	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₄×Q₈×C₆₁	—	4	1	no	—
76	C₄×Q₈×C₆₁	—	4	1	no	—
77	#117×C₆₁	—	4	1	no	—
78	#117×C₆₁	—	4	1	no	—
79	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
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	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₈×C₆₁	—	4	1	no	—
103	C₄⋊C₈×C₆₁	—	4	1	no	—
104	C₄⋊C₈×C₆₁	—	4	1	no	—
105	C₄⋊C₈×C₆₁	—	4	1	no	—
106	C₄⋊C₈×C₆₁	—	4	1	no	—
107	C₄⋊C₈×C₆₁	—	4	1	no	—
108	C₂²⋊C₈×C₆₁	—	2	2	no	—
109	C₂²⋊C₈×C₆₁	—	2	2	no	—
110	#97×C₆₁	—	4	1	no	—
111	#97×C₆₁	—	4	1	no	—
112	C₄×C₈×C₆₁	—	4	1	no	—
113	C₄×C₈×C₆₁	—	4	1	no	—
114	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
115	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
116	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
117	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
118	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
119	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
120	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
121	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
122	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
123	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
124	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
125	C₂×Q₈×C₆₁	—	2	2	no	—
126	C₂×Q₈×C₆₁	—	4	1	no	—
127	C₂×Q₈×C₆₁	—	4	1	no	—
128	C₂²×C₄×C₆₁	—	2	1	no	—
129	C₂²×C₄×C₆₁	—	2	1	no	—
130	C₂²×C₄×C₆₁	—	2	1	no	—
131	C₂²×C₄×C₆₁	—	2	1	no	—
132	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
146	C₂²×C₄×C₆₁	#41	1	2	yes	—
147	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
148	M₁₆×C₆₁	—	4	4	no	—
149	C₂×C₈×C₆₁	—	2	1	no	—
150	C₂×C₈×C₆₁	—	2	1	no	—
151	C₂×C₈×C₆₁	—	2	1	no	—
152	C₂×C₈×C₆₁	—	2	1	no	—
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	—	2	1	no	—
163	C₂×C₈×C₆₁	—	2	1	no	—
164	C₂×C₈×C₆₁	—	2	1	no	—
165	C₂×C₈×C₆₁	—	2	1	no	—
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	—	4	4	no	—
172	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
173	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
174	C₄⋊C₄×C₆₁	—	2	2	no	—
175	C₄⋊C₄×C₆₁	—	2	2	no	—
176	C₄⋊C₄×C₆₁	—	2	2	no	—
177	C₄⋊C₄×C₆₁	—	4	1	no	—
178	C₄⋊C₄×C₆₁	—	4	1	no	—
179	C₄⋊C₄×C₆₁	—	4	1	no	—
180	C₄⋊C₄×C₆₁	—	4	1	no	—
181	C₄⋊C₄×C₆₁	—	4	1	no	—
182	C₄⋊C₄×C₆₁	—	4	1	no	—
183	C₄⋊C₄×C₆₁	—	4	1	no	—
184	C₄⋊C₄×C₆₁	—	4	1	no	—
185	C₂²⋊C₄×C₆₁	—	2	2	no	—
186	C₂²⋊C₄×C₆₁	—	2	2	no	—
187	C₂²⋊C₄×C₆₁	—	4	1	no	—
188	C₂²⋊C₄×C₆₁	—	4	1	no	—
189	C₂²⋊C₄×C₆₁	—	4	1	no	—
190	C₂²⋊C₄×C₆₁	—	4	1	no	—
191	C₄²×C₆₁	—	2	1	no	—
192	C₄²×C₆₁	—	2	1	no	—
193	C₄²×C₆₁	—	2	1	no	—
194	C₄²×C₆₁	—	2	1	no	—
195	C₄²×C₆₁	—	2	1	no	—
196	C₄²×C₆₁	—	2	1	no	—
197	C₄²×C₆₁	—	2	1	no	—
198	C₄²×C₆₁	—	2	1	no	—
199	C₄²×C₆₁	—	2	1	no	—
200	C₄²×C₆₁	—	2	1	no	—
201	C₄²×C₆₁	—	2	1	no	—
202	C₄²×C₆₁	—	2	1	no	—
203	C₂³×C₆₁	—	2	1	no	—
204	C₂³×C₆₁	—	2	1	no	—
205	C₂³×C₆₁	—	2	1	no	—
206	C₂³×C₆₁	—	2	1	no	—
207	C₂³×C₆₁	—	2	1	no	—
208	C₂³×C₆₁	—	2	1	no	—
209	C₂³×C₆₁	C₄×Q₈	1	2	yes	—
210	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
211	Q₈×C₆₁	—	4	4	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	1	no	—
216	C₂×C₄×C₆₁	—	2	1	no	—
217	C₂×C₄×C₆₁	—	2	1	no	—
218	C₂×C₄×C₆₁	—	2	2	no	—
219	C₂×C₄×C₆₁	—	2	2	no	—
220	C₂×C₄×C₆₁	—	2	2	no	—
221	C₂×C₄×C₆₁	—	2	2	no	—
222	C₂×C₄×C₆₁	—	2	1	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	—	2	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	1	no	—
239	C₂×C₄×C₆₁	—	2	1	no	—
240	C₂×C₄×C₆₁	—	2	1	no	—
241	C₂×C₄×C₆₁	—	2	1	no	—
242	C₂×C₄×C₆₁	—	2	1	no	—
243	C₂×C₄×C₆₁	—	2	1	no	—
244	C₂×C₄×C₆₁	—	2	1	no	—
245	C₂×C₄×C₆₁	—	2	1	no	—
246	C₂×C₄×C₆₁	—	2	1	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	C₄×Q₈	1	2	yes	—
253	C₂×C₄×C₆₁	#131	1	2	yes	—
254	C₂×C₄×C₆₁	#131	1	2	yes	—
255	C₂×C₄×C₆₁	C₂²×Q₈	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	H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
260	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
261	C₈×C₆₁	—	2	2	no	—
262	C₈×C₆₁	—	2	2	no	—
263	C₈×C₆₁	—	4	1	no	—
264	C₈×C₆₁	—	4	1	no	—
265	C₈×C₆₁	—	4	1	no	—
266	C₈×C₆₁	—	4	1	no	—
267	C₈×C₆₁	—	4	1	no	—
268	C₈×C₆₁	—	4	1	no	—
269	#22292	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂²×C₆₁	—	2	2	no	—
271	C₂²×C₆₁	—	2	2	no	—
272	C₂²×C₆₁	—	2	2	no	—
273	C₂²×C₆₁	—	4	1	no	—
274	C₂²×C₆₁	—	4	1	no	—
275	C₂²×C₆₁	—	4	1	no	—
276	C₂²×C₆₁	—	4	1	no	—
277	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
278	C₂²×C₆₁	#518	1	1	yes	characteristic
279	C₂²×C₆₁	#520	1	1	yes	characteristic
280	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
281	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
282	C₂²×C₆₁	#568	1	1	yes	characteristic
283	C₂²×C₆₁	#568	1	1	yes	characteristic
284	C₄×C₆₁	—	2	2	no	—
285	C₄×C₆₁	—	2	2	no	—
286	C₄×C₆₁	—	2	2	no	—
287	C₄×C₆₁	—	4	1	no	—
288	C₄×C₆₁	—	4	1	no	—
289	C₄×C₆₁	—	4	1	no	—
290	C₄×C₆₁	—	4	1	no	—
291	C₄×C₆₁	—	4	1	no	—
292	C₄×C₆₁	—	4	1	no	—
293	C₄×C₆₁	—	4	1	no	—
294	C₄×C₆₁	—	4	1	no	—
295	C₄×C₆₁	—	4	1	no	—
296	C₄×C₆₁	—	4	1	no	—
297	C₄×C₆₁	#538	1	1	yes	characteristic
298	C₄×C₆₁	#539	1	1	yes	characteristic
299	C₄×C₆₁	#543	1	1	yes	characteristic
300	C₄×C₆₁	#544	1	1	yes	characteristic
301	#2396	C₂×C₆₁	1	1	yes	characteristic
302	#1985	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#1984	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#1940	C₂×C₆₁	1	1	yes	characteristic, complemented
305	#1939	C₂×C₆₁	1	1	yes	characteristic, complemented
306	#1917	C₂×C₆₁	1	1	yes	characteristic, complemented
307	#1917	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#1916	C₂×C₆₁	1	1	yes	characteristic, complemented
309	#1916	C₂×C₆₁	1	1	yes	characteristic, complemented
310	#1851	C₂×C₆₁	1	1	yes	characteristic
311	#1851	C₂×C₆₁	1	1	yes	characteristic
312	#1798	C₂×C₆₁	1	1	yes	characteristic
313	#1752	C₂×C₆₁	1	1	yes	characteristic
314	C₈×C₂²⋊C₄	C₂×C₆₁	1	1	yes	characteristic
315	#1729	C₂×C₆₁	1	1	yes	characteristic
316	C₂×C₆₁	—	4	2	no	—
317	C₂×C₆₁	#2329	1	1	yes	characteristic
318	C₂×C₆₁	#2960	1	1	yes	characteristic
319	C₂×C₆₁	#2961	1	1	yes	characteristic
320	C₂×C₆₁	#2976	1	1	yes	characteristic
321	C₂×C₆₁	#2981	1	1	yes	characteristic
322	C₂×C₆₁	#2992	1	1	yes	characteristic
323	C₂×C₆₁	#2993	1	1	yes	characteristic
324	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
325	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
326	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
327	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
328	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
329	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
330	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
331	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
333	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
334	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
335	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
336	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
337	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
338	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
339	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
340	#400	C₄×C₆₁	1	1	yes	characteristic
341	#399	C₄×C₆₁	1	1	yes	characteristic
342	#398	C₄×C₆₁	1	1	yes	characteristic
343	#398	C₄×C₆₁	1	1	yes	characteristic
344	#396	C₄×C₆₁	1	1	yes	characteristic
345	#396	C₄×C₆₁	1	1	yes	characteristic
346	#395	C₄×C₆₁	1	1	yes	characteristic
347	#393	C₄×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	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
352	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
353	#336	C₂²×C₆₁	1	1	yes	characteristic
354	#336	C₂²×C₆₁	1	1	yes	characteristic
355	#336	C₂²×C₆₁	1	1	yes	characteristic
356	#336	C₂²×C₆₁	1	1	yes	characteristic
357	#336	C₂²×C₆₁	1	1	yes	characteristic
358	#336	C₂²×C₆₁	1	1	yes	characteristic
359	#336	C₂²×C₆₁	1	1	yes	characteristic
360	#336	C₂²×C₆₁	1	1	yes	characteristic
361	#336	C₂²×C₆₁	1	1	yes	characteristic
362	#336	C₂²×C₆₁	1	1	yes	characteristic
363	#336	C₂²×C₆₁	1	1	yes	characteristic
364	#336	C₂²×C₆₁	1	1	yes	characteristic
365	#336	C₂²×C₆₁	1	1	yes	characteristic
366	#336	C₂²×C₆₁	1	1	yes	characteristic
367	C₆₁	#22292	1	1	yes	characteristic, direct product factor, Sylow
368	C₂²×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
369	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
370	C₂×M₁₆	—	2	2	no	—
371	C₂×M₁₆	—	4	1	no	—
372	C₂×M₁₆	—	4	1	no	—
373	C₂²×C₈	—	4	1	no	—
374	C₂²×C₈	—	4	1	no	—
375	C₂²×C₈	Q₈×C₆₁	1	4	yes	complemented
376	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
380	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
382	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
383	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
384	C₄×Q₈	—	4	1	no	—
385	C₄×Q₈	—	4	1	no	—
386	#117	—	4	1	no	—
387	#117	—	4	1	no	—
388	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
400	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
401	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
402	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
404	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
411	C₄⋊C₈	—	4	1	no	—
412	C₄⋊C₈	—	4	1	no	—
413	C₄⋊C₈	—	4	1	no	—
414	C₄⋊C₈	—	4	1	no	—
415	C₄⋊C₈	—	4	1	no	—
416	C₄⋊C₈	—	4	1	no	—
417	C₂²⋊C₈	—	2	2	no	—
418	C₂²⋊C₈	—	2	2	no	—
419	#97	—	4	1	no	—
420	#97	—	4	1	no	—
421	C₄×C₈	—	4	1	no	—
422	C₄×C₈	—	4	1	no	—
423	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
424	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
425	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
426	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
427	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
428	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
429	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
430	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
431	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
432	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
433	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
434	C₂×Q₈	—	2	2	no	—
435	C₂×Q₈	—	4	1	no	—
436	C₂×Q₈	—	4	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
442	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
443	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
444	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
445	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
446	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
447	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
448	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
449	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
450	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
451	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
454	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
455	C₂²×C₄	#41×C₆₁	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
457	M₁₆	—	4	4	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	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	4	4	no	—
481	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
482	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	4	1	no	—
487	C₄⋊C₄	—	4	1	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	—	4	1	no	—
492	C₄⋊C₄	—	4	1	no	—
493	C₄⋊C₄	—	4	1	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	—	4	1	no	—
498	C₂²⋊C₄	—	4	1	no	—
499	C₂²⋊C₄	—	4	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	C₄×Q₈×C₆₁	1	2	yes	—
519	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
520	Q₈	—	4	4	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	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	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
562	C₂×C₄	#131×C₆₁	1	2	yes	—
563	C₂×C₄	#131×C₆₁	1	2	yes	—
564	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
565	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
566	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
567	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
568	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
569	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
570	C₈	—	2	2	no	—
571	C₈	—	2	2	no	—
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	—	4	1	no	—
577	C₈	—	4	1	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	C₂×C₄×Q₈×C₆₁	1	1	yes	characteristic
586	C₂²	C₆₁×#518	1	1	yes	characteristic
587	C₂²	C₆₁×#520	1	1	yes	characteristic
588	C₂²	C₂×#131×C₆₁	1	1	yes	characteristic
589	C₂²	C₂×#131×C₆₁	1	1	yes	characteristic
590	C₂²	C₆₁×#568	1	1	yes	characteristic
591	C₂²	C₆₁×#568	1	1	yes	characteristic
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	1	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	1	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	C₆₁×#538	1	1	yes	characteristic
606	C₄	C₆₁×#539	1	1	yes	characteristic
607	C₄	C₆₁×#543	1	1	yes	characteristic
608	C₄	C₆₁×#544	1	1	yes	characteristic
609	C₂	—	4	2	no	—
610	C₂	C₆₁×#2329	1	1	yes	characteristic
611	C₂	C₆₁×#2960	1	1	yes	characteristic
612	C₂	C₆₁×#2961	1	1	yes	characteristic
613	C₂	C₆₁×#2976	1	1	yes	characteristic
614	C₂	C₆₁×#2981	1	1	yes	characteristic
615	C₂	C₆₁×#2992	1	1	yes	characteristic
616	C₂	C₆₁×#2993	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₆₁×#1985	C₂	semidirect
3. C₆₁×#1984	C₂	semidirect
4. C₆₁×#1940	C₂	semidirect
5. C₆₁×#1939	C₂	semidirect
6. C₆₁×#1917	C₂	semidirect
7. C₆₁×#1917	C₂	semidirect
8. C₆₁×#1916	C₂	semidirect
9. C₆₁×#1916	C₂	semidirect
66. C₂²×C₈×C₆₁	Q₈	semidirect
269. #22292	C₆₁	direct (N × H)
302. #1985	C₂×C₆₁	semidirect
303. #1984	C₂×C₆₁	semidirect
304. #1940	C₂×C₆₁	semidirect
305. #1939	C₂×C₆₁	semidirect
306. #1917	C₂×C₆₁	semidirect
307. #1917	C₂×C₆₁	semidirect
308. #1916	C₂×C₆₁	semidirect
309. #1916	C₂×C₆₁	semidirect
367. C₆₁	#22292	direct (N × H)
375. C₂²×C₈	Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2396	2
C₆₁×#1985	2
C₆₁×#1984	2
C₆₁×#1940	2
C₆₁×#1939	2
C₆₁×#1917	2
C₆₁×#1916	2
C₆₁×#1851	2
C₆₁×#1798	2
C₆₁×#1752	2
C₈×C₂²⋊C₄×C₆₁	2
C₆₁×#1729	2
#22292	61

Subgroups — C61×#22292 #29525725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22292 #29525725