Subgroups — C₆₁×#18010 #29521443

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3108	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2969	C₂	1	1	yes	maximal, characteristic
3	Q₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1587	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1582	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1579	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
14	C₆₁×#1479	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₆₁×#529	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#483	C₄	1	1	yes	characteristic
19	C₆₁×#480	C₄	1	1	yes	characteristic
20	C₆₁×#479	C₄	1	1	yes	characteristic
21	C₆₁×#479	C₄	1	1	yes	characteristic
22	C₆₁×#464	C₄	1	1	yes	characteristic
23	C₆₁×#464	C₄	1	1	yes	characteristic
24	C₆₁×#451	C₄	1	1	yes	characteristic
25	C₆₁×#448	C₄	1	1	yes	characteristic
26	C₆₁×#446	—	2	1	no	—
27	C₆₁×#446	—	2	1	no	—
28	C₆₁×#446	C₂²	1	1	yes	characteristic
29	C₆₁×#446	C₂²	1	1	yes	characteristic
30	C₆₁×#446	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₄×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
37	C₄×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#436	—	2	1	no	—
39	C₆₁×#436	C₂²	1	1	yes	characteristic
40	C₆₁×#435	—	2	1	no	—
41	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#433	—	2	1	no	—
43	C₆₁×#431	C₂²	1	1	yes	characteristic
44	C₆₁×#431	C₂²	1	1	yes	characteristic
45	C₆₁×#431	C₂²	1	1	yes	characteristic
46	C₆₁×#423	—	2	1	no	—
47	C₆₁×#423	C₂²	1	1	yes	characteristic
48	C₆₁×#405	—	2	1	no	—
49	C₄×M₁₆×C₆₁	—	2	1	no	—
50	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
51	C₆₁×#335	C₂²	1	1	yes	characteristic
52	C₆₁×#335	C₂²	1	1	yes	characteristic
53	C₆₁×#334	C₂²	1	1	yes	characteristic
54	C₆₁×#334	C₂²	1	1	yes	characteristic
55	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
61	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₆₁×#322	C₂²	1	1	yes	characteristic
64	C₆₁×#322	C₂²	1	1	yes	characteristic
65	C₆₁×#322	C₂²	1	1	yes	characteristic
66	C₈²×C₆₁	C₂²	1	1	yes	characteristic
67	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₂×M₁₆×C₆₁	—	2	1	no	—
73	C₂×M₁₆×C₆₁	—	2	1	no	—
74	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
75	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₆₁	—	2	1	no	—
77	#126×C₆₁	—	2	1	no	—
78	#125×C₆₁	C₂×C₄	1	1	yes	characteristic
79	#124×C₆₁	C₂×C₄	1	1	yes	characteristic
80	#123×C₆₁	—	2	1	no	—
81	C₂²⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
82	#121×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₆₁	—	2	1	no	—
84	C₄×Q₈×C₆₁	—	2	1	no	—
85	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
88	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₆₁	—	2	1	no	—
91	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
92	#117×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×C₄²×C₆₁	—	2	1	no	—
96	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
97	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
98	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
99	#106×C₆₁	C₂×C₄	1	1	yes	characteristic
100	#106×C₆₁	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₈×C₆₁	—	2	1	no	—
102	C₄⋊C₈×C₆₁	—	2	1	no	—
103	C₄⋊C₈×C₆₁	—	2	1	no	—
104	C₄⋊C₈×C₆₁	—	2	1	no	—
105	C₄⋊C₈×C₆₁	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
110	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
111	C₄⋊C₈×C₆₁	C₂³	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	#103×C₆₁	C₂×C₄	1	1	yes	characteristic
116	#103×C₆₁	C₂×C₄	1	1	yes	characteristic
117	#103×C₆₁	C₂×C₄	1	1	yes	characteristic
118	#103×C₆₁	C₂×C₄	1	1	yes	characteristic
119	#103×C₆₁	C₂×C₄	1	1	yes	characteristic
120	#103×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	C₂²⋊C₈×C₆₁	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
128	C₂²⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
129	#97×C₆₁	—	2	1	no	—
130	#97×C₆₁	—	2	1	no	—
131	#97×C₆₁	—	2	1	no	—
132	#97×C₆₁	D₄	1	1	yes	characteristic
133	#97×C₆₁	D₄	1	1	yes	characteristic
134	C₄×C₈×C₆₁	—	2	1	no	—
135	C₄×C₈×C₆₁	—	2	1	no	—
136	C₄×C₈×C₆₁	—	2	1	no	—
137	C₄×C₈×C₆₁	—	2	1	no	—
138	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
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	C₂×Q₈×C₆₁	—	2	1	no	—
145	C₂×Q₈×C₆₁	—	2	2	no	—
146	C₂×Q₈×C₆₁	—	4	1	no	—
147	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄×C₆₁	—	2	1	no	—
151	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₁	—	2	1	no	—
153	C₂²×C₄×C₆₁	—	2	1	no	—
154	C₂²×C₄×C₆₁	—	2	1	no	—
155	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
158	Q₁₆×C₆₁	—	2	2	no	—
159	Q₁₆×C₆₁	—	4	1	no	—
160	SD₁₆×C₆₁	—	2	2	no	—
161	SD₁₆×C₆₁	—	4	1	no	—
162	M₁₆×C₆₁	—	2	2	no	—
163	M₁₆×C₆₁	—	4	1	no	—
164	M₁₆×C₆₁	—	4	2	no	—
165	M₁₆×C₆₁	—	4	2	no	—
166	C₂×C₈×C₆₁	—	2	1	no	—
167	C₂×C₈×C₆₁	—	2	1	no	—
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	—	4	2	no	—
173	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
174	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
181	C₄⋊C₄×C₆₁	—	2	1	no	—
182	C₄⋊C₄×C₆₁	—	2	1	no	—
183	C₄⋊C₄×C₆₁	—	2	1	no	—
184	C₄⋊C₄×C₆₁	—	2	1	no	—
185	C₄⋊C₄×C₆₁	—	2	1	no	—
186	C₄⋊C₄×C₆₁	—	2	1	no	—
187	C₄⋊C₄×C₆₁	—	2	1	no	—
188	C₄⋊C₄×C₆₁	—	2	1	no	—
189	C₄⋊C₄×C₆₁	—	2	1	no	—
190	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
194	C₂²⋊C₄×C₆₁	—	2	1	no	—
195	C₂²⋊C₄×C₆₁	—	2	1	no	—
196	C₂²⋊C₄×C₆₁	—	2	1	no	—
197	C₂²⋊C₄×C₆₁	—	2	1	no	—
198	C₂²⋊C₄×C₆₁	—	2	1	no	—
199	C₄²×C₆₁	—	2	1	no	—
200	C₄²×C₆₁	—	2	1	no	—
201	C₄²×C₆₁	—	2	1	no	—
202	C₄²×C₆₁	—	2	1	no	—
203	C₄²×C₆₁	—	2	1	no	—
204	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₆₁	#41	1	1	yes	characteristic
207	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
208	C₂³×C₆₁	—	2	1	no	—
209	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
210	Q₈×C₆₁	—	2	1	no	—
211	Q₈×C₆₁	—	2	1	no	—
212	Q₈×C₆₁	—	2	1	no	—
213	Q₈×C₆₁	—	2	1	no	—
214	Q₈×C₆₁	—	2	1	no	—
215	Q₈×C₆₁	—	4	2	no	—
216	Q₈×C₆₁	#131	1	2	yes	—
217	D₄×C₆₁	—	2	1	no	—
218	D₄×C₆₁	—	2	1	no	—
219	D₄×C₆₁	—	8	1	no	—
220	C₂×C₄×C₆₁	—	2	1	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	—	2	1	no	—
227	C₂×C₄×C₆₁	—	2	1	no	—
228	C₂×C₄×C₆₁	—	2	1	no	—
229	C₂×C₄×C₆₁	—	2	1	no	—
230	C₂×C₄×C₆₁	—	2	1	no	—
231	C₂×C₄×C₆₁	—	4	1	no	—
232	C₂×C₄×C₆₁	—	4	2	no	—
233	C₂×C₄×C₆₁	—	4	1	no	—
234	C₂×C₄×C₆₁	—	4	1	no	—
235	C₂×C₄×C₆₁	—	8	1	no	—
236	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
242	C₈×C₆₁	—	2	1	no	—
243	C₈×C₆₁	—	2	1	no	—
244	C₈×C₆₁	—	2	1	no	—
245	C₈×C₆₁	—	2	1	no	—
246	C₈×C₆₁	—	2	1	no	—
247	C₈×C₆₁	—	2	1	no	—
248	C₈×C₆₁	—	2	1	no	—
249	C₈×C₆₁	—	4	1	no	—
250	C₈×C₆₁	—	4	1	no	—
251	C₈×C₆₁	—	4	1	no	—
252	C₈×C₆₁	—	4	1	no	—
253	C₈×C₆₁	—	4	1	no	—
254	C₈×C₆₁	—	4	1	no	—
255	C₈×C₆₁	#137	1	2	yes	—
256	#18010	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂²×C₆₁	—	2	1	no	—
258	C₂²×C₆₁	—	2	1	no	—
259	C₂²×C₆₁	—	2	1	no	—
260	C₂²×C₆₁	—	4	1	no	—
261	C₂²×C₆₁	—	4	1	no	—
262	C₂²×C₆₁	—	4	1	no	—
263	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
264	C₄×C₆₁	—	2	1	no	—
265	C₄×C₆₁	—	2	2	no	—
266	C₄×C₆₁	—	2	1	no	—
267	C₄×C₆₁	—	2	1	no	—
268	C₄×C₆₁	—	2	1	no	—
269	C₄×C₆₁	—	4	1	no	—
270	C₄×C₆₁	—	4	1	no	—
271	C₄×C₆₁	—	4	1	no	—
272	C₄×C₆₁	—	4	1	no	—
273	C₄×C₆₁	—	4	1	no	—
274	C₄×C₆₁	—	4	1	no	—
275	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
276	C₄×C₆₁	#568	1	1	yes	characteristic
277	C₄×C₆₁	C₂×#137	1	1	yes	characteristic
278	C₄×C₆₁	#575	1	1	yes	characteristic
279	#3108	C₂×C₆₁	1	1	yes	characteristic
280	#2969	C₂×C₆₁	1	1	yes	characteristic
281	Q₈×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
282	#1592	C₂×C₆₁	1	1	yes	characteristic, complemented
283	#1591	C₂×C₆₁	1	1	yes	characteristic, complemented
284	#1588	C₂×C₆₁	1	1	yes	characteristic, complemented
285	#1587	C₂×C₆₁	1	1	yes	characteristic, complemented
286	#1584	C₂×C₆₁	1	1	yes	characteristic, complemented
287	#1582	C₂×C₆₁	1	1	yes	characteristic, complemented
288	#1579	C₂×C₆₁	1	1	yes	characteristic, complemented
289	C₈×SD₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
290	#1574	C₂×C₆₁	1	1	yes	characteristic, complemented
291	#1495	C₂×C₆₁	1	1	yes	characteristic
292	#1479	C₂×C₆₁	1	1	yes	characteristic, complemented
293	#1453	C₂×C₆₁	1	1	yes	characteristic, complemented
294	C₂×C₆₁	—	4	1	no	—
295	C₂×C₆₁	—	8	1	no	—
296	C₂×C₆₁	#2933	1	1	yes	characteristic
297	C₂×C₆₁	C₄×#137	1	1	yes	characteristic
298	C₂×C₆₁	#2957	1	1	yes	characteristic
299	#529	C₂²×C₆₁	1	1	yes	characteristic
300	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
301	#483	C₄×C₆₁	1	1	yes	characteristic
302	#480	C₄×C₆₁	1	1	yes	characteristic
303	#479	C₄×C₆₁	1	1	yes	characteristic
304	#479	C₄×C₆₁	1	1	yes	characteristic
305	#464	C₄×C₆₁	1	1	yes	characteristic
306	#464	C₄×C₆₁	1	1	yes	characteristic
307	#451	C₄×C₆₁	1	1	yes	characteristic
308	#448	C₄×C₆₁	1	1	yes	characteristic
309	#446	—	2	1	no	—
310	#446	—	2	1	no	—
311	#446	C₂²×C₆₁	1	1	yes	characteristic
312	#446	C₂²×C₆₁	1	1	yes	characteristic
313	#446	C₂²×C₆₁	1	1	yes	characteristic
314	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
315	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
316	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
317	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
318	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
319	C₄×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
320	C₄×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
321	#436	—	2	1	no	—
322	#436	C₂²×C₆₁	1	1	yes	characteristic
323	#435	—	2	1	no	—
324	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic
325	#433	—	2	1	no	—
326	#431	C₂²×C₆₁	1	1	yes	characteristic
327	#431	C₂²×C₆₁	1	1	yes	characteristic
328	#431	C₂²×C₆₁	1	1	yes	characteristic
329	#423	—	2	1	no	—
330	#423	C₂²×C₆₁	1	1	yes	characteristic
331	#405	—	2	1	no	—
332	C₄×M₁₆	—	2	1	no	—
333	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
334	#335	C₂²×C₆₁	1	1	yes	characteristic
335	#335	C₂²×C₆₁	1	1	yes	characteristic
336	#334	C₂²×C₆₁	1	1	yes	characteristic
337	#334	C₂²×C₆₁	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
339	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
342	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
344	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
345	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
346	#322	C₂²×C₆₁	1	1	yes	characteristic
347	#322	C₂²×C₆₁	1	1	yes	characteristic
348	#322	C₂²×C₆₁	1	1	yes	characteristic
349	C₈²	C₂²×C₆₁	1	1	yes	characteristic
350	C₆₁	#18010	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
352	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
353	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
354	C₂×SD₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
355	C₂×SD₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
359	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
360	C₂²×C₈	—	2	1	no	—
361	#126	—	2	1	no	—
362	#125	C₂×C₄×C₆₁	1	1	yes	characteristic
363	#124	C₂×C₄×C₆₁	1	1	yes	characteristic
364	#123	—	2	1	no	—
365	C₂²⋊Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
366	#121	C₂×C₄×C₆₁	1	1	yes	characteristic
367	C₄×Q₈	—	2	1	no	—
368	C₄×Q₈	—	2	1	no	—
369	C₄×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
370	C₄×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
371	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
372	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
373	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
376	#117	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
379	C₂×C₄²	—	2	1	no	—
380	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
381	#107	C₂×C₄×C₆₁	1	1	yes	characteristic
382	#107	C₂×C₄×C₆₁	1	1	yes	characteristic
383	#106	C₂×C₄×C₆₁	1	1	yes	characteristic
384	#106	C₂×C₄×C₆₁	1	1	yes	characteristic
385	C₄⋊C₈	—	2	1	no	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	—	2	1	no	—
393	C₄⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
394	C₄⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
395	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
396	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
397	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
398	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
399	#103	C₂×C₄×C₆₁	1	1	yes	characteristic
400	#103	C₂×C₄×C₆₁	1	1	yes	characteristic
401	#103	C₂×C₄×C₆₁	1	1	yes	characteristic
402	#103	C₂×C₄×C₆₁	1	1	yes	characteristic
403	#103	C₂×C₄×C₆₁	1	1	yes	characteristic
404	#103	C₂×C₄×C₆₁	1	1	yes	characteristic
405	#102	C₂×C₄×C₆₁	1	1	yes	characteristic
406	#102	C₂×C₄×C₆₁	1	1	yes	characteristic
407	C₂²⋊C₈	—	2	1	no	—
408	C₂²⋊C₈	—	2	1	no	—
409	C₂²⋊C₈	—	2	1	no	—
410	C₂²⋊C₈	—	2	1	no	—
411	C₂²⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
412	C₂²⋊C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
413	#97	—	2	1	no	—
414	#97	—	2	1	no	—
415	#97	—	2	1	no	—
416	#97	D₄×C₆₁	1	1	yes	characteristic
417	#97	D₄×C₆₁	1	1	yes	characteristic
418	C₄×C₈	—	2	1	no	—
419	C₄×C₈	—	2	1	no	—
420	C₄×C₈	—	2	1	no	—
421	C₄×C₈	—	2	1	no	—
422	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
426	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
427	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	4	1	no	—
431	C₂×Q₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
432	C₂×Q₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
433	C₂×Q₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
441	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
442	Q₁₆	—	2	2	no	—
443	Q₁₆	—	4	1	no	—
444	SD₁₆	—	2	2	no	—
445	SD₁₆	—	4	1	no	—
446	M₁₆	—	2	2	no	—
447	M₁₆	—	4	1	no	—
448	M₁₆	—	4	2	no	—
449	M₁₆	—	4	2	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
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	—	4	2	no	—
457	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
458	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
459	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
460	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
461	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
462	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
463	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
464	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
465	C₄⋊C₄	—	2	1	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	C₂²×C₄×C₆₁	1	1	yes	characteristic
475	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
476	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
477	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
478	C₂²⋊C₄	—	2	1	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	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
489	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
490	C₄²	#41×C₆₁	1	1	yes	characteristic
491	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
492	C₂³	—	2	1	no	—
493	C₂³	C₄×D₄×C₆₁	1	1	yes	characteristic
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	2	1	no	—
498	Q₈	—	2	1	no	—
499	Q₈	—	4	2	no	—
500	Q₈	#131×C₆₁	1	2	yes	—
501	D₄	—	2	1	no	—
502	D₄	—	2	1	no	—
503	D₄	—	8	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	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	—	4	1	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	8	1	no	—
520	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
521	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
522	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
523	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
524	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
526	C₈	—	2	1	no	—
527	C₈	—	2	1	no	—
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	2	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	1	no	—
539	C₈	#137×C₆₁	1	2	yes	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
547	C₄	—	2	1	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	4	1	no	—
558	C₄	C₂×#131×C₆₁	1	1	yes	characteristic
559	C₄	C₆₁×#568	1	1	yes	characteristic
560	C₄	C₂×#137×C₆₁	1	1	yes	characteristic
561	C₄	C₆₁×#575	1	1	yes	characteristic
562	C₂	—	4	1	no	—
563	C₂	—	8	1	no	—
564	C₂	C₆₁×#2933	1	1	yes	characteristic
565	C₂	C₄×#137×C₆₁	1	1	yes	characteristic
566	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	#121×C₆₁	C₂×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
3. Q₈×M₁₆×C₆₁	C₂	semidirect
4. C₆₁×#1592	C₂	semidirect
5. C₆₁×#1591	C₂	semidirect
6. C₆₁×#1588	C₂	semidirect
7. C₆₁×#1587	C₂	semidirect
8. C₆₁×#1584	C₂	semidirect
9. C₆₁×#1582	C₂	semidirect
10. C₆₁×#1579	C₂	semidirect
11. C₈×SD₁₆×C₆₁	C₂	semidirect
12. C₆₁×#1574	C₂	semidirect
14. C₆₁×#1479	C₂	semidirect
15. C₆₁×#1453	C₂	semidirect
256. #18010	C₆₁	direct (N × H)
281. Q₈×M₁₆	C₂×C₆₁	semidirect
282. #1592	C₂×C₆₁	semidirect
283. #1591	C₂×C₆₁	semidirect
284. #1588	C₂×C₆₁	semidirect
285. #1587	C₂×C₆₁	semidirect
286. #1584	C₂×C₆₁	semidirect
287. #1582	C₂×C₆₁	semidirect
288. #1579	C₂×C₆₁	semidirect
289. C₈×SD₁₆	C₂×C₆₁	semidirect
290. #1574	C₂×C₆₁	semidirect
292. #1479	C₂×C₆₁	semidirect
293. #1453	C₂×C₆₁	semidirect
350. C₆₁	#18010	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×#3108	2
C₆₁×#2969	2
Q₈×M₁₆×C₆₁	2
C₆₁×#1592	2
C₆₁×#1591	2
C₆₁×#1588	2
C₆₁×#1587	2
C₆₁×#1584	2
C₆₁×#1582	2
C₆₁×#1579	2
C₈×SD₁₆×C₆₁	2
C₆₁×#1574	2
C₆₁×#1495	2
C₆₁×#1479	2
C₆₁×#1453	2
#18010	61

Subgroups — C61×#18010 #29521443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18010 #29521443