Subgroups — C₆₁×#17450 #29520883

Order 15616.

1246 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1489	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1488	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1479	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1478	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₆₁	C₄	1	1	yes	characteristic
9	C₄×#41×C₆₁	—	2	1	no	—
10	C₄×#41×C₆₁	—	2	1	no	—
11	C₄×#41×C₆₁	—	2	1	no	—
12	C₄×#41×C₆₁	—	2	1	no	—
13	C₄×#41×C₆₁	C₄	1	2	yes	—
14	C₄×#41×C₆₁	C₄	1	2	yes	—
15	C₄×#41×C₆₁	C₂²	1	2	yes	—
16	C₄×#41×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₁	—	2	1	no	—
20	C₂×C₄×D₄×C₆₁	—	2	1	no	—
21	C₂×C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₁	—	2	1	no	—
24	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
25	C₂²×C₄²×C₆₁	—	2	1	no	—
26	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#432	—	2	1	no	—
28	C₆₁×#432	—	2	1	no	—
29	C₆₁×#432	—	2	1	no	—
30	C₆₁×#432	C₂²	1	2	yes	complemented
31	C₆₁×#431	—	2	1	no	—
32	C₆₁×#431	C₂²	1	2	yes	—
33	C₆₁×#431	C₂²	1	2	yes	—
34	C₆₁×#431	C₂²	1	2	yes	complemented
35	C₆₁×#423	—	2	1	no	—
36	C₆₁×#423	C₂²	1	2	yes	complemented
37	C₆₁×#407	—	2	1	no	—
38	C₆₁×#407	—	2	1	no	—
39	C₄×M₁₆×C₆₁	—	2	1	no	—
40	C₄×M₁₆×C₆₁	C₂²	1	2	yes	complemented
41	C₆₁×Q₈⋊C₈	C₂²	1	8	yes	complemented
42	C₆₁×D₄⋊C₈	C₂²	1	8	yes	complemented
43	C₂×#41×C₆₁	—	2	4	no	—
44	C₂×#41×C₆₁	—	2	4	no	—
45	C₂×#41×C₆₁	—	2	1	no	—
46	C₂×#41×C₆₁	—	2	1	no	—
47	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
48	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
49	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₆₁	—	2	1	no	—
51	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₁	—	2	1	no	—
53	C₂³×C₄×C₆₁	—	2	1	no	—
54	C₂³×C₄×C₆₁	—	2	1	no	—
55	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₆₁	—	2	2	no	—
59	C₂×M₁₆×C₆₁	—	2	2	no	—
60	C₄×Q₈×C₆₁	—	2	1	no	—
61	C₄×Q₈×C₆₁	—	2	1	no	—
62	C₄×Q₈×C₆₁	—	2	1	no	—
63	C₄×Q₈×C₆₁	—	2	1	no	—
64	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
65	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
66	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
67	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
68	C₄×D₄×C₆₁	—	2	2	no	—
69	C₄×D₄×C₆₁	—	2	2	no	—
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	—	2	2	no	—
74	C₄×D₄×C₆₁	—	2	2	no	—
75	C₄×D₄×C₆₁	—	2	2	no	—
76	C₄×D₄×C₆₁	—	2	1	no	—
77	C₄×D₄×C₆₁	—	2	1	no	—
78	C₄×D₄×C₆₁	—	2	1	no	—
79	C₄×D₄×C₆₁	—	2	1	no	—
80	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
81	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
82	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
83	C₄×D₄×C₆₁	C₂³	1	2	yes	—
84	#117×C₆₁	—	2	1	no	—
85	#117×C₆₁	—	2	1	no	—
86	#117×C₆₁	—	2	2	no	—
87	#117×C₆₁	—	2	2	no	—
88	#117×C₆₁	—	2	2	no	—
89	#117×C₆₁	—	2	2	no	—
90	#117×C₆₁	C₂×C₄	1	2	yes	—
91	#117×C₆₁	C₂×C₄	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
99	C₂×C₄²×C₆₁	—	2	1	no	—
100	C₂×C₄²×C₆₁	—	2	1	no	—
101	C₂×C₄²×C₆₁	—	2	1	no	—
102	C₂×C₄²×C₆₁	—	2	2	no	—
103	C₂×C₄²×C₆₁	—	2	2	no	—
104	C₂×C₄²×C₆₁	—	2	2	no	—
105	C₂×C₄²×C₆₁	—	2	2	no	—
106	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
118	C₄⋊C₈×C₆₁	—	2	2	no	—
119	C₄⋊C₈×C₆₁	—	2	2	no	—
120	C₄⋊C₈×C₆₁	—	2	2	no	—
121	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
122	C₂²⋊C₈×C₆₁	—	2	2	no	—
123	C₂²⋊C₈×C₆₁	—	2	2	no	—
124	C₂²⋊C₈×C₆₁	—	2	2	no	—
125	C₂²⋊C₈×C₆₁	—	2	2	no	—
126	#97×C₆₁	—	2	2	no	—
127	#97×C₆₁	—	2	2	no	—
128	C₄×C₈×C₆₁	—	2	2	no	—
129	C₄×C₈×C₆₁	C₂³	1	4	yes	—
130	C₂⁴×C₆₁	—	2	1	no	—
131	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
132	#41×C₆₁	—	2	8	no	—
133	#41×C₆₁	—	2	8	no	—
134	#41×C₆₁	—	2	4	no	—
135	#41×C₆₁	—	2	4	no	—
136	#41×C₆₁	—	2	4	no	—
137	#41×C₆₁	—	2	4	no	—
138	C₂×Q₈×C₆₁	—	2	1	no	—
139	C₂×Q₈×C₆₁	—	2	2	no	—
140	C₂×Q₈×C₆₁	M₁₆	1	4	yes	complemented
141	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
142	C₂×D₄×C₆₁	—	2	8	no	—
143	C₂×D₄×C₆₁	—	2	2	no	—
144	C₂×D₄×C₆₁	—	2	2	no	—
145	C₂×D₄×C₆₁	—	2	1	no	—
146	C₂×D₄×C₆₁	—	2	2	no	—
147	C₂×D₄×C₆₁	M₁₆	1	4	yes	complemented
148	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
149	C₂²×C₄×C₆₁	—	2	8	no	—
150	C₂²×C₄×C₆₁	—	2	1	no	—
151	C₂²×C₄×C₆₁	—	2	2	no	—
152	C₂²×C₄×C₆₁	—	2	2	no	—
153	C₂²×C₄×C₆₁	—	2	2	no	—
154	C₂²×C₄×C₆₁	—	2	2	no	—
155	C₂²×C₄×C₆₁	—	2	1	no	—
156	C₂²×C₄×C₆₁	—	2	1	no	—
157	C₂²×C₄×C₆₁	—	2	2	no	—
158	C₂²×C₄×C₆₁	—	2	2	no	—
159	C₂²×C₄×C₆₁	—	2	2	no	—
160	C₂²×C₄×C₆₁	—	2	2	no	—
161	C₂²×C₄×C₆₁	—	4	2	no	—
162	C₂²×C₄×C₆₁	—	4	2	no	—
163	C₂²×C₄×C₆₁	—	4	4	no	—
164	C₂²×C₄×C₆₁	—	4	4	no	—
165	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
183	M₁₆×C₆₁	—	4	4	no	—
184	M₁₆×C₆₁	—	4	4	no	—
185	C₂×C₈×C₆₁	—	2	4	no	—
186	C₂×C₈×C₆₁	—	2	4	no	—
187	C₄⋊C₄×C₆₁	—	2	2	no	—
188	C₄⋊C₄×C₆₁	—	2	2	no	—
189	C₄⋊C₄×C₆₁	—	2	2	no	—
190	C₄⋊C₄×C₆₁	—	2	1	no	—
191	C₄⋊C₄×C₆₁	—	2	1	no	—
192	C₄⋊C₄×C₆₁	—	2	2	no	—
193	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
194	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
195	C₂²⋊C₄×C₆₁	—	2	2	no	—
196	C₂²⋊C₄×C₆₁	—	2	2	no	—
197	C₂²⋊C₄×C₆₁	—	2	2	no	—
198	C₂²⋊C₄×C₆₁	—	2	2	no	—
199	C₂²⋊C₄×C₆₁	—	2	2	no	—
200	C₂²⋊C₄×C₆₁	—	2	2	no	—
201	C₄²×C₆₁	—	2	2	no	—
202	C₄²×C₆₁	—	2	2	no	—
203	C₄²×C₆₁	—	2	2	no	—
204	C₄²×C₆₁	—	2	2	no	—
205	C₄²×C₆₁	—	2	2	no	—
206	C₄²×C₆₁	—	2	2	no	—
207	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
210	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
211	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
212	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
213	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
214	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
215	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
223	C₂³×C₆₁	—	2	2	no	—
224	C₂³×C₆₁	—	2	2	no	—
225	C₂³×C₆₁	—	2	4	no	—
226	C₂³×C₆₁	—	4	1	no	—
227	C₂³×C₆₁	—	4	1	no	—
228	C₂³×C₆₁	—	4	2	no	—
229	C₂³×C₆₁	—	4	2	no	—
230	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	Q₈×C₆₁	—	2	2	no	—
234	Q₈×C₆₁	—	2	4	no	—
235	Q₈×C₆₁	C₂×M₁₆	1	4	yes	—
236	D₄×C₆₁	—	2	8	no	—
237	D₄×C₆₁	—	2	8	no	—
238	D₄×C₆₁	—	2	2	no	—
239	D₄×C₆₁	—	2	4	no	—
240	D₄×C₆₁	C₂×M₁₆	1	4	yes	—
241	C₂×C₄×C₆₁	—	2	8	no	—
242	C₂×C₄×C₆₁	—	2	8	no	—
243	C₂×C₄×C₆₁	—	2	2	no	—
244	C₂×C₄×C₆₁	—	2	2	no	—
245	C₂×C₄×C₆₁	—	2	4	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	2	no	—
251	C₂×C₄×C₆₁	—	2	2	no	—
252	C₂×C₄×C₆₁	—	2	2	no	—
253	C₂×C₄×C₆₁	—	2	2	no	—
254	C₂×C₄×C₆₁	—	2	2	no	—
255	C₂×C₄×C₆₁	—	2	2	no	—
256	C₂×C₄×C₆₁	—	2	2	no	—
257	C₂×C₄×C₆₁	—	2	2	no	—
258	C₂×C₄×C₆₁	—	2	2	no	—
259	C₂×C₄×C₆₁	—	2	2	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	4	no	—
267	C₂×C₄×C₆₁	—	4	4	no	—
268	C₂×C₄×C₆₁	—	4	2	no	—
269	C₂×C₄×C₆₁	—	4	2	no	—
270	C₂×C₄×C₆₁	—	4	4	no	—
271	C₂×C₄×C₆₁	—	4	4	no	—
272	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
282	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
283	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
284	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
286	C₈×C₆₁	—	4	4	no	—
287	C₈×C₆₁	—	4	4	no	—
288	#17450	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
289	C₂²×C₆₁	—	2	1	no	—
290	C₂²×C₆₁	—	2	1	no	—
291	C₂²×C₆₁	—	2	1	no	—
292	C₂²×C₆₁	—	2	1	no	—
293	C₂²×C₆₁	—	2	4	no	—
294	C₂²×C₆₁	—	2	1	no	—
295	C₂²×C₆₁	—	2	1	no	—
296	C₂²×C₆₁	—	2	1	no	—
297	C₂²×C₆₁	—	2	1	no	—
298	C₂²×C₆₁	—	4	4	no	—
299	C₂²×C₆₁	—	4	8	no	—
300	C₂²×C₆₁	—	4	2	no	—
301	C₂²×C₆₁	—	4	2	no	—
302	C₂²×C₆₁	#407	1	2	yes	—
303	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
304	C₄×C₆₁	—	2	1	no	—
305	C₄×C₆₁	—	2	1	no	—
306	C₄×C₆₁	—	2	4	no	—
307	C₄×C₆₁	—	2	1	no	—
308	C₄×C₆₁	—	2	1	no	—
309	C₄×C₆₁	—	2	1	no	—
310	C₄×C₆₁	—	2	1	no	—
311	C₄×C₆₁	—	2	1	no	—
312	C₄×C₆₁	—	2	1	no	—
313	C₄×C₆₁	—	4	2	no	—
314	C₄×C₆₁	—	4	2	no	—
315	C₄×C₆₁	#407	1	2	yes	—
316	C₄×C₆₁	#421	1	2	yes	—
317	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
318	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
319	C₄×C₆₁	#575	1	2	yes	—
320	C₂×C₄×#41	C₂×C₆₁	1	1	yes	characteristic
321	#2920	C₂×C₆₁	1	1	yes	characteristic, complemented
322	#2920	C₂×C₆₁	1	1	yes	characteristic, complemented
323	#1489	C₂×C₆₁	1	4	yes	complemented
324	#1488	C₂×C₆₁	1	4	yes	complemented
325	#1479	C₂×C₆₁	1	2	yes	complemented
326	#1478	C₂×C₆₁	1	2	yes	complemented
327	C₂×C₆₁	—	2	2	no	—
328	C₂×C₆₁	—	4	1	no	—
329	C₂×C₆₁	—	4	1	no	—
330	C₂×C₆₁	—	4	4	no	—
331	C₂×C₆₁	C₂×#407	1	1	yes	characteristic
332	C₂×C₆₁	#2896	1	1	yes	characteristic
333	C₂×C₆₁	C₂×#421	1	1	yes	characteristic
334	C₂²×#41	C₄×C₆₁	1	1	yes	characteristic
335	C₄×#41	—	2	1	no	—
336	C₄×#41	—	2	1	no	—
337	C₄×#41	—	2	1	no	—
338	C₄×#41	—	2	1	no	—
339	C₄×#41	C₄×C₆₁	1	2	yes	—
340	C₄×#41	C₄×C₆₁	1	2	yes	—
341	C₄×#41	C₂²×C₆₁	1	2	yes	—
342	C₄×#41	C₂²×C₆₁	1	2	yes	—
343	C₂×C₄×Q₈	C₄×C₆₁	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
345	C₂×C₄×D₄	—	2	1	no	—
346	C₂×C₄×D₄	—	2	1	no	—
347	C₂×C₄×D₄	C₄×C₆₁	1	1	yes	characteristic
348	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
349	C₂×#117	—	2	1	no	—
350	C₂×#117	C₄×C₆₁	1	1	yes	characteristic
351	C₂²×C₄²	—	2	1	no	—
352	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
353	#432	—	2	1	no	—
354	#432	—	2	1	no	—
355	#432	—	2	1	no	—
356	#432	C₂²×C₆₁	1	2	yes	complemented
357	#431	—	2	1	no	—
358	#431	C₂²×C₆₁	1	2	yes	—
359	#431	C₂²×C₆₁	1	2	yes	—
360	#431	C₂²×C₆₁	1	2	yes	complemented
361	#423	—	2	1	no	—
362	#423	C₂²×C₆₁	1	2	yes	complemented
363	#407	—	2	1	no	—
364	#407	—	2	1	no	—
365	C₄×M₁₆	—	2	1	no	—
366	C₄×M₁₆	C₂²×C₆₁	1	2	yes	complemented
367	Q₈⋊C₈	C₂²×C₆₁	1	8	yes	complemented
368	D₄⋊C₈	C₂²×C₆₁	1	8	yes	complemented
369	C₆₁	#17450	1	1	yes	characteristic, direct product factor, Sylow
370	C₂×#41	—	2	4	no	—
371	C₂×#41	—	2	4	no	—
372	C₂×#41	—	2	1	no	—
373	C₂×#41	—	2	1	no	—
374	C₂×#41	C₂×C₄×C₆₁	1	2	yes	—
375	C₂×#41	C₂×C₄×C₆₁	1	2	yes	—
376	C₂²×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
377	C₂²×D₄	—	2	1	no	—
378	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
379	C₂³×C₄	—	2	1	no	—
380	C₂³×C₄	—	2	1	no	—
381	C₂³×C₄	—	2	1	no	—
382	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
383	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
384	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
385	C₂×M₁₆	—	2	2	no	—
386	C₂×M₁₆	—	2	2	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
392	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
393	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
394	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
408	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
409	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
410	C₄×D₄	C₂³×C₆₁	1	2	yes	—
411	#117	—	2	1	no	—
412	#117	—	2	1	no	—
413	#117	—	2	2	no	—
414	#117	—	2	2	no	—
415	#117	—	2	2	no	—
416	#117	—	2	2	no	—
417	#117	C₂×C₄×C₆₁	1	2	yes	—
418	#117	C₂×C₄×C₆₁	1	2	yes	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	—	2	1	no	—
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	—	2	2	no	—
430	C₂×C₄²	—	2	2	no	—
431	C₂×C₄²	—	2	2	no	—
432	C₂×C₄²	—	2	2	no	—
433	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
434	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
435	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
436	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
437	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
438	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
439	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
440	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
441	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
442	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
445	C₄⋊C₈	—	2	2	no	—
446	C₄⋊C₈	—	2	2	no	—
447	C₄⋊C₈	—	2	2	no	—
448	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
449	C₂²⋊C₈	—	2	2	no	—
450	C₂²⋊C₈	—	2	2	no	—
451	C₂²⋊C₈	—	2	2	no	—
452	C₂²⋊C₈	—	2	2	no	—
453	#97	—	2	2	no	—
454	#97	—	2	2	no	—
455	C₄×C₈	—	2	2	no	—
456	C₄×C₈	C₂³×C₆₁	1	4	yes	—
457	C₂⁴	—	2	1	no	—
458	C₂⁴	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
459	#41	—	2	8	no	—
460	#41	—	2	8	no	—
461	#41	—	2	4	no	—
462	#41	—	2	4	no	—
463	#41	—	2	4	no	—
464	#41	—	2	4	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	2	no	—
467	C₂×Q₈	M₁₆×C₆₁	1	4	yes	complemented
468	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
469	C₂×D₄	—	2	8	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	M₁₆×C₆₁	1	4	yes	complemented
475	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
476	C₂²×C₄	—	2	8	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	4	2	no	—
489	C₂²×C₄	—	4	2	no	—
490	C₂²×C₄	—	4	4	no	—
491	C₂²×C₄	—	4	4	no	—
492	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
493	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
494	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
495	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
496	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
497	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
498	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
499	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
500	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
501	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
502	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
503	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
504	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
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	M₁₆	—	4	4	no	—
511	M₁₆	—	4	4	no	—
512	C₂×C₈	—	2	4	no	—
513	C₂×C₈	—	2	4	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
521	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	2	no	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	2	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	2	2	no	—
531	C₄²	—	2	2	no	—
532	C₄²	—	2	2	no	—
533	C₄²	—	2	2	no	—
534	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
535	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
536	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
537	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
538	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
539	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
540	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
541	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
542	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
543	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
544	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
545	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
546	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
547	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
548	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
549	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
550	C₂³	—	2	2	no	—
551	C₂³	—	2	4	no	—
552	C₂³	—	2	2	no	—
553	C₂³	—	4	1	no	—
554	C₂³	—	4	1	no	—
555	C₂³	—	4	2	no	—
556	C₂³	—	4	2	no	—
557	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
558	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
559	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
560	Q₈	—	2	2	no	—
561	Q₈	—	2	4	no	—
562	Q₈	C₂×M₁₆×C₆₁	1	4	yes	—
563	D₄	—	2	8	no	—
564	D₄	—	2	8	no	—
565	D₄	—	2	2	no	—
566	D₄	—	2	4	no	—
567	D₄	C₂×M₁₆×C₆₁	1	4	yes	—
568	C₂×C₄	—	2	8	no	—
569	C₂×C₄	—	2	8	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	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	4	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	4	4	no	—
598	C₂×C₄	—	4	4	no	—
599	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
600	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
601	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
602	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
603	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
604	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
605	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
606	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
607	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
608	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
609	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
610	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
611	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
612	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
613	C₈	—	4	4	no	—
614	C₈	—	4	4	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	1	no	—
619	C₂²	—	2	4	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	2	1	no	—
622	C₂²	—	2	1	no	—
623	C₂²	—	2	1	no	—
624	C₂²	—	4	4	no	—
625	C₂²	—	4	8	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	C₆₁×#407	1	2	yes	—
629	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
630	C₄	—	2	1	no	—
631	C₄	—	2	1	no	—
632	C₄	—	2	4	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	2	1	no	—
637	C₄	—	2	1	no	—
638	C₄	—	2	1	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	2	no	—
641	C₄	C₆₁×#407	1	2	yes	—
642	C₄	C₆₁×#421	1	2	yes	—
643	C₄	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
644	C₄	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
645	C₄	C₆₁×#575	1	2	yes	—
646	C₂	—	2	2	no	—
647	C₂	—	4	1	no	—
648	C₂	—	4	1	no	—
649	C₂	—	4	4	no	—
650	C₂	C₂×C₆₁×#407	1	1	yes	characteristic
651	C₂	C₆₁×#2896	1	1	yes	characteristic
652	C₂	C₂×C₆₁×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₄²	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₆₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2920	C₂	semidirect
3. C₆₁×#2920	C₂	semidirect
4. C₆₁×#1489	C₂	semidirect
5. C₆₁×#1488	C₂	semidirect
6. C₆₁×#1479	C₂	semidirect
7. C₆₁×#1478	C₂	semidirect
30. C₆₁×#432	C₂²	semidirect
34. C₆₁×#431	C₂²	semidirect
36. C₆₁×#423	C₂²	semidirect
40. C₄×M₁₆×C₆₁	C₂²	semidirect
41. C₆₁×Q₈⋊C₈	C₂²	semidirect
42. C₆₁×D₄⋊C₈	C₂²	semidirect
140. C₂×Q₈×C₆₁	M₁₆	semidirect
147. C₂×D₄×C₆₁	M₁₆	semidirect
288. #17450	C₆₁	direct (N × H)
321. #2920	C₂×C₆₁	semidirect
322. #2920	C₂×C₆₁	semidirect
323. #1489	C₂×C₆₁	semidirect
324. #1488	C₂×C₆₁	semidirect
325. #1479	C₂×C₆₁	semidirect
326. #1478	C₂×C₆₁	semidirect
356. #432	C₂²×C₆₁	semidirect
360. #431	C₂²×C₆₁	semidirect
362. #423	C₂²×C₆₁	semidirect
366. C₄×M₁₆	C₂²×C₆₁	semidirect
367. Q₈⋊C₈	C₂²×C₆₁	semidirect
368. D₄⋊C₈	C₂²×C₆₁	semidirect
369. C₆₁	#17450	direct (N × H)
467. C₂×Q₈	M₁₆×C₆₁	semidirect
474. C₂×D₄	M₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₁	2
C₆₁×#2920	2
C₆₁×#1489	2
C₆₁×#1488	2
C₆₁×#1479	2
C₆₁×#1478	2
#17450	61

Subgroups — C61×#17450 #29520883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#17450 #29520883