Subgroups — C₆₁×#25990 #29529423

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2440	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2344	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2088	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2088	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2058	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1920	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1920	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1853	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1850	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1823	C₂	1	1	yes	maximal, characteristic
14	C₆₁×#1807	C₂	1	1	yes	maximal, characteristic
15	C₂×C₆₁×#340	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
19	C₂×#117×C₆₁	—	2	1	no	—
20	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×#106×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×#106×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#398	—	2	1	no	—
30	C₆₁×#395	—	2	1	no	—
31	C₆₁×#389	C₂²	1	1	yes	characteristic
32	C₆₁×#389	C₂²	1	1	yes	characteristic
33	C₆₁×#389	C₂²	1	1	yes	characteristic
34	C₆₁×#389	C₂²	1	1	yes	characteristic
35	C₆₁×#387	—	2	1	no	—
36	C₆₁×#387	—	2	1	no	—
37	C₆₁×#385	—	2	1	no	—
38	C₆₁×#384	C₂²	1	1	yes	characteristic
39	C₆₁×#381	—	2	1	no	—
40	C₆₁×#380	C₂²	1	1	yes	characteristic
41	C₆₁×#380	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
44	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
45	C₆₁×#340	C₂²	1	1	yes	characteristic
46	C₆₁×#340	C₂²	1	1	yes	characteristic
47	C₆₁×#340	C₂²	1	1	yes	characteristic
48	C₆₁×#340	C₂²	1	1	yes	characteristic
49	C₆₁×#340	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×#336	C₂²	1	1	yes	characteristic
58	C₆₁×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₆₁	—	2	1	no	—
60	C₂³×C₄×C₆₁	—	2	1	no	—
61	C₂³×C₄×C₆₁	—	2	1	no	—
62	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₁	—	2	2	no	—
64	C₂²×C₈×C₆₁	—	2	2	no	—
65	C₂²×C₈×C₆₁	—	4	1	no	—
66	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
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	C₄⋊Q₈×C₆₁	—	2	2	no	—
71	C₄⋊Q₈×C₆₁	—	4	1	no	—
72	C₂²⋊Q₈×C₆₁	—	4	4	no	—
73	#117×C₆₁	—	4	4	no	—
74	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₁	—	2	2	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	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
86	C₂×C₄⋊C₄×C₆₁	Q₈	1	4	yes	complemented
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×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	D₄	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₆₁	—	2	1	no	—
100	C₂×C₄²×C₆₁	—	2	1	no	—
101	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
103	#106×C₆₁	—	2	2	no	—
104	#106×C₆₁	—	2	2	no	—
105	#106×C₆₁	—	4	1	no	—
106	#106×C₆₁	—	4	1	no	—
107	C₄⋊C₈×C₆₁	—	2	2	no	—
108	C₄⋊C₈×C₆₁	—	4	1	no	—
109	C₂²⋊C₈×C₆₁	—	2	2	no	—
110	#95×C₆₁	—	2	1	no	—
111	#95×C₆₁	—	2	1	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	D₄	1	1	yes	characteristic
119	#95×C₆₁	D₄	1	1	yes	characteristic
120	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Q₈×C₆₁	—	4	2	no	—
122	C₂×Q₈×C₆₁	—	4	2	no	—
123	C₂×Q₈×C₆₁	—	8	1	no	—
124	C₂²×C₄×C₆₁	—	2	1	no	—
125	C₂²×C₄×C₆₁	—	2	1	no	—
126	C₂²×C₄×C₆₁	—	2	1	no	—
127	C₂²×C₄×C₆₁	—	2	1	no	—
128	C₂²×C₄×C₆₁	—	2	4	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	—	2	1	no	—
133	C₂²×C₄×C₆₁	—	2	1	no	—
134	C₂²×C₄×C₆₁	—	2	1	no	—
135	C₂²×C₄×C₆₁	—	2	1	no	—
136	C₂²×C₄×C₆₁	—	2	1	no	—
137	C₂²×C₄×C₆₁	—	4	2	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	—	8	1	no	—
142	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
148	C₂²×C₄×C₆₁	#41	1	2	yes	—
149	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
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	—	4	4	no	—
162	C₂×C₈×C₆₁	#41	1	2	yes	—
163	C₄⋊C₄×C₆₁	—	2	4	no	—
164	C₄⋊C₄×C₆₁	—	2	1	no	—
165	C₄⋊C₄×C₆₁	—	2	1	no	—
166	C₄⋊C₄×C₆₁	—	2	1	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	2	no	—
178	C₄⋊C₄×C₆₁	—	4	2	no	—
179	C₄⋊C₄×C₆₁	—	4	2	no	—
180	C₄⋊C₄×C₆₁	—	4	2	no	—
181	C₄⋊C₄×C₆₁	—	4	2	no	—
182	C₄⋊C₄×C₆₁	—	4	2	no	—
183	C₄⋊C₄×C₆₁	—	4	2	no	—
184	C₄⋊C₄×C₆₁	—	8	1	no	—
185	C₄⋊C₄×C₆₁	C₂×Q₈	1	2	yes	—
186	C₄⋊C₄×C₆₁	C₂×Q₈	1	2	yes	—
187	C₄⋊C₄×C₆₁	#41	1	2	yes	—
188	C₂²⋊C₄×C₆₁	—	2	2	no	—
189	C₂²⋊C₄×C₆₁	—	4	2	no	—
190	C₂²⋊C₄×C₆₁	—	4	2	no	—
191	C₂²⋊C₄×C₆₁	SD₁₆	1	4	yes	—
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	—	4	2	no	—
197	C₄²×C₆₁	—	4	2	no	—
198	C₂³×C₆₁	—	2	1	no	—
199	C₂³×C₆₁	—	2	1	no	—
200	C₂³×C₆₁	—	2	1	no	—
201	C₂³×C₆₁	—	2	1	no	—
202	C₂³×C₆₁	C₂²⋊Q₈	1	2	yes	—
203	C₂³×C₆₁	C₂×SD₁₆	1	2	yes	—
204	C₂³×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
205	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₆₁	—	8	4	no	—
207	C₂×C₄×C₆₁	—	2	1	no	—
208	C₂×C₄×C₆₁	—	2	1	no	—
209	C₂×C₄×C₆₁	—	2	1	no	—
210	C₂×C₄×C₆₁	—	2	1	no	—
211	C₂×C₄×C₆₁	—	2	1	no	—
212	C₂×C₄×C₆₁	—	2	1	no	—
213	C₂×C₄×C₆₁	—	2	2	no	—
214	C₂×C₄×C₆₁	—	2	2	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	2	no	—
219	C₂×C₄×C₆₁	—	2	1	no	—
220	C₂×C₄×C₆₁	—	2	2	no	—
221	C₂×C₄×C₆₁	—	2	1	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	—	4	1	no	—
227	C₂×C₄×C₆₁	—	4	1	no	—
228	C₂×C₄×C₆₁	—	4	1	no	—
229	C₂×C₄×C₆₁	—	4	1	no	—
230	C₂×C₄×C₆₁	—	4	1	no	—
231	C₂×C₄×C₆₁	—	4	1	no	—
232	C₂×C₄×C₆₁	—	4	8	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	C₂²⋊Q₈	1	2	yes	—
250	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
251	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
252	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
253	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
255	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
257	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
258	C₈×C₆₁	—	2	2	no	—
259	C₈×C₆₁	—	4	1	no	—
260	C₈×C₆₁	—	4	1	no	—
261	C₈×C₆₁	—	4	1	no	—
262	#25990	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂²×C₆₁	—	2	2	no	—
264	C₂²×C₆₁	—	2	2	no	—
265	C₂²×C₆₁	—	2	2	no	—
266	C₂²×C₆₁	—	2	2	no	—
267	C₂²×C₆₁	—	2	2	no	—
268	C₂²×C₆₁	—	2	2	no	—
269	C₂²×C₆₁	#476	1	4	yes	—
270	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
271	C₂²×C₆₁	#546	1	1	yes	characteristic
272	C₂²×C₆₁	#547	1	1	yes	characteristic
273	C₂²×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
274	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂²×C₆₁	#577	1	1	yes	characteristic
276	C₂²×C₆₁	#578	1	1	yes	characteristic
277	C₄×C₆₁	—	2	2	no	—
278	C₄×C₆₁	—	2	2	no	—
279	C₄×C₆₁	—	2	2	no	—
280	C₄×C₆₁	—	4	1	no	—
281	C₄×C₆₁	—	4	1	no	—
282	C₄×C₆₁	—	4	2	no	—
283	C₄×C₆₁	—	4	2	no	—
284	C₄×C₆₁	—	4	2	no	—
285	C₄×C₆₁	—	8	1	no	—
286	C₄×C₆₁	—	8	1	no	—
287	C₄×C₆₁	—	8	1	no	—
288	C₄×C₆₁	—	8	1	no	—
289	C₄×C₆₁	—	8	1	no	—
290	C₄×C₆₁	#533	1	1	yes	characteristic
291	C₄×C₆₁	#544	1	1	yes	characteristic
292	C₄×C₆₁	#551	1	1	yes	characteristic
293	C₄×C₆₁	#552	1	1	yes	characteristic
294	#2440	C₂×C₆₁	1	1	yes	characteristic
295	#2344	C₂×C₆₁	1	1	yes	characteristic
296	#2100	C₂×C₆₁	1	1	yes	characteristic, complemented
297	#2088	C₂×C₆₁	1	1	yes	characteristic, complemented
298	#2088	C₂×C₆₁	1	1	yes	characteristic, complemented
299	#2058	C₂×C₆₁	1	1	yes	characteristic, complemented
300	#1987	C₂×C₆₁	1	1	yes	characteristic, complemented
301	#1948	C₂×C₆₁	1	1	yes	characteristic, complemented
302	#1920	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#1920	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#1853	C₂×C₆₁	1	1	yes	characteristic
305	#1850	C₂×C₆₁	1	1	yes	characteristic
306	#1823	C₂×C₆₁	1	1	yes	characteristic
307	#1807	C₂×C₆₁	1	1	yes	characteristic
308	C₂×#340	C₂×C₆₁	1	1	yes	characteristic
309	C₂×C₆₁	—	2	4	no	—
310	C₂×C₆₁	#2438	1	1	yes	characteristic
311	C₂×C₆₁	C₂×#476	1	1	yes	characteristic
312	C₂×C₆₁	#3082	1	1	yes	characteristic
313	C₂×C₆₁	#3298	1	1	yes	characteristic
314	C₂×C₆₁	#3310	1	1	yes	characteristic
315	C₂×C₆₁	#3335	1	1	yes	characteristic
316	C₂×C₆₁	#3343	1	1	yes	characteristic
317	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
318	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
319	C₂×C₂²⋊Q₈	—	2	1	no	—
320	C₂×#117	—	2	1	no	—
321	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
322	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
323	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
324	C₂×#106	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×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
329	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
330	#398	—	2	1	no	—
331	#395	—	2	1	no	—
332	#389	C₂²×C₆₁	1	1	yes	characteristic
333	#389	C₂²×C₆₁	1	1	yes	characteristic
334	#389	C₂²×C₆₁	1	1	yes	characteristic
335	#389	C₂²×C₆₁	1	1	yes	characteristic
336	#387	—	2	1	no	—
337	#387	—	2	1	no	—
338	#385	—	2	1	no	—
339	#384	C₂²×C₆₁	1	1	yes	characteristic
340	#381	—	2	1	no	—
341	#380	C₂²×C₆₁	1	1	yes	characteristic
342	#380	C₂²×C₆₁	1	1	yes	characteristic
343	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
344	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
345	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
346	#340	C₂²×C₆₁	1	1	yes	characteristic
347	#340	C₂²×C₆₁	1	1	yes	characteristic
348	#340	C₂²×C₆₁	1	1	yes	characteristic
349	#340	C₂²×C₆₁	1	1	yes	characteristic
350	#340	C₂²×C₆₁	1	1	yes	characteristic
351	#340	C₂²×C₆₁	1	1	yes	characteristic
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	#336	C₂²×C₆₁	1	1	yes	characteristic
359	#336	C₂²×C₆₁	1	1	yes	characteristic
360	C₆₁	#25990	1	1	yes	characteristic, direct product factor, Sylow
361	C₂²×Q₈	—	2	1	no	—
362	C₂³×C₄	—	2	1	no	—
363	C₂³×C₄	—	2	1	no	—
364	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
365	C₂²×C₈	—	2	2	no	—
366	C₂²×C₈	—	2	2	no	—
367	C₂²×C₈	—	4	1	no	—
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	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
372	C₄⋊Q₈	—	2	2	no	—
373	C₄⋊Q₈	—	4	1	no	—
374	C₂²⋊Q₈	—	4	4	no	—
375	#117	—	4	4	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
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	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	4	1	no	—
388	C₂×C₄⋊C₄	Q₈×C₆₁	1	4	yes	complemented
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×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
400	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
401	C₂×C₄²	—	2	1	no	—
402	C₂×C₄²	—	2	1	no	—
403	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
405	#106	—	2	2	no	—
406	#106	—	2	2	no	—
407	#106	—	4	1	no	—
408	#106	—	4	1	no	—
409	C₄⋊C₈	—	2	2	no	—
410	C₄⋊C₈	—	4	1	no	—
411	C₂²⋊C₈	—	2	2	no	—
412	#95	—	2	1	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	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	D₄×C₆₁	1	1	yes	characteristic
421	#95	D₄×C₆₁	1	1	yes	characteristic
422	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
423	C₂×Q₈	—	4	2	no	—
424	C₂×Q₈	—	4	2	no	—
425	C₂×Q₈	—	8	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
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	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	8	1	no	—
444	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
449	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
450	C₂²×C₄	#41×C₆₁	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
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	—	4	4	no	—
464	C₂×C₈	#41×C₆₁	1	2	yes	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	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	—	4	2	no	—
480	C₄⋊C₄	—	4	2	no	—
481	C₄⋊C₄	—	4	2	no	—
482	C₄⋊C₄	—	4	2	no	—
483	C₄⋊C₄	—	4	2	no	—
484	C₄⋊C₄	—	4	2	no	—
485	C₄⋊C₄	—	4	2	no	—
486	C₄⋊C₄	—	8	1	no	—
487	C₄⋊C₄	C₂×Q₈×C₆₁	1	2	yes	—
488	C₄⋊C₄	C₂×Q₈×C₆₁	1	2	yes	—
489	C₄⋊C₄	#41×C₆₁	1	2	yes	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₂²⋊C₄	SD₁₆×C₆₁	1	4	yes	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	4	2	no	—
499	C₄²	—	4	2	no	—
500	C₂³	—	2	1	no	—
501	C₂³	—	2	1	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	C₂²⋊Q₈×C₆₁	1	2	yes	—
505	C₂³	C₂×SD₁₆×C₆₁	1	2	yes	—
506	C₂³	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
507	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
508	Q₈	—	8	4	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	2	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	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	8	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	—	8	4	no	—
551	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
552	C₂×C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
553	C₂×C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
554	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
555	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
556	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
557	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
558	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
559	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
560	C₈	—	2	2	no	—
561	C₈	—	4	1	no	—
562	C₈	—	4	1	no	—
563	C₈	—	4	1	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	C₆₁×#476	1	4	yes	—
571	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
572	C₂²	C₆₁×#546	1	1	yes	characteristic
573	C₂²	C₆₁×#547	1	1	yes	characteristic
574	C₂²	C₂²×SD₁₆×C₆₁	1	1	yes	characteristic
575	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
576	C₂²	C₆₁×#577	1	1	yes	characteristic
577	C₂²	C₆₁×#578	1	1	yes	characteristic
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	1	no	—
586	C₄	—	8	1	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	1	no	—
589	C₄	—	8	1	no	—
590	C₄	—	8	1	no	—
591	C₄	C₆₁×#533	1	1	yes	characteristic
592	C₄	C₆₁×#544	1	1	yes	characteristic
593	C₄	C₆₁×#551	1	1	yes	characteristic
594	C₄	C₆₁×#552	1	1	yes	characteristic
595	C₂	—	2	4	no	—
596	C₂	C₆₁×#2438	1	1	yes	characteristic
597	C₂	C₂×C₆₁×#476	1	1	yes	characteristic
598	C₂	C₆₁×#3082	1	1	yes	characteristic
599	C₂	C₆₁×#3298	1	1	yes	characteristic
600	C₂	C₆₁×#3310	1	1	yes	characteristic
601	C₂	C₆₁×#3335	1	1	yes	characteristic
602	C₂	C₆₁×#3343	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₂²×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×#2100	C₂	semidirect
4. C₆₁×#2088	C₂	semidirect
5. C₆₁×#2088	C₂	semidirect
6. C₆₁×#2058	C₂	semidirect
7. C₆₁×#1987	C₂	semidirect
8. C₆₁×#1948	C₂	semidirect
9. C₆₁×#1920	C₂	semidirect
10. C₆₁×#1920	C₂	semidirect
86. C₂×C₄⋊C₄×C₆₁	Q₈	semidirect
262. #25990	C₆₁	direct (N × H)
296. #2100	C₂×C₆₁	semidirect
297. #2088	C₂×C₆₁	semidirect
298. #2088	C₂×C₆₁	semidirect
299. #2058	C₂×C₆₁	semidirect
300. #1987	C₂×C₆₁	semidirect
301. #1948	C₂×C₆₁	semidirect
302. #1920	C₂×C₆₁	semidirect
303. #1920	C₂×C₆₁	semidirect
360. C₆₁	#25990	direct (N × H)
388. C₂×C₄⋊C₄	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×#2440	2
C₆₁×#2344	2
C₆₁×#2100	2
C₆₁×#2088	2
C₆₁×#2058	2
C₆₁×#1987	2
C₆₁×#1948	2
C₆₁×#1920	2
C₆₁×#1853	2
C₆₁×#1850	2
C₆₁×#1823	2
C₆₁×#1807	2
C₂×C₆₁×#340	2
#25990	61

Subgroups — C61×#25990 #29529423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25990 #29529423