Subgroups — C₆₁×#24334 #29527767

Order 15616.

1042 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#478	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2580	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2058	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2025	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#2023	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#1991	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1880	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1880	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1865	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1863	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#557	—	2	4	no	—
18	C₆₁×#557	—	2	4	no	—
19	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#125×C₆₁	—	2	1	no	—
22	C₂×#125×C₆₁	—	2	1	no	—
23	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
26	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
31	C₆₁×#478	—	2	4	no	—
32	C₄×SD₁₆×C₆₁	—	2	4	no	—
33	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
42	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#399	—	2	1	no	—
45	C₆₁×#393	—	2	1	no	—
46	C₆₁×#390	C₂²	1	1	yes	characteristic
47	C₆₁×#390	C₂²	1	1	yes	characteristic
48	C₆₁×#389	C₂²	1	1	yes	characteristic
49	C₆₁×#389	C₂²	1	1	yes	characteristic
50	C₆₁×#385	—	2	1	no	—
51	C₆₁×#385	—	2	1	no	—
52	C₆₁×#382	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×#336	C₂²	1	1	yes	characteristic
60	C₆₁×#336	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	D₄	1	1	yes	characteristic
65	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₁	—	2	1	no	—
68	C₂×SD₁₆×C₆₁	—	2	2	no	—
69	C₂×SD₁₆×C₆₁	—	2	2	no	—
70	C₂×SD₁₆×C₆₁	Q₈	1	4	yes	complemented
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	—	2	8	no	—
76	C₄⋊Q₈×C₆₁	—	2	2	no	—
77	C₄⋊Q₈×C₆₁	—	2	2	no	—
78	#125×C₆₁	—	2	8	no	—
79	#125×C₆₁	—	2	8	no	—
80	#125×C₆₁	—	2	2	no	—
81	#125×C₆₁	—	2	2	no	—
82	#125×C₆₁	—	2	2	no	—
83	#125×C₆₁	D₄	1	4	yes	—
84	C₄×Q₈×C₆₁	—	2	8	no	—
85	C₄×Q₈×C₆₁	—	2	2	no	—
86	C₄×Q₈×C₆₁	—	2	2	no	—
87	C₄×Q₈×C₆₁	—	2	2	no	—
88	C₄×Q₈×C₆₁	—	2	2	no	—
89	C₄×Q₈×C₆₁	—	2	2	no	—
90	C₄×Q₈×C₆₁	—	2	2	no	—
91	C₄×Q₈×C₆₁	—	2	2	no	—
92	C₄×Q₈×C₆₁	D₄	1	4	yes	—
93	C₄×D₄×C₆₁	—	2	2	no	—
94	C₄×D₄×C₆₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₂×C₄⋊C₄×C₆₁	—	2	1	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	—	2	1	no	—
105	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₁	D₄	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	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
119	C₂×C₄²×C₆₁	—	2	1	no	—
120	C₂×C₄²×C₆₁	—	2	1	no	—
121	C₂×C₄²×C₆₁	—	2	1	no	—
122	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
125	#107×C₆₁	—	2	2	no	—
126	#106×C₆₁	—	2	2	no	—
127	#106×C₆₁	—	2	2	no	—
128	C₄⋊C₈×C₆₁	—	2	2	no	—
129	#103×C₆₁	—	2	2	no	—
130	#103×C₆₁	—	2	2	no	—
131	#103×C₆₁	—	2	2	no	—
132	#102×C₆₁	—	2	2	no	—
133	#102×C₆₁	—	2	2	no	—
134	#102×C₆₁	—	2	2	no	—
135	C₄×C₈×C₆₁	—	2	2	no	—
136	#95×C₆₁	—	2	1	no	—
137	#95×C₆₁	—	2	1	no	—
138	#95×C₆₁	—	2	1	no	—
139	#95×C₆₁	—	2	1	no	—
140	C₂⁴×C₆₁	—	2	1	no	—
141	C₂×Q₈×C₆₁	—	2	4	no	—
142	C₂×Q₈×C₆₁	—	2	2	no	—
143	C₂×Q₈×C₆₁	—	2	2	no	—
144	C₂×Q₈×C₆₁	—	2	1	no	—
145	C₂×Q₈×C₆₁	—	2	1	no	—
146	C₂×Q₈×C₆₁	—	2	1	no	—
147	C₂×Q₈×C₆₁	—	4	1	no	—
148	C₂×Q₈×C₆₁	—	4	2	no	—
149	C₂×Q₈×C₆₁	—	4	2	no	—
150	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
151	C₂×Q₈×C₆₁	C₂×Q₈	1	2	yes	—
152	C₂×Q₈×C₆₁	#41	1	2	yes	—
153	C₂×D₄×C₆₁	—	2	1	no	—
154	C₂×D₄×C₆₁	—	2	1	no	—
155	C₂×D₄×C₆₁	—	2	1	no	—
156	C₂×D₄×C₆₁	—	2	1	no	—
157	C₂×D₄×C₆₁	—	2	1	no	—
158	C₂×D₄×C₆₁	C₂×Q₈	1	2	yes	—
159	C₂²×C₄×C₆₁	—	2	1	no	—
160	C₂²×C₄×C₆₁	—	2	1	no	—
161	C₂²×C₄×C₆₁	—	2	1	no	—
162	C₂²×C₄×C₆₁	—	2	1	no	—
163	C₂²×C₄×C₆₁	—	2	1	no	—
164	C₂²×C₄×C₆₁	—	2	1	no	—
165	C₂²×C₄×C₆₁	—	2	1	no	—
166	C₂²×C₄×C₆₁	—	4	2	no	—
167	C₂²×C₄×C₆₁	—	8	1	no	—
168	C₂²×C₄×C₆₁	—	8	1	no	—
169	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
176	SD₁₆×C₆₁	—	2	8	no	—
177	C₂×C₈×C₆₁	—	2	1	no	—
178	C₂×C₈×C₆₁	—	2	1	no	—
179	C₂×C₈×C₆₁	—	2	1	no	—
180	C₂×C₈×C₆₁	—	2	1	no	—
181	C₂×C₈×C₆₁	—	2	1	no	—
182	C₂×C₈×C₆₁	—	2	1	no	—
183	C₂×C₈×C₆₁	—	2	1	no	—
184	C₂×C₈×C₆₁	—	2	1	no	—
185	C₂×C₈×C₆₁	—	2	1	no	—
186	C₂×C₈×C₆₁	—	2	1	no	—
187	C₂×C₈×C₆₁	—	2	1	no	—
188	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
189	C₄⋊C₄×C₆₁	—	2	4	no	—
190	C₄⋊C₄×C₆₁	—	2	4	no	—
191	C₄⋊C₄×C₆₁	—	2	4	no	—
192	C₄⋊C₄×C₆₁	—	2	4	no	—
193	C₄⋊C₄×C₆₁	—	2	4	no	—
194	C₄⋊C₄×C₆₁	—	2	4	no	—
195	C₄⋊C₄×C₆₁	—	2	4	no	—
196	C₄⋊C₄×C₆₁	—	2	4	no	—
197	C₄⋊C₄×C₆₁	—	2	2	no	—
198	C₄⋊C₄×C₆₁	—	2	2	no	—
199	C₄⋊C₄×C₆₁	—	2	2	no	—
200	C₄⋊C₄×C₆₁	—	2	1	no	—
201	C₄⋊C₄×C₆₁	—	2	1	no	—
202	C₄⋊C₄×C₆₁	—	2	1	no	—
203	C₄⋊C₄×C₆₁	—	2	1	no	—
204	C₄⋊C₄×C₆₁	—	2	1	no	—
205	C₄⋊C₄×C₆₁	—	2	1	no	—
206	C₄⋊C₄×C₆₁	—	2	1	no	—
207	C₄⋊C₄×C₆₁	—	2	1	no	—
208	C₄⋊C₄×C₆₁	—	2	1	no	—
209	C₄⋊C₄×C₆₁	—	8	1	no	—
210	C₄⋊C₄×C₆₁	—	8	1	no	—
211	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
212	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
213	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
214	C₂²⋊C₄×C₆₁	—	4	2	no	—
215	C₂²⋊C₄×C₆₁	—	4	2	no	—
216	C₂²⋊C₄×C₆₁	—	4	2	no	—
217	C₂²⋊C₄×C₆₁	—	8	1	no	—
218	C₂²⋊C₄×C₆₁	—	8	1	no	—
219	C₄²×C₆₁	—	2	4	no	—
220	C₄²×C₆₁	—	2	4	no	—
221	C₄²×C₆₁	—	2	4	no	—
222	C₄²×C₆₁	—	2	1	no	—
223	C₄²×C₆₁	—	2	1	no	—
224	C₄²×C₆₁	—	2	1	no	—
225	C₄²×C₆₁	—	2	1	no	—
226	C₄²×C₆₁	C₂×D₄	1	2	yes	—
227	C₄²×C₆₁	C₂×D₄	1	2	yes	—
228	C₂³×C₆₁	—	2	2	no	—
229	C₂³×C₆₁	—	4	1	no	—
230	C₂³×C₆₁	—	4	1	no	—
231	C₂³×C₆₁	—	4	1	no	—
232	C₂³×C₆₁	—	4	1	no	—
233	C₂³×C₆₁	—	4	1	no	—
234	C₂³×C₆₁	—	4	1	no	—
235	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
236	Q₈×C₆₁	—	2	2	no	—
237	Q₈×C₆₁	—	2	2	no	—
238	Q₈×C₆₁	—	2	2	no	—
239	Q₈×C₆₁	—	4	4	no	—
240	Q₈×C₆₁	—	4	8	no	—
241	Q₈×C₆₁	C₂²⋊Q₈	1	4	yes	—
242	D₄×C₆₁	—	2	2	no	—
243	D₄×C₆₁	—	2	2	no	—
244	D₄×C₆₁	—	4	1	no	—
245	D₄×C₆₁	—	4	1	no	—
246	C₂×C₄×C₆₁	—	2	2	no	—
247	C₂×C₄×C₆₁	—	2	1	no	—
248	C₂×C₄×C₆₁	—	2	2	no	—
249	C₂×C₄×C₆₁	—	2	2	no	—
250	C₂×C₄×C₆₁	—	2	1	no	—
251	C₂×C₄×C₆₁	—	2	1	no	—
252	C₂×C₄×C₆₁	—	2	1	no	—
253	C₂×C₄×C₆₁	—	2	1	no	—
254	C₂×C₄×C₆₁	—	2	1	no	—
255	C₂×C₄×C₆₁	—	2	1	no	—
256	C₂×C₄×C₆₁	—	2	1	no	—
257	C₂×C₄×C₆₁	—	2	2	no	—
258	C₂×C₄×C₆₁	—	2	1	no	—
259	C₂×C₄×C₆₁	—	2	2	no	—
260	C₂×C₄×C₆₁	—	2	2	no	—
261	C₂×C₄×C₆₁	—	2	1	no	—
262	C₂×C₄×C₆₁	—	2	1	no	—
263	C₂×C₄×C₆₁	—	2	1	no	—
264	C₂×C₄×C₆₁	—	2	1	no	—
265	C₂×C₄×C₆₁	—	2	1	no	—
266	C₂×C₄×C₆₁	—	2	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	—	4	1	no	—
272	C₂×C₄×C₆₁	—	4	1	no	—
273	C₂×C₄×C₆₁	—	4	1	no	—
274	C₂×C₄×C₆₁	—	4	1	no	—
275	C₂×C₄×C₆₁	—	4	1	no	—
276	C₂×C₄×C₆₁	—	4	1	no	—
277	C₂×C₄×C₆₁	—	4	1	no	—
278	C₂×C₄×C₆₁	—	4	1	no	—
279	C₂×C₄×C₆₁	—	4	1	no	—
280	C₂×C₄×C₆₁	—	4	1	no	—
281	C₂×C₄×C₆₁	—	4	1	no	—
282	C₂×C₄×C₆₁	—	8	4	no	—
283	C₂×C₄×C₆₁	#120	1	2	yes	—
284	C₂×C₄×C₆₁	#120	1	2	yes	—
285	C₂×C₄×C₆₁	#135	1	2	yes	—
286	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
289	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
290	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
291	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
292	C₈×C₆₁	—	2	2	no	—
293	C₈×C₆₁	—	4	1	no	—
294	C₈×C₆₁	—	4	1	no	—
295	C₈×C₆₁	—	4	1	no	—
296	#24334	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₂²×C₆₁	—	4	2	no	—
298	C₂²×C₆₁	—	4	2	no	—
299	C₂²×C₆₁	—	4	2	no	—
300	C₂²×C₆₁	—	8	1	no	—
301	C₂²×C₆₁	—	8	1	no	—
302	C₂²×C₆₁	—	8	1	no	—
303	C₂²×C₆₁	—	8	1	no	—
304	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
305	C₂²×C₆₁	#548	1	1	yes	characteristic
306	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
307	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
308	C₂²×C₆₁	#575	1	1	yes	characteristic
309	C₂²×C₆₁	#577	1	1	yes	characteristic
310	C₂²×C₆₁	#577	1	1	yes	characteristic
311	C₄×C₆₁	—	2	4	no	—
312	C₄×C₆₁	—	2	2	no	—
313	C₄×C₆₁	—	2	2	no	—
314	C₄×C₆₁	—	2	2	no	—
315	C₄×C₆₁	—	4	1	no	—
316	C₄×C₆₁	—	4	2	no	—
317	C₄×C₆₁	—	4	2	no	—
318	C₄×C₆₁	—	4	1	no	—
319	C₄×C₆₁	—	4	1	no	—
320	C₄×C₆₁	—	4	2	no	—
321	C₄×C₆₁	—	8	1	no	—
322	C₄×C₆₁	—	8	1	no	—
323	C₄×C₆₁	—	8	1	no	—
324	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
325	C₄×C₆₁	#536	1	1	yes	characteristic
326	C₄×C₆₁	#548	1	1	yes	characteristic
327	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
328	C₂×#557	C₂×C₆₁	1	1	yes	characteristic, complemented
329	C₂×#478	C₂×C₆₁	1	1	yes	characteristic
330	C₂×C₄×SD₁₆	C₂×C₆₁	1	1	yes	characteristic
331	#2580	C₂×C₆₁	1	1	yes	characteristic
332	#2059	C₂×C₆₁	1	1	yes	characteristic, complemented
333	#2058	C₂×C₆₁	1	1	yes	characteristic, complemented
334	#2025	C₂×C₆₁	1	1	yes	characteristic
335	#2023	C₂×C₆₁	1	1	yes	characteristic
336	#1991	C₂×C₆₁	1	1	yes	characteristic, complemented
337	#1984	C₂×C₆₁	1	1	yes	characteristic, complemented
338	#1880	C₂×C₆₁	1	1	yes	characteristic
339	#1880	C₂×C₆₁	1	1	yes	characteristic
340	#1865	C₂×C₆₁	1	1	yes	characteristic, complemented
341	#1863	C₂×C₆₁	1	1	yes	characteristic, complemented
342	#1805	C₂×C₆₁	1	1	yes	characteristic, complemented
343	C₂×C₆₁	—	8	2	no	—
344	C₂×C₆₁	#2409	1	1	yes	characteristic
345	C₂×C₆₁	#3002	1	1	yes	characteristic
346	C₂×C₆₁	#3010	1	1	yes	characteristic
347	C₂×C₆₁	#3365	1	1	yes	characteristic
348	C₂×C₆₁	#3371	1	1	yes	characteristic
349	C₂×C₆₁	#3381	1	1	yes	characteristic
350	C₂×C₆₁	#3388	1	1	yes	characteristic
351	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
352	#557	—	2	4	no	—
353	#557	—	2	4	no	—
354	C₂×C₄⋊Q₈	—	2	1	no	—
355	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
356	C₂×#125	—	2	1	no	—
357	C₂×#125	—	2	1	no	—
358	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
359	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
360	C₂×C₄×Q₈	—	2	1	no	—
361	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
362	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
363	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
364	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
365	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
366	#478	—	2	4	no	—
367	C₄×SD₁₆	—	2	4	no	—
368	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
369	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
370	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
371	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
372	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
373	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
374	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
375	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
376	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
377	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
378	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
379	#399	—	2	1	no	—
380	#393	—	2	1	no	—
381	#390	C₂²×C₆₁	1	1	yes	characteristic
382	#390	C₂²×C₆₁	1	1	yes	characteristic
383	#389	C₂²×C₆₁	1	1	yes	characteristic
384	#389	C₂²×C₆₁	1	1	yes	characteristic
385	#385	—	2	1	no	—
386	#385	—	2	1	no	—
387	#382	C₂²×C₆₁	1	1	yes	characteristic
388	#340	C₂²×C₆₁	1	1	yes	characteristic
389	#340	C₂²×C₆₁	1	1	yes	characteristic
390	#340	C₂²×C₆₁	1	1	yes	characteristic
391	#340	C₂²×C₆₁	1	1	yes	characteristic
392	#340	C₂²×C₆₁	1	1	yes	characteristic
393	#340	C₂²×C₆₁	1	1	yes	characteristic
394	#336	C₂²×C₆₁	1	1	yes	characteristic
395	#336	C₂²×C₆₁	1	1	yes	characteristic
396	#336	C₂²×C₆₁	1	1	yes	characteristic
397	#336	C₂²×C₆₁	1	1	yes	characteristic
398	C₆₁	#24334	1	1	yes	characteristic, direct product factor, Sylow
399	C₂²×Q₈	—	2	1	no	—
400	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
402	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
403	C₂³×C₄	—	2	1	no	—
404	C₂×SD₁₆	—	2	2	no	—
405	C₂×SD₁₆	—	2	2	no	—
406	C₂×SD₁₆	Q₈×C₆₁	1	4	yes	complemented
407	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
408	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
409	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
410	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
411	C₄⋊Q₈	—	2	8	no	—
412	C₄⋊Q₈	—	2	2	no	—
413	C₄⋊Q₈	—	2	2	no	—
414	#125	—	2	8	no	—
415	#125	—	2	8	no	—
416	#125	—	2	2	no	—
417	#125	—	2	2	no	—
418	#125	—	2	2	no	—
419	#125	D₄×C₆₁	1	4	yes	—
420	C₄×Q₈	—	2	8	no	—
421	C₄×Q₈	—	2	2	no	—
422	C₄×Q₈	—	2	2	no	—
423	C₄×Q₈	—	2	2	no	—
424	C₄×Q₈	—	2	2	no	—
425	C₄×Q₈	—	2	2	no	—
426	C₄×Q₈	—	2	2	no	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×Q₈	D₄×C₆₁	1	4	yes	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	2	2	no	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	2	1	no	—
434	C₂×C₄⋊C₄	—	2	1	no	—
435	C₂×C₄⋊C₄	—	2	1	no	—
436	C₂×C₄⋊C₄	—	2	1	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	—	2	1	no	—
440	C₂×C₄⋊C₄	—	2	1	no	—
441	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
442	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
443	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
447	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
448	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
449	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
450	C₂×C₂²⋊C₄	—	2	1	no	—
451	C₂×C₂²⋊C₄	—	2	1	no	—
452	C₂×C₂²⋊C₄	—	2	1	no	—
453	C₂×C₂²⋊C₄	—	2	1	no	—
454	C₂×C₂²⋊C₄	—	2	1	no	—
455	C₂×C₄²	—	2	1	no	—
456	C₂×C₄²	—	2	1	no	—
457	C₂×C₄²	—	2	1	no	—
458	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
459	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
460	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
461	#107	—	2	2	no	—
462	#106	—	2	2	no	—
463	#106	—	2	2	no	—
464	C₄⋊C₈	—	2	2	no	—
465	#103	—	2	2	no	—
466	#103	—	2	2	no	—
467	#103	—	2	2	no	—
468	#102	—	2	2	no	—
469	#102	—	2	2	no	—
470	#102	—	2	2	no	—
471	C₄×C₈	—	2	2	no	—
472	#95	—	2	1	no	—
473	#95	—	2	1	no	—
474	#95	—	2	1	no	—
475	#95	—	2	1	no	—
476	C₂⁴	—	2	1	no	—
477	C₂×Q₈	—	2	4	no	—
478	C₂×Q₈	—	2	2	no	—
479	C₂×Q₈	—	2	2	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	4	1	no	—
484	C₂×Q₈	—	4	2	no	—
485	C₂×Q₈	—	4	2	no	—
486	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
487	C₂×Q₈	C₂×Q₈×C₆₁	1	2	yes	—
488	C₂×Q₈	#41×C₆₁	1	2	yes	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	C₂×Q₈×C₆₁	1	2	yes	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	4	2	no	—
503	C₂²×C₄	—	8	1	no	—
504	C₂²×C₄	—	8	1	no	—
505	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
509	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
510	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
512	SD₁₆	—	2	8	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	4	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	4	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	2	no	—
534	C₄⋊C₄	—	2	2	no	—
535	C₄⋊C₄	—	2	2	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	1	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	8	1	no	—
546	C₄⋊C₄	—	8	1	no	—
547	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
548	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
549	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
550	C₂²⋊C₄	—	4	2	no	—
551	C₂²⋊C₄	—	4	2	no	—
552	C₂²⋊C₄	—	4	2	no	—
553	C₂²⋊C₄	—	8	1	no	—
554	C₂²⋊C₄	—	8	1	no	—
555	C₄²	—	2	4	no	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	4	no	—
558	C₄²	—	2	1	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	2	1	no	—
562	C₄²	C₂×D₄×C₆₁	1	2	yes	—
563	C₄²	C₂×D₄×C₆₁	1	2	yes	—
564	C₂³	—	2	2	no	—
565	C₂³	—	4	1	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	1	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
572	Q₈	—	2	2	no	—
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	4	4	no	—
576	Q₈	—	4	8	no	—
577	Q₈	C₂²⋊Q₈×C₆₁	1	4	yes	—
578	D₄	—	2	2	no	—
579	D₄	—	2	2	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	8	4	no	—
619	C₂×C₄	#120×C₆₁	1	2	yes	—
620	C₂×C₄	#120×C₆₁	1	2	yes	—
621	C₂×C₄	#135×C₆₁	1	2	yes	—
622	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
623	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
624	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
625	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
626	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
627	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
628	C₈	—	2	2	no	—
629	C₈	—	4	1	no	—
630	C₈	—	4	1	no	—
631	C₈	—	4	1	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	8	1	no	—
636	C₂²	—	8	1	no	—
637	C₂²	—	8	1	no	—
638	C₂²	—	8	1	no	—
639	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
640	C₂²	C₆₁×#548	1	1	yes	characteristic
641	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
642	C₂²	C₂×#135×C₆₁	1	1	yes	characteristic
643	C₂²	C₆₁×#575	1	1	yes	characteristic
644	C₂²	C₆₁×#577	1	1	yes	characteristic
645	C₂²	C₆₁×#577	1	1	yes	characteristic
646	C₄	—	2	4	no	—
647	C₄	—	2	2	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	4	1	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	1	no	—
654	C₄	—	4	1	no	—
655	C₄	—	4	2	no	—
656	C₄	—	8	1	no	—
657	C₄	—	8	1	no	—
658	C₄	—	8	1	no	—
659	C₄	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
660	C₄	C₆₁×#536	1	1	yes	characteristic
661	C₄	C₆₁×#548	1	1	yes	characteristic
662	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
663	C₂	—	8	2	no	—
664	C₂	C₆₁×#2409	1	1	yes	characteristic
665	C₂	C₆₁×#3002	1	1	yes	characteristic
666	C₂	C₆₁×#3010	1	1	yes	characteristic
667	C₂	C₆₁×#3365	1	1	yes	characteristic
668	C₂	C₆₁×#3371	1	1	yes	characteristic
669	C₂	C₆₁×#3381	1	1	yes	characteristic
670	C₂	C₆₁×#3388	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂²×D₄×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#557	C₂	semidirect
5. C₆₁×#2059	C₂	semidirect
6. C₆₁×#2058	C₂	semidirect
9. C₆₁×#1991	C₂	semidirect
10. C₆₁×#1984	C₂	semidirect
13. C₆₁×#1865	C₂	semidirect
14. C₆₁×#1863	C₂	semidirect
15. C₆₁×#1805	C₂	semidirect
70. C₂×SD₁₆×C₆₁	Q₈	semidirect
296. #24334	C₆₁	direct (N × H)
328. C₂×#557	C₂×C₆₁	semidirect
332. #2059	C₂×C₆₁	semidirect
333. #2058	C₂×C₆₁	semidirect
336. #1991	C₂×C₆₁	semidirect
337. #1984	C₂×C₆₁	semidirect
340. #1865	C₂×C₆₁	semidirect
341. #1863	C₂×C₆₁	semidirect
342. #1805	C₂×C₆₁	semidirect
398. C₆₁	#24334	direct (N × H)
406. C₂×SD₁₆	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×#557	2
C₂×C₆₁×#478	2
C₂×C₄×SD₁₆×C₆₁	2
C₆₁×#2580	2
C₆₁×#2059	2
C₆₁×#2058	2
C₆₁×#2025	2
C₆₁×#2023	2
C₆₁×#1991	2
C₆₁×#1984	2
C₆₁×#1880	2
C₆₁×#1865	2
C₆₁×#1863	2
C₆₁×#1805	2
#24334	61

Subgroups — C61×#24334 #29527767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24334 #29527767