Subgroups — C₆₁×#18008 #29521441

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3105	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2969	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1586	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1583	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1581	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1580	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₁×#1495	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1478	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₆₁	—	2	1	no	—
17	C₆₁×#529	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#484	C₄	1	1	yes	characteristic
20	C₆₁×#481	C₄	1	1	yes	characteristic
21	C₆₁×#478	C₄	1	1	yes	characteristic
22	C₆₁×#478	C₄	1	1	yes	characteristic
23	C₆₁×#463	C₄	1	1	yes	characteristic
24	C₆₁×#463	C₄	1	1	yes	characteristic
25	C₆₁×#450	C₄	1	1	yes	characteristic
26	C₆₁×#447	C₄	1	1	yes	characteristic
27	C₆₁×#446	C₂²	1	1	yes	characteristic, complemented
28	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
29	C₄×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
30	C₄×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₆₁×#436	—	2	1	no	—
34	C₆₁×#436	C₂²	1	1	yes	characteristic
35	C₆₁×#436	C₂²	1	1	yes	characteristic
36	C₆₁×#436	C₂²	1	1	yes	characteristic, complemented
37	C₆₁×#435	—	2	1	no	—
38	C₆₁×#435	—	2	1	no	—
39	C₆₁×#435	—	2	1	no	—
40	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
42	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
43	C₆₁×#433	—	2	1	no	—
44	C₆₁×#431	C₂²	1	1	yes	characteristic
45	C₆₁×#431	C₂²	1	1	yes	characteristic
46	C₆₁×#431	C₂²	1	1	yes	characteristic
47	C₆₁×#423	C₂²	1	1	yes	characteristic
48	C₆₁×#407	—	2	1	no	—
49	C₆₁×#405	—	2	1	no	—
50	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
51	C₆₁×#335	C₂²	1	1	yes	characteristic, complemented
52	C₆₁×#335	C₂²	1	1	yes	characteristic, complemented
53	C₆₁×#334	C₂²	1	1	yes	characteristic, complemented
54	C₆₁×#334	C₂²	1	1	yes	characteristic, complemented
55	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
62	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₆₁×#322	C₂²	1	1	yes	characteristic, complemented
64	C₆₁×#322	C₂²	1	1	yes	characteristic, complemented
65	C₆₁×#322	C₂²	1	1	yes	characteristic, complemented
66	C₈²×C₆₁	C₂²	1	1	yes	characteristic, complemented
67	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₆₁	—	2	1	no	—
69	C₂×SD₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
71	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆×C₆₁	—	2	1	no	—
74	C₂×M₁₆×C₆₁	—	2	1	no	—
75	C₂×M₁₆×C₆₁	—	2	1	no	—
76	C₂×M₁₆×C₆₁	—	2	1	no	—
77	C₂×M₁₆×C₆₁	—	4	4	no	—
78	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
79	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₆₁	—	2	1	no	—
81	C₂²×C₈×C₆₁	—	2	1	no	—
82	C₂²×C₈×C₆₁	—	2	1	no	—
83	#126×C₆₁	—	2	1	no	—
84	#125×C₆₁	C₂×C₄	1	1	yes	characteristic
85	#124×C₆₁	C₂×C₄	1	1	yes	characteristic
86	#123×C₆₁	—	2	1	no	—
87	C₂²⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
88	#121×C₆₁	C₂×C₄	1	1	yes	characteristic
89	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₆₁	—	2	1	no	—
91	C₄×D₄×C₆₁	—	2	1	no	—
92	C₄×D₄×C₆₁	—	2	1	no	—
93	C₄×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
95	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
98	#117×C₆₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
101	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
102	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
103	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
104	#106×C₆₁	C₂×C₄	1	1	yes	characteristic
105	#106×C₆₁	C₂×C₄	1	1	yes	characteristic
106	C₄⋊C₈×C₆₁	—	2	1	no	—
107	C₄⋊C₈×C₆₁	—	2	1	no	—
108	C₄⋊C₈×C₆₁	—	2	1	no	—
109	C₄⋊C₈×C₆₁	—	2	1	no	—
110	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
111	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
112	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
113	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
114	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
115	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
116	#103×C₆₁	C₂×C₄	1	1	yes	characteristic
117	#103×C₆₁	C₂×C₄	1	1	yes	characteristic
118	#102×C₆₁	C₂×C₄	1	1	yes	characteristic
119	#102×C₆₁	C₂×C₄	1	1	yes	characteristic
120	#102×C₆₁	C₂×C₄	1	1	yes	characteristic
121	#102×C₆₁	C₂×C₄	1	1	yes	characteristic
122	#102×C₆₁	C₂×C₄	1	1	yes	characteristic
123	#102×C₆₁	C₂×C₄	1	1	yes	characteristic
124	C₂²⋊C₈×C₆₁	—	2	1	no	—
125	C₂²⋊C₈×C₆₁	—	2	1	no	—
126	C₂²⋊C₈×C₆₁	—	2	1	no	—
127	C₂²⋊C₈×C₆₁	—	2	1	no	—
128	C₂²⋊C₈×C₆₁	—	2	1	no	—
129	C₂²⋊C₈×C₆₁	—	2	1	no	—
130	C₂²⋊C₈×C₆₁	—	2	1	no	—
131	C₂²⋊C₈×C₆₁	—	2	1	no	—
132	C₂²⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
133	C₂²⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
134	#97×C₆₁	—	2	1	no	—
135	#97×C₆₁	D₄	1	1	yes	characteristic
136	#97×C₆₁	D₄	1	1	yes	characteristic
137	C₄×C₈×C₆₁	—	2	1	no	—
138	C₄×C₈×C₆₁	—	2	1	no	—
139	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
145	C₂⁴×C₆₁	—	2	1	no	—
146	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
147	C₂×D₄×C₆₁	—	2	2	no	—
148	C₂×D₄×C₆₁	—	2	1	no	—
149	C₂×D₄×C₆₁	—	2	1	no	—
150	C₂×D₄×C₆₁	—	4	1	no	—
151	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
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	—	2	1	no	—
159	C₂²×C₄×C₆₁	—	2	1	no	—
160	C₂²×C₄×C₆₁	—	4	2	no	—
161	C₂²×C₄×C₆₁	—	4	2	no	—
162	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
165	SD₁₆×C₆₁	—	2	2	no	—
166	SD₁₆×C₆₁	—	4	1	no	—
167	D₈×C₆₁	—	2	2	no	—
168	D₈×C₆₁	—	4	1	no	—
169	M₁₆×C₆₁	—	2	2	no	—
170	M₁₆×C₆₁	—	4	1	no	—
171	M₁₆×C₆₁	—	4	2	no	—
172	M₁₆×C₆₁	—	4	2	no	—
173	M₁₆×C₆₁	—	4	2	no	—
174	M₁₆×C₆₁	—	4	2	no	—
175	C₂×C₈×C₆₁	—	2	1	no	—
176	C₂×C₈×C₆₁	—	2	1	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	—	4	2	no	—
182	C₂×C₈×C₆₁	—	4	2	no	—
183	C₂×C₈×C₆₁	—	4	2	no	—
184	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
185	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
186	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
187	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
188	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
191	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
201	C₂²⋊C₄×C₆₁	—	2	1	no	—
202	C₂²⋊C₄×C₆₁	—	2	1	no	—
203	C₂²⋊C₄×C₆₁	—	2	1	no	—
204	C₂²⋊C₄×C₆₁	—	2	1	no	—
205	C₂²⋊C₄×C₆₁	—	2	1	no	—
206	C₂²⋊C₄×C₆₁	—	2	1	no	—
207	C₂²⋊C₄×C₆₁	—	2	1	no	—
208	C₂²⋊C₄×C₆₁	—	2	1	no	—
209	C₂²⋊C₄×C₆₁	—	2	1	no	—
210	C₄²×C₆₁	—	2	1	no	—
211	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₆₁	#41	1	1	yes	characteristic
214	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
215	C₂³×C₆₁	—	2	1	no	—
216	C₂³×C₆₁	—	2	1	no	—
217	C₂³×C₆₁	—	2	1	no	—
218	C₂³×C₆₁	—	4	2	no	—
219	C₂³×C₆₁	—	4	2	no	—
220	C₂³×C₆₁	—	8	1	no	—
221	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
222	Q₈×C₆₁	—	2	1	no	—
223	Q₈×C₆₁	—	2	1	no	—
224	D₄×C₆₁	—	2	1	no	—
225	D₄×C₆₁	—	2	1	no	—
226	D₄×C₆₁	—	2	1	no	—
227	D₄×C₆₁	—	2	1	no	—
228	D₄×C₆₁	—	2	1	no	—
229	D₄×C₆₁	—	4	2	no	—
230	D₄×C₆₁	—	8	1	no	—
231	D₄×C₆₁	#131	1	2	yes	—
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	—	4	1	no	—
242	C₂×C₄×C₆₁	—	4	2	no	—
243	C₂×C₄×C₆₁	—	4	2	no	—
244	C₂×C₄×C₆₁	—	4	2	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	—	4	1	no	—
249	C₂×C₄×C₆₁	—	4	1	no	—
250	C₂×C₄×C₆₁	—	4	1	no	—
251	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
253	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
255	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
257	C₈×C₆₁	—	2	1	no	—
258	C₈×C₆₁	—	2	1	no	—
259	C₈×C₆₁	—	2	1	no	—
260	C₈×C₆₁	—	2	1	no	—
261	C₈×C₆₁	—	2	1	no	—
262	C₈×C₆₁	—	2	1	no	—
263	C₈×C₆₁	—	2	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	C₈×C₆₁	—	4	1	no	—
270	C₈×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
271	#18008	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂²×C₆₁	—	2	1	no	—
273	C₂²×C₆₁	—	2	2	no	—
274	C₂²×C₆₁	—	2	1	no	—
275	C₂²×C₆₁	—	2	1	no	—
276	C₂²×C₆₁	—	4	1	no	—
277	C₂²×C₆₁	—	4	1	no	—
278	C₂²×C₆₁	—	4	1	no	—
279	C₂²×C₆₁	—	4	1	no	—
280	C₂²×C₆₁	—	4	1	no	—
281	C₂²×C₆₁	—	4	1	no	—
282	C₂²×C₆₁	—	4	1	no	—
283	C₂²×C₆₁	—	4	1	no	—
284	C₂²×C₆₁	—	8	4	no	—
285	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
286	C₄×C₆₁	—	2	1	no	—
287	C₄×C₆₁	—	2	1	no	—
288	C₄×C₆₁	—	2	1	no	—
289	C₄×C₆₁	—	2	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	C₂×#131	1	1	yes	characteristic
296	C₄×C₆₁	#568	1	1	yes	characteristic
297	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
298	C₄×C₆₁	#575	1	1	yes	characteristic
299	#3105	C₂×C₆₁	1	1	yes	characteristic
300	#2969	C₂×C₆₁	1	1	yes	characteristic, complemented
301	D₄×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
302	#1592	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#1589	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#1586	C₂×C₆₁	1	1	yes	characteristic, complemented
305	#1585	C₂×C₆₁	1	1	yes	characteristic, complemented
306	#1583	C₂×C₆₁	1	1	yes	characteristic, complemented
307	#1581	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#1580	C₂×C₆₁	1	1	yes	characteristic, complemented
309	C₈×SD₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
310	#1574	C₂×C₆₁	1	1	yes	characteristic, complemented
311	#1495	C₂×C₆₁	1	1	yes	characteristic, complemented
312	#1478	C₂×C₆₁	1	1	yes	characteristic, complemented
313	#1453	C₂×C₆₁	1	1	yes	characteristic, complemented
314	C₂×C₆₁	—	4	1	no	—
315	C₂×C₆₁	—	4	2	no	—
316	C₂×C₆₁	—	8	1	no	—
317	C₂×C₆₁	—	8	1	no	—
318	C₂×C₆₁	#2933	1	1	yes	characteristic
319	C₂×C₆₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
320	C₂×C₆₁	#2957	1	1	yes	characteristic
321	C₂²×M₁₆	—	2	1	no	—
322	#529	C₂²×C₆₁	1	1	yes	characteristic
323	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
324	#484	C₄×C₆₁	1	1	yes	characteristic
325	#481	C₄×C₆₁	1	1	yes	characteristic
326	#478	C₄×C₆₁	1	1	yes	characteristic
327	#478	C₄×C₆₁	1	1	yes	characteristic
328	#463	C₄×C₆₁	1	1	yes	characteristic
329	#463	C₄×C₆₁	1	1	yes	characteristic
330	#450	C₄×C₆₁	1	1	yes	characteristic
331	#447	C₄×C₆₁	1	1	yes	characteristic
332	#446	C₂²×C₆₁	1	1	yes	characteristic, complemented
333	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
334	C₄×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
335	C₄×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
336	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
337	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
338	#436	—	2	1	no	—
339	#436	C₂²×C₆₁	1	1	yes	characteristic
340	#436	C₂²×C₆₁	1	1	yes	characteristic
341	#436	C₂²×C₆₁	1	1	yes	characteristic, complemented
342	#435	—	2	1	no	—
343	#435	—	2	1	no	—
344	#435	—	2	1	no	—
345	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic
346	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic
347	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic, complemented
348	#433	—	2	1	no	—
349	#431	C₂²×C₆₁	1	1	yes	characteristic
350	#431	C₂²×C₆₁	1	1	yes	characteristic
351	#431	C₂²×C₆₁	1	1	yes	characteristic
352	#423	C₂²×C₆₁	1	1	yes	characteristic
353	#407	—	2	1	no	—
354	#405	—	2	1	no	—
355	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
356	#335	C₂²×C₆₁	1	1	yes	characteristic, complemented
357	#335	C₂²×C₆₁	1	1	yes	characteristic, complemented
358	#334	C₂²×C₆₁	1	1	yes	characteristic, complemented
359	#334	C₂²×C₆₁	1	1	yes	characteristic, complemented
360	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
361	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
362	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
363	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
364	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
365	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
366	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
367	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
368	#322	C₂²×C₆₁	1	1	yes	characteristic, complemented
369	#322	C₂²×C₆₁	1	1	yes	characteristic, complemented
370	#322	C₂²×C₆₁	1	1	yes	characteristic, complemented
371	C₈²	C₂²×C₆₁	1	1	yes	characteristic, complemented
372	C₆₁	#18008	1	1	yes	characteristic, direct product factor, Sylow
373	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
374	C₂³×C₄	—	2	1	no	—
375	C₂×SD₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
376	C₂×SD₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
377	C₂×D₈	C₂×C₄×C₆₁	1	1	yes	characteristic
378	C₂×D₈	C₂×C₄×C₆₁	1	1	yes	characteristic
379	C₂×M₁₆	—	2	1	no	—
380	C₂×M₁₆	—	2	1	no	—
381	C₂×M₁₆	—	2	1	no	—
382	C₂×M₁₆	—	2	1	no	—
383	C₂×M₁₆	—	4	4	no	—
384	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
385	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
386	C₂²×C₈	—	2	1	no	—
387	C₂²×C₈	—	2	1	no	—
388	C₂²×C₈	—	2	1	no	—
389	#126	—	2	1	no	—
390	#125	C₂×C₄×C₆₁	1	1	yes	characteristic
391	#124	C₂×C₄×C₆₁	1	1	yes	characteristic
392	#123	—	2	1	no	—
393	C₂²⋊Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
394	#121	C₂×C₄×C₆₁	1	1	yes	characteristic
395	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	—	2	1	no	—
399	C₄×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
400	C₄×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
401	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
402	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
403	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
404	#117	—	2	1	no	—
405	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
408	#107	C₂×C₄×C₆₁	1	1	yes	characteristic
409	#107	C₂×C₄×C₆₁	1	1	yes	characteristic
410	#106	C₂×C₄×C₆₁	1	1	yes	characteristic
411	#106	C₂×C₄×C₆₁	1	1	yes	characteristic
412	C₄⋊C₈	—	2	1	no	—
413	C₄⋊C₈	—	2	1	no	—
414	C₄⋊C₈	—	2	1	no	—
415	C₄⋊C₈	—	2	1	no	—
416	C₄⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
417	C₄⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
418	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
419	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
420	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
421	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
422	#103	C₂×C₄×C₆₁	1	1	yes	characteristic
423	#103	C₂×C₄×C₆₁	1	1	yes	characteristic
424	#102	C₂×C₄×C₆₁	1	1	yes	characteristic
425	#102	C₂×C₄×C₆₁	1	1	yes	characteristic
426	#102	C₂×C₄×C₆₁	1	1	yes	characteristic
427	#102	C₂×C₄×C₆₁	1	1	yes	characteristic
428	#102	C₂×C₄×C₆₁	1	1	yes	characteristic
429	#102	C₂×C₄×C₆₁	1	1	yes	characteristic
430	C₂²⋊C₈	—	2	1	no	—
431	C₂²⋊C₈	—	2	1	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	C₂²⋊C₈	—	2	1	no	—
434	C₂²⋊C₈	—	2	1	no	—
435	C₂²⋊C₈	—	2	1	no	—
436	C₂²⋊C₈	—	2	1	no	—
437	C₂²⋊C₈	—	2	1	no	—
438	C₂²⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
439	C₂²⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
440	#97	—	2	1	no	—
441	#97	D₄×C₆₁	1	1	yes	characteristic
442	#97	D₄×C₆₁	1	1	yes	characteristic
443	C₄×C₈	—	2	1	no	—
444	C₄×C₈	—	2	1	no	—
445	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
446	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
447	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
449	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
450	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
451	C₂⁴	—	2	1	no	—
452	C₂×Q₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	4	1	no	—
457	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
458	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
459	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
470	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
471	SD₁₆	—	2	2	no	—
472	SD₁₆	—	4	1	no	—
473	D₈	—	2	2	no	—
474	D₈	—	4	1	no	—
475	M₁₆	—	2	2	no	—
476	M₁₆	—	4	1	no	—
477	M₁₆	—	4	2	no	—
478	M₁₆	—	4	2	no	—
479	M₁₆	—	4	2	no	—
480	M₁₆	—	4	2	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	4	2	no	—
488	C₂×C₈	—	4	2	no	—
489	C₂×C₈	—	4	2	no	—
490	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
491	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
492	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
493	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
494	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
495	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
496	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
497	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
504	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
505	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
506	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	2	1	no	—
510	C₂²⋊C₄	—	2	1	no	—
511	C₂²⋊C₄	—	2	1	no	—
512	C₂²⋊C₄	—	2	1	no	—
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	1	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₄²	—	2	1	no	—
517	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
518	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
519	C₄²	#41×C₆₁	1	1	yes	characteristic
520	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	C₂³	—	2	1	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	2	no	—
526	C₂³	—	8	1	no	—
527	C₂³	C₄×D₄×C₆₁	1	1	yes	characteristic
528	Q₈	—	2	1	no	—
529	Q₈	—	2	1	no	—
530	D₄	—	2	1	no	—
531	D₄	—	2	1	no	—
532	D₄	—	2	1	no	—
533	D₄	—	2	1	no	—
534	D₄	—	2	1	no	—
535	D₄	—	4	2	no	—
536	D₄	—	8	1	no	—
537	D₄	#131×C₆₁	1	2	yes	—
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	—	4	1	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
558	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
559	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
560	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
561	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
562	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
563	C₈	—	2	1	no	—
564	C₈	—	2	1	no	—
565	C₈	—	2	1	no	—
566	C₈	—	2	1	no	—
567	C₈	—	2	1	no	—
568	C₈	—	2	1	no	—
569	C₈	—	2	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	—	4	1	no	—
588	C₂²	—	4	1	no	—
589	C₂²	—	8	4	no	—
590	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	1	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	C₂×#131×C₆₁	1	1	yes	characteristic
601	C₄	C₆₁×#568	1	1	yes	characteristic
602	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
603	C₄	C₆₁×#575	1	1	yes	characteristic
604	C₂	—	4	1	no	—
605	C₂	—	4	2	no	—
606	C₂	—	8	1	no	—
607	C₂	—	8	1	no	—
608	C₂	C₆₁×#2933	1	1	yes	characteristic
609	C₂	C₄×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
610	C₂	C₆₁×#2957	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	C₂×C₄×D₄
prime radical	C₆₁×#447	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₄×M₁₆×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2969	C₂	semidirect
3. D₄×M₁₆×C₆₁	C₂	semidirect
4. C₆₁×#1592	C₂	semidirect
5. C₆₁×#1589	C₂	semidirect
6. C₆₁×#1586	C₂	semidirect
7. C₆₁×#1585	C₂	semidirect
8. C₆₁×#1583	C₂	semidirect
9. C₆₁×#1581	C₂	semidirect
10. C₆₁×#1580	C₂	semidirect
11. C₈×SD₁₆×C₆₁	C₂	semidirect
12. C₆₁×#1574	C₂	semidirect
13. C₆₁×#1495	C₂	semidirect
14. C₆₁×#1478	C₂	semidirect
15. C₆₁×#1453	C₂	semidirect
27. C₆₁×#446	C₂²	semidirect
28. C₈×Q₈×C₆₁	C₂²	semidirect
36. C₆₁×#436	C₂²	semidirect
42. C₈×D₄×C₆₁	C₂²	semidirect
51. C₆₁×#335	C₂²	semidirect
52. C₆₁×#335	C₂²	semidirect
53. C₆₁×#334	C₂²	semidirect
54. C₆₁×#334	C₂²	semidirect
55. C₆₁×Q₈⋊C₈	C₂²	semidirect
56. C₆₁×Q₈⋊C₈	C₂²	semidirect
61. C₆₁×D₄⋊C₈	C₂²	semidirect
62. C₆₁×D₄⋊C₈	C₂²	semidirect
63. C₆₁×#322	C₂²	semidirect
64. C₆₁×#322	C₂²	semidirect
65. C₆₁×#322	C₂²	semidirect
66. C₈²×C₆₁	C₂²	semidirect
271. #18008	C₆₁	direct (N × H)
300. #2969	C₂×C₆₁	semidirect
301. D₄×M₁₆	C₂×C₆₁	semidirect
302. #1592	C₂×C₆₁	semidirect
303. #1589	C₂×C₆₁	semidirect
304. #1586	C₂×C₆₁	semidirect
305. #1585	C₂×C₆₁	semidirect
306. #1583	C₂×C₆₁	semidirect
307. #1581	C₂×C₆₁	semidirect
308. #1580	C₂×C₆₁	semidirect
309. C₈×SD₁₆	C₂×C₆₁	semidirect
310. #1574	C₂×C₆₁	semidirect
311. #1495	C₂×C₆₁	semidirect
312. #1478	C₂×C₆₁	semidirect
313. #1453	C₂×C₆₁	semidirect
332. #446	C₂²×C₆₁	semidirect
333. C₈×Q₈	C₂²×C₆₁	semidirect
341. #436	C₂²×C₆₁	semidirect
347. C₈×D₄	C₂²×C₆₁	semidirect
356. #335	C₂²×C₆₁	semidirect
357. #335	C₂²×C₆₁	semidirect
358. #334	C₂²×C₆₁	semidirect
359. #334	C₂²×C₆₁	semidirect
360. Q₈⋊C₈	C₂²×C₆₁	semidirect
361. Q₈⋊C₈	C₂²×C₆₁	semidirect
366. D₄⋊C₈	C₂²×C₆₁	semidirect
367. D₄⋊C₈	C₂²×C₆₁	semidirect
368. #322	C₂²×C₆₁	semidirect
369. #322	C₂²×C₆₁	semidirect
370. #322	C₂²×C₆₁	semidirect
371. C₈²	C₂²×C₆₁	semidirect
372. C₆₁	#18008	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3105	2
C₆₁×#2969	2
D₄×M₁₆×C₆₁	2
C₆₁×#1592	2
C₆₁×#1589	2
C₆₁×#1586	2
C₆₁×#1585	2
C₆₁×#1583	2
C₆₁×#1581	2
C₆₁×#1580	2
C₈×SD₁₆×C₆₁	2
C₆₁×#1574	2
C₆₁×#1495	2
C₆₁×#1478	2
C₆₁×#1453	2
#18008	61

Subgroups — C61×#18008 #29521441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18008 #29521441