Subgroups — C₆₁×#25822 #29529255

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#544	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#501	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2629	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2059	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2058	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1854	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1854	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1853	C₂	1	1	yes	maximal, characteristic
14	C₆₁×#1852	C₂	1	1	yes	maximal, characteristic
15	C₆₁×#1750	C₂	1	1	yes	maximal, characteristic
16	C₆₁×#544	—	2	4	no	—
17	C₆₁×#544	—	2	4	no	—
18	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
19	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#125×C₆₁	—	2	1	no	—
22	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
27	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#501	—	2	4	no	—
31	C₆₁×#499	—	2	4	no	—
32	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
42	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#398	—	2	1	no	—
45	C₆₁×#398	—	2	1	no	—
46	C₆₁×#396	—	2	1	no	—
47	C₆₁×#396	—	2	1	no	—
48	C₆₁×#389	C₂²	1	1	yes	characteristic
49	C₆₁×#389	C₂²	1	1	yes	characteristic
50	C₆₁×#384	C₂²	1	1	yes	characteristic
51	C₆₁×#380	C₂²	1	1	yes	characteristic
52	C₆₁×#380	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×#340	C₂²	1	1	yes	characteristic
59	C₆₁×#340	C₂²	1	1	yes	characteristic
60	C₆₁×#340	C₂²	1	1	yes	characteristic
61	C₆₁×#336	C₂²	1	1	yes	characteristic
62	C₆₁×#336	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₆₁	—	2	1	no	—
64	C₂²×Q₈×C₆₁	—	2	1	no	—
65	C₂³×C₄×C₆₁	—	2	1	no	—
66	C₂³×C₄×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	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₁	—	2	8	no	—
72	C₄⋊Q₈×C₆₁	—	2	2	no	—
73	C₄⋊Q₈×C₆₁	—	2	2	no	—
74	C₄⋊Q₈×C₆₁	—	2	2	no	—
75	C₄⋊Q₈×C₆₁	D₄	1	4	yes	—
76	#125×C₆₁	—	2	8	no	—
77	#125×C₆₁	—	2	2	no	—
78	#125×C₆₁	—	2	2	no	—
79	#125×C₆₁	—	2	2	no	—
80	#125×C₆₁	—	2	2	no	—
81	#125×C₆₁	—	2	2	no	—
82	#125×C₆₁	D₄	1	4	yes	—
83	C₂²⋊Q₈×C₆₁	—	2	8	no	—
84	C₂²⋊Q₈×C₆₁	—	2	8	no	—
85	#117×C₆₁	—	2	2	no	—
86	#117×C₆₁	—	2	2	no	—
87	#117×C₆₁	—	2	2	no	—
88	#117×C₆₁	—	2	2	no	—
89	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
95	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₆₁	—	2	1	no	—
120	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
122	#107×C₆₁	—	2	2	no	—
123	#107×C₆₁	—	2	2	no	—
124	#107×C₆₁	—	2	2	no	—
125	#107×C₆₁	—	2	2	no	—
126	#106×C₆₁	—	2	2	no	—
127	#106×C₆₁	—	2	2	no	—
128	#106×C₆₁	—	2	2	no	—
129	#106×C₆₁	—	2	2	no	—
130	C₂²⋊C₈×C₆₁	Q₈	1	4	yes	complemented
131	C₂²⋊C₈×C₆₁	Q₈	1	4	yes	complemented
132	#97×C₆₁	—	2	2	no	—
133	C₄×C₈×C₆₁	—	2	2	no	—
134	#95×C₆₁	—	2	1	no	—
135	#95×C₆₁	—	2	1	no	—
136	#95×C₆₁	—	2	1	no	—
137	#95×C₆₁	—	2	1	no	—
138	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₆₁	—	2	4	no	—
140	C₂×Q₈×C₆₁	—	2	4	no	—
141	C₂×Q₈×C₆₁	—	8	1	no	—
142	C₂×Q₈×C₆₁	—	8	1	no	—
143	C₂×Q₈×C₆₁	—	8	1	no	—
144	C₂×Q₈×C₆₁	—	8	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	—	2	1	no	—
151	C₂²×C₄×C₆₁	—	2	1	no	—
152	C₂²×C₄×C₆₁	—	2	1	no	—
153	C₂²×C₄×C₆₁	—	2	4	no	—
154	C₂²×C₄×C₆₁	—	2	1	no	—
155	C₂²×C₄×C₆₁	—	2	1	no	—
156	C₂²×C₄×C₆₁	—	2	1	no	—
157	C₂²×C₄×C₆₁	—	2	1	no	—
158	C₂²×C₄×C₆₁	—	8	1	no	—
159	C₂²×C₄×C₆₁	—	8	1	no	—
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	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
166	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
167	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
176	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
177	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
178	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
179	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
180	C₄⋊C₄×C₆₁	—	2	4	no	—
181	C₄⋊C₄×C₆₁	—	2	4	no	—
182	C₄⋊C₄×C₆₁	—	2	4	no	—
183	C₄⋊C₄×C₆₁	—	2	4	no	—
184	C₄⋊C₄×C₆₁	—	2	4	no	—
185	C₄⋊C₄×C₆₁	—	2	4	no	—
186	C₄⋊C₄×C₆₁	—	2	4	no	—
187	C₄⋊C₄×C₆₁	—	2	2	no	—
188	C₄⋊C₄×C₆₁	—	2	2	no	—
189	C₄⋊C₄×C₆₁	—	2	1	no	—
190	C₄⋊C₄×C₆₁	—	2	1	no	—
191	C₄⋊C₄×C₆₁	—	2	1	no	—
192	C₄⋊C₄×C₆₁	—	2	1	no	—
193	C₄⋊C₄×C₆₁	—	2	1	no	—
194	C₄⋊C₄×C₆₁	—	2	1	no	—
195	C₄⋊C₄×C₆₁	—	2	1	no	—
196	C₄⋊C₄×C₆₁	—	2	1	no	—
197	C₄⋊C₄×C₆₁	—	2	1	no	—
198	C₄⋊C₄×C₆₁	—	2	1	no	—
199	C₄⋊C₄×C₆₁	—	2	1	no	—
200	C₄⋊C₄×C₆₁	—	2	1	no	—
201	C₄⋊C₄×C₆₁	—	4	2	no	—
202	C₄⋊C₄×C₆₁	—	4	2	no	—
203	C₄⋊C₄×C₆₁	—	8	1	no	—
204	C₄⋊C₄×C₆₁	—	8	1	no	—
205	C₄⋊C₄×C₆₁	—	8	1	no	—
206	C₄⋊C₄×C₆₁	—	8	1	no	—
207	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
208	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
210	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
211	C₂²⋊C₄×C₆₁	—	2	4	no	—
212	C₂²⋊C₄×C₆₁	—	2	2	no	—
213	C₂²⋊C₄×C₆₁	—	2	2	no	—
214	C₂²⋊C₄×C₆₁	—	8	1	no	—
215	C₂²⋊C₄×C₆₁	—	8	1	no	—
216	C₄²×C₆₁	—	2	4	no	—
217	C₄²×C₆₁	—	2	1	no	—
218	C₄²×C₆₁	—	2	1	no	—
219	C₄²×C₆₁	C₂×D₄	1	2	yes	—
220	C₄²×C₆₁	C₂×D₄	1	2	yes	—
221	C₂³×C₆₁	—	2	1	no	—
222	C₂³×C₆₁	—	2	1	no	—
223	C₂³×C₆₁	—	2	1	no	—
224	C₂³×C₆₁	—	2	1	no	—
225	C₂³×C₆₁	—	2	1	no	—
226	C₂³×C₆₁	—	2	1	no	—
227	C₂³×C₆₁	C₄⋊Q₈	1	2	yes	—
228	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
229	Q₈×C₆₁	—	8	4	no	—
230	Q₈×C₆₁	—	8	4	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	2	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	1	no	—
242	C₂×C₄×C₆₁	—	2	2	no	—
243	C₂×C₄×C₆₁	—	2	1	no	—
244	C₂×C₄×C₆₁	—	2	2	no	—
245	C₂×C₄×C₆₁	—	2	2	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	—	4	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	1	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	—	4	1	no	—
271	C₂×C₄×C₆₁	—	8	4	no	—
272	C₂×C₄×C₆₁	#120	1	2	yes	—
273	C₂×C₄×C₆₁	#120	1	2	yes	—
274	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
275	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
278	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
279	C₂×C₄×C₆₁	2₋¹⁺⁴	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	—	2	2	no	—
285	#25822	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	—	4	1	no	—
290	C₂²×C₆₁	—	4	1	no	—
291	C₂²×C₆₁	—	4	1	no	—
292	C₂²×C₆₁	—	4	1	no	—
293	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
294	C₂²×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
295	C₂²×C₆₁	#536	1	1	yes	characteristic
296	C₂²×C₆₁	#576	1	1	yes	characteristic
297	C₂²×C₆₁	#577	1	1	yes	characteristic
298	C₂²×C₆₁	#577	1	1	yes	characteristic
299	C₂²×C₆₁	#578	1	1	yes	characteristic
300	C₄×C₆₁	—	2	2	no	—
301	C₄×C₆₁	—	2	2	no	—
302	C₄×C₆₁	—	2	2	no	—
303	C₄×C₆₁	—	4	1	no	—
304	C₄×C₆₁	—	4	2	no	—
305	C₄×C₆₁	—	4	2	no	—
306	C₄×C₆₁	—	4	2	no	—
307	C₄×C₆₁	—	4	1	no	—
308	C₄×C₆₁	—	4	2	no	—
309	C₄×C₆₁	—	8	1	no	—
310	C₄×C₆₁	—	8	1	no	—
311	C₄×C₆₁	—	8	1	no	—
312	C₄×C₆₁	—	8	1	no	—
313	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
314	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
315	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
316	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
317	C₂×#544	C₂×C₆₁	1	1	yes	characteristic
318	C₂×#501	C₂×C₆₁	1	1	yes	characteristic, complemented
319	C₂×#499	C₂×C₆₁	1	1	yes	characteristic, complemented
320	#2629	C₂×C₆₁	1	1	yes	characteristic
321	#2059	C₂×C₆₁	1	1	yes	characteristic, complemented
322	#2059	C₂×C₆₁	1	1	yes	characteristic, complemented
323	#2058	C₂×C₆₁	1	1	yes	characteristic, complemented
324	#2057	C₂×C₆₁	1	1	yes	characteristic, complemented
325	#1992	C₂×C₆₁	1	1	yes	characteristic, complemented
326	#1990	C₂×C₆₁	1	1	yes	characteristic, complemented
327	#1854	C₂×C₆₁	1	1	yes	characteristic
328	#1854	C₂×C₆₁	1	1	yes	characteristic
329	#1853	C₂×C₆₁	1	1	yes	characteristic
330	#1852	C₂×C₆₁	1	1	yes	characteristic
331	#1750	C₂×C₆₁	1	1	yes	characteristic
332	C₂×C₆₁	—	4	2	no	—
333	C₂×C₆₁	#2434	1	1	yes	characteristic
334	C₂×C₆₁	#3012	1	1	yes	characteristic
335	C₂×C₆₁	#3013	1	1	yes	characteristic
336	C₂×C₆₁	#3163	1	1	yes	characteristic
337	C₂×C₆₁	#3164	1	1	yes	characteristic
338	C₂×C₆₁	#3241	1	1	yes	characteristic
339	C₂×C₆₁	#3242	1	1	yes	characteristic
340	#544	—	2	4	no	—
341	#544	—	2	4	no	—
342	C₂×C₄⋊Q₈	—	2	1	no	—
343	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
344	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
345	C₂×#125	—	2	1	no	—
346	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
347	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
348	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
349	C₂×C₂²⋊Q₈	—	2	1	no	—
350	C₂×C₂²⋊Q₈	—	2	1	no	—
351	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
352	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
353	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
354	#501	—	2	4	no	—
355	#499	—	2	4	no	—
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×#107	C₂²×C₆₁	1	1	yes	characteristic
360	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
361	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
362	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
363	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
364	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
365	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
366	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
367	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
368	#398	—	2	1	no	—
369	#398	—	2	1	no	—
370	#396	—	2	1	no	—
371	#396	—	2	1	no	—
372	#389	C₂²×C₆₁	1	1	yes	characteristic
373	#389	C₂²×C₆₁	1	1	yes	characteristic
374	#384	C₂²×C₆₁	1	1	yes	characteristic
375	#380	C₂²×C₆₁	1	1	yes	characteristic
376	#380	C₂²×C₆₁	1	1	yes	characteristic
377	#340	C₂²×C₆₁	1	1	yes	characteristic
378	#340	C₂²×C₆₁	1	1	yes	characteristic
379	#340	C₂²×C₆₁	1	1	yes	characteristic
380	#340	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	#336	C₂²×C₆₁	1	1	yes	characteristic
386	#336	C₂²×C₆₁	1	1	yes	characteristic
387	C₆₁	#25822	1	1	yes	characteristic, direct product factor, Sylow
388	C₂²×Q₈	—	2	1	no	—
389	C₂²×Q₈	—	2	1	no	—
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
392	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
393	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
394	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
395	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
396	C₄⋊Q₈	—	2	8	no	—
397	C₄⋊Q₈	—	2	2	no	—
398	C₄⋊Q₈	—	2	2	no	—
399	C₄⋊Q₈	—	2	2	no	—
400	C₄⋊Q₈	D₄×C₆₁	1	4	yes	—
401	#125	—	2	8	no	—
402	#125	—	2	2	no	—
403	#125	—	2	2	no	—
404	#125	—	2	2	no	—
405	#125	—	2	2	no	—
406	#125	—	2	2	no	—
407	#125	D₄×C₆₁	1	4	yes	—
408	C₂²⋊Q₈	—	2	8	no	—
409	C₂²⋊Q₈	—	2	8	no	—
410	#117	—	2	2	no	—
411	#117	—	2	2	no	—
412	#117	—	2	2	no	—
413	#117	—	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	—	2	1	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	2	no	—
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
430	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
438	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
443	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
444	C₂×C₄²	—	2	1	no	—
445	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
447	#107	—	2	2	no	—
448	#107	—	2	2	no	—
449	#107	—	2	2	no	—
450	#107	—	2	2	no	—
451	#106	—	2	2	no	—
452	#106	—	2	2	no	—
453	#106	—	2	2	no	—
454	#106	—	2	2	no	—
455	C₂²⋊C₈	Q₈×C₆₁	1	4	yes	complemented
456	C₂²⋊C₈	Q₈×C₆₁	1	4	yes	complemented
457	#97	—	2	2	no	—
458	C₄×C₈	—	2	2	no	—
459	#95	—	2	1	no	—
460	#95	—	2	1	no	—
461	#95	—	2	1	no	—
462	#95	—	2	1	no	—
463	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
464	C₂×Q₈	—	2	4	no	—
465	C₂×Q₈	—	2	4	no	—
466	C₂×Q₈	—	8	1	no	—
467	C₂×Q₈	—	8	1	no	—
468	C₂×Q₈	—	8	1	no	—
469	C₂×Q₈	—	8	1	no	—
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	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	8	1	no	—
484	C₂²×C₄	—	8	1	no	—
485	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
490	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
491	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
502	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
503	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
504	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	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	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	4	2	no	—
527	C₄⋊C₄	—	4	2	no	—
528	C₄⋊C₄	—	8	1	no	—
529	C₄⋊C₄	—	8	1	no	—
530	C₄⋊C₄	—	8	1	no	—
531	C₄⋊C₄	—	8	1	no	—
532	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
533	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
534	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
535	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	8	1	no	—
540	C₂²⋊C₄	—	8	1	no	—
541	C₄²	—	2	4	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	C₂×D₄×C₆₁	1	2	yes	—
545	C₄²	C₂×D₄×C₆₁	1	2	yes	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	1	no	—
552	C₂³	C₄⋊Q₈×C₆₁	1	2	yes	—
553	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
554	Q₈	—	8	4	no	—
555	Q₈	—	8	4	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	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	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	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	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	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	8	4	no	—
597	C₂×C₄	#120×C₆₁	1	2	yes	—
598	C₂×C₄	#120×C₆₁	1	2	yes	—
599	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
600	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
601	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
602	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
603	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
604	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
605	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
606	C₈	—	2	2	no	—
607	C₈	—	2	2	no	—
608	C₈	—	2	2	no	—
609	C₈	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	2	2	no	—
613	C₂²	—	4	1	no	—
614	C₂²	—	4	1	no	—
615	C₂²	—	4	1	no	—
616	C₂²	—	4	1	no	—
617	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
618	C₂²	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
619	C₂²	C₆₁×#536	1	1	yes	characteristic
620	C₂²	C₆₁×#576	1	1	yes	characteristic
621	C₂²	C₆₁×#577	1	1	yes	characteristic
622	C₂²	C₆₁×#577	1	1	yes	characteristic
623	C₂²	C₆₁×#578	1	1	yes	characteristic
624	C₄	—	2	2	no	—
625	C₄	—	2	2	no	—
626	C₄	—	2	2	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	1	no	—
632	C₄	—	4	2	no	—
633	C₄	—	8	1	no	—
634	C₄	—	8	1	no	—
635	C₄	—	8	1	no	—
636	C₄	—	8	1	no	—
637	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
638	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
639	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
640	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
641	C₂	—	4	2	no	—
642	C₂	C₆₁×#2434	1	1	yes	characteristic
643	C₂	C₆₁×#3012	1	1	yes	characteristic
644	C₂	C₆₁×#3013	1	1	yes	characteristic
645	C₂	C₆₁×#3163	1	1	yes	characteristic
646	C₂	C₆₁×#3164	1	1	yes	characteristic
647	C₂	C₆₁×#3241	1	1	yes	characteristic
648	C₂	C₆₁×#3242	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 ◁ C₂×#117×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#501	C₂	semidirect
3. C₂×C₆₁×#499	C₂	semidirect
5. C₆₁×#2059	C₂	semidirect
6. C₆₁×#2059	C₂	semidirect
7. C₆₁×#2058	C₂	semidirect
8. C₆₁×#2057	C₂	semidirect
9. C₆₁×#1992	C₂	semidirect
10. C₆₁×#1990	C₂	semidirect
130. C₂²⋊C₈×C₆₁	Q₈	semidirect
131. C₂²⋊C₈×C₆₁	Q₈	semidirect
285. #25822	C₆₁	direct (N × H)
318. C₂×#501	C₂×C₆₁	semidirect
319. C₂×#499	C₂×C₆₁	semidirect
321. #2059	C₂×C₆₁	semidirect
322. #2059	C₂×C₆₁	semidirect
323. #2058	C₂×C₆₁	semidirect
324. #2057	C₂×C₆₁	semidirect
325. #1992	C₂×C₆₁	semidirect
326. #1990	C₂×C₆₁	semidirect
387. C₆₁	#25822	direct (N × H)
455. C₂²⋊C₈	Q₈×C₆₁	semidirect
456. 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
C₂×C₆₁×#544	2
C₂×C₆₁×#501	2
C₂×C₆₁×#499	2
C₆₁×#2629	2
C₆₁×#2059	2
C₆₁×#2058	2
C₆₁×#2057	2
C₆₁×#1992	2
C₆₁×#1990	2
C₆₁×#1854	2
C₆₁×#1853	2
C₆₁×#1852	2
C₆₁×#1750	2
#25822	61

Subgroups — C61×#25822 #29529255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25822 #29529255