Subgroups — C₆₁×#17523 #29520956

Order 15616.

936 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×#3482	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1497	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1488	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1481	C₂	1	2	yes	maximal, complemented
10	C₆₁×#1477	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1476	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#557	C₄	1	1	yes	characteristic
13	C₆₁×#556	C₄	1	2	yes	—
14	C₆₁×#556	C₄	1	1	yes	characteristic
15	C₆₁×#552	—	2	1	no	—
16	C₆₁×#552	—	2	2	no	—
17	D₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
18	C₆₁×#548	—	2	1	no	—
19	C₆₁×#547	C₄	1	1	yes	characteristic
20	C₆₁×#547	C₄	1	2	yes	—
21	C₆₁×#533	C₂²	1	2	yes	—
22	C₆₁×#533	C₂²	1	1	yes	characteristic
23	C₆₁×#529	—	2	1	no	—
24	C₆₁×#529	—	2	2	no	—
25	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
26	C₄×#41×C₆₁	C₂²	1	2	yes	—
27	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₆₁	—	2	1	no	—
29	C₆₁×#446	—	2	1	no	—
30	C₆₁×#435	—	2	2	no	—
31	C₈×D₄×C₆₁	—	2	1	no	—
32	C₆₁×#433	—	2	2	no	—
33	C₆₁×#432	C₂²	1	2	yes	—
34	C₆₁×#431	C₂²	1	1	yes	characteristic
35	C₆₁×#431	C₂²	1	2	yes	—
36	C₆₁×#431	C₂²	1	2	yes	—
37	C₆₁×#424	—	2	1	no	—
38	C₆₁×#423	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
40	C₆₁×#408	—	2	1	no	—
41	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
43	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
46	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
47	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
48	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
52	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
55	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₆₁	—	2	1	no	—
57	C₂×M₁₆×C₆₁	—	2	1	no	—
58	C₂×M₁₆×C₆₁	—	2	1	no	—
59	C₂²×C₈×C₆₁	—	2	1	no	—
60	C₂²×C₈×C₆₁	—	2	2	no	—
61	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
62	C₄⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
64	C₄⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
65	#126×C₆₁	—	2	2	no	—
66	#126×C₆₁	—	2	2	no	—
67	#126×C₆₁	—	2	2	no	—
68	#125×C₆₁	—	2	2	no	—
69	#125×C₆₁	—	2	1	no	—
70	#125×C₆₁	—	2	1	no	—
71	#125×C₆₁	—	2	2	no	—
72	#124×C₆₁	—	2	1	no	—
73	#124×C₆₁	—	2	2	no	—
74	#123×C₆₁	—	2	2	no	—
75	#123×C₆₁	—	2	2	no	—
76	#123×C₆₁	—	2	2	no	—
77	C₂²⋊Q₈×C₆₁	—	2	1	no	—
78	C₂²⋊Q₈×C₆₁	—	2	2	no	—
79	C₂²⋊Q₈×C₆₁	—	2	2	no	—
80	C₂²⋊Q₈×C₆₁	—	2	1	no	—
81	C₂²⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
82	C₂²⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
83	C₂²⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
84	#121×C₆₁	—	2	1	no	—
85	#121×C₆₁	—	2	2	no	—
86	C₄×Q₈×C₆₁	—	2	1	no	—
87	C₄×Q₈×C₆₁	—	2	2	no	—
88	C₄×Q₈×C₆₁	—	2	1	no	—
89	C₄×Q₈×C₆₁	—	2	1	no	—
90	C₄×Q₈×C₆₁	—	2	2	no	—
91	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
92	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
93	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
97	C₄×D₄×C₆₁	—	2	1	no	—
98	C₄×D₄×C₆₁	—	2	2	no	—
99	C₄×D₄×C₆₁	—	2	2	no	—
100	C₄×D₄×C₆₁	—	2	1	no	—
101	C₄×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
102	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
103	C₄×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
105	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
106	C₄×D₄×C₆₁	D₄	1	2	yes	—
107	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
108	#117×C₆₁	—	2	2	no	—
109	#117×C₆₁	—	2	1	no	—
110	#117×C₆₁	—	2	1	no	—
111	#117×C₆₁	—	2	2	no	—
112	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
113	#117×C₆₁	C₂×C₄	1	2	yes	—
114	#117×C₆₁	D₄	1	1	yes	characteristic
115	#117×C₆₁	D₄	1	2	yes	—
116	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
117	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
118	C₂×C₄²×C₆₁	—	2	1	no	—
119	C₂×C₄²×C₆₁	—	2	2	no	—
120	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
122	C₄⋊C₈×C₆₁	—	2	2	no	—
123	C₄⋊C₈×C₆₁	—	2	1	no	—
124	C₄⋊C₈×C₆₁	—	2	2	no	—
125	C₄⋊C₈×C₆₁	—	2	2	no	—
126	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
127	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
128	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
129	C₂²⋊C₈×C₆₁	—	2	2	no	—
130	C₂²⋊C₈×C₆₁	—	2	2	no	—
131	C₂²⋊C₈×C₆₁	—	2	1	no	—
132	C₂²⋊C₈×C₆₁	—	2	2	no	—
133	#97×C₆₁	—	2	1	no	—
134	#97×C₆₁	—	2	2	no	—
135	C₄×C₈×C₆₁	—	2	1	no	—
136	C₄×C₈×C₆₁	—	2	1	no	—
137	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₆₁	C₂³	1	2	yes	—
140	#41×C₆₁	—	2	2	no	—
141	#41×C₆₁	—	2	4	no	—
142	#41×C₆₁	—	4	1	no	—
143	#41×C₆₁	—	4	2	no	—
144	#41×C₆₁	—	4	2	no	—
145	C₂×Q₈×C₆₁	—	2	2	no	—
146	C₂×Q₈×C₆₁	—	2	1	no	—
147	C₂×Q₈×C₆₁	—	4	4	no	—
148	C₂×Q₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
151	C₂×D₄×C₆₁	—	2	1	no	—
152	C₂×D₄×C₆₁	—	2	2	no	—
153	C₂×D₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
154	C₂×D₄×C₆₁	C₂²⋊C₄	1	2	yes	—
155	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
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	1	no	—
160	C₂²×C₄×C₆₁	—	2	1	no	—
161	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
163	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
167	M₁₆×C₆₁	—	4	2	no	—
168	M₁₆×C₆₁	—	8	1	no	—
169	C₂×C₈×C₆₁	—	2	1	no	—
170	C₂×C₈×C₆₁	—	2	1	no	—
171	C₂×C₈×C₆₁	—	2	2	no	—
172	C₂×C₈×C₆₁	—	2	1	no	—
173	C₂×C₈×C₆₁	—	2	1	no	—
174	C₂×C₈×C₆₁	—	2	2	no	—
175	C₂×C₈×C₆₁	—	4	2	no	—
176	C₂×C₈×C₆₁	—	8	2	no	—
177	C₄⋊C₄×C₆₁	—	2	2	no	—
178	C₄⋊C₄×C₆₁	—	2	1	no	—
179	C₄⋊C₄×C₆₁	—	2	2	no	—
180	C₄⋊C₄×C₆₁	—	2	2	no	—
181	C₄⋊C₄×C₆₁	—	2	2	no	—
182	C₄⋊C₄×C₆₁	—	2	1	no	—
183	C₄⋊C₄×C₆₁	—	2	2	no	—
184	C₄⋊C₄×C₆₁	—	2	1	no	—
185	C₄⋊C₄×C₆₁	—	2	2	no	—
186	C₄⋊C₄×C₆₁	—	2	1	no	—
187	C₄⋊C₄×C₆₁	—	2	2	no	—
188	C₄⋊C₄×C₆₁	—	2	1	no	—
189	C₄⋊C₄×C₆₁	—	2	2	no	—
190	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
191	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
193	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
194	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
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	1	no	—
201	C₂²⋊C₄×C₆₁	—	2	2	no	—
202	C₂²⋊C₄×C₆₁	—	2	1	no	—
203	C₂²⋊C₄×C₆₁	—	2	2	no	—
204	C₂²⋊C₄×C₆₁	—	2	1	no	—
205	C₂²⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
206	C₂²⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
207	C₂²⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
208	C₄²×C₆₁	—	2	1	no	—
209	C₄²×C₆₁	—	2	2	no	—
210	C₄²×C₆₁	—	2	2	no	—
211	C₄²×C₆₁	—	2	1	no	—
212	C₄²×C₆₁	—	2	1	no	—
213	C₄²×C₆₁	—	2	2	no	—
214	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
215	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₆₁	C₂×D₄	1	2	yes	—
218	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₆₁	C₂×D₄	1	2	yes	—
220	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
221	C₂³×C₆₁	—	2	1	no	—
222	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
224	Q₈×C₆₁	—	2	2	no	—
225	Q₈×C₆₁	—	2	1	no	—
226	Q₈×C₆₁	—	2	2	no	—
227	Q₈×C₆₁	—	2	1	no	—
228	Q₈×C₆₁	—	2	2	no	—
229	Q₈×C₆₁	—	4	4	no	—
230	Q₈×C₆₁	—	4	2	no	—
231	D₄×C₆₁	—	2	1	no	—
232	D₄×C₆₁	—	2	2	no	—
233	D₄×C₆₁	—	4	2	no	—
234	D₄×C₆₁	—	4	2	no	—
235	D₄×C₆₁	—	4	4	no	—
236	D₄×C₆₁	#131	1	2	yes	—
237	C₂×C₄×C₆₁	—	2	1	no	—
238	C₂×C₄×C₆₁	—	2	2	no	—
239	C₂×C₄×C₆₁	—	2	1	no	—
240	C₂×C₄×C₆₁	—	2	2	no	—
241	C₂×C₄×C₆₁	—	2	2	no	—
242	C₂×C₄×C₆₁	—	2	1	no	—
243	C₂×C₄×C₆₁	—	2	2	no	—
244	C₂×C₄×C₆₁	—	2	1	no	—
245	C₂×C₄×C₆₁	—	2	2	no	—
246	C₂×C₄×C₆₁	—	2	1	no	—
247	C₂×C₄×C₆₁	—	2	2	no	—
248	C₂×C₄×C₆₁	—	2	1	no	—
249	C₂×C₄×C₆₁	—	2	2	no	—
250	C₂×C₄×C₆₁	—	2	1	no	—
251	C₂×C₄×C₆₁	—	4	2	no	—
252	C₂×C₄×C₆₁	—	4	2	no	—
253	C₂×C₄×C₆₁	—	4	4	no	—
254	C₂×C₄×C₆₁	—	4	4	no	—
255	C₂×C₄×C₆₁	—	4	1	no	—
256	C₂×C₄×C₆₁	—	4	1	no	—
257	C₂×C₄×C₆₁	—	4	2	no	—
258	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
259	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
261	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
263	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
265	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
267	C₈×C₆₁	—	4	1	no	—
268	C₈×C₆₁	—	4	1	no	—
269	C₈×C₆₁	—	4	2	no	—
270	C₈×C₆₁	—	4	1	no	—
271	C₈×C₆₁	—	4	2	no	—
272	C₈×C₆₁	—	4	1	no	—
273	#17523	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂²×C₆₁	—	2	1	no	—
275	C₂²×C₆₁	—	2	2	no	—
276	C₂²×C₆₁	—	2	2	no	—
277	C₂²×C₆₁	—	2	2	no	—
278	C₂²×C₆₁	—	2	1	no	—
279	C₂²×C₆₁	—	2	1	no	—
280	C₂²×C₆₁	—	2	2	no	—
281	C₂²×C₆₁	—	4	1	no	—
282	C₂²×C₆₁	—	4	1	no	—
283	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
284	C₄×C₆₁	—	2	1	no	—
285	C₄×C₆₁	—	2	2	no	—
286	C₄×C₆₁	—	2	2	no	—
287	C₄×C₆₁	—	2	1	no	—
288	C₄×C₆₁	—	2	1	no	—
289	C₄×C₆₁	—	2	2	no	—
290	C₄×C₆₁	—	4	2	no	—
291	C₄×C₆₁	—	4	1	no	—
292	C₄×C₆₁	—	4	1	no	—
293	C₄×C₆₁	—	4	2	no	—
294	C₄×C₆₁	—	4	1	no	—
295	C₄×C₆₁	#421	1	2	yes	—
296	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
297	C₄×C₆₁	#568	1	1	yes	characteristic
298	C₄×C₆₁	#577	1	1	yes	characteristic
299	C₄×C₆₁	#578	1	1	yes	characteristic
300	#3482	C₂×C₆₁	1	1	yes	characteristic
301	#2990	C₂×C₆₁	1	1	yes	characteristic, complemented
302	#2922	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#1497	C₂×C₆₁	1	2	yes	complemented
304	#1493	C₂×C₆₁	1	1	yes	characteristic, complemented
305	#1489	C₂×C₆₁	1	1	yes	characteristic, complemented
306	#1488	C₂×C₆₁	1	2	yes	complemented
307	#1485	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#1481	C₂×C₆₁	1	2	yes	complemented
309	#1477	C₂×C₆₁	1	2	yes	complemented
310	#1476	C₂×C₆₁	1	1	yes	characteristic, complemented
311	C₂×C₆₁	—	4	1	no	—
312	C₂×C₆₁	—	4	2	no	—
313	C₂×C₆₁	—	4	2	no	—
314	C₂×C₆₁	#2880	1	1	yes	characteristic
315	C₂×C₆₁	#2898	1	1	yes	characteristic
316	C₂×C₆₁	C₂×#421	1	1	yes	characteristic
317	#557	C₄×C₆₁	1	1	yes	characteristic
318	#556	C₄×C₆₁	1	2	yes	—
319	#556	C₄×C₆₁	1	1	yes	characteristic
320	#552	—	2	1	no	—
321	#552	—	2	2	no	—
322	D₄×Q₈	C₄×C₆₁	1	1	yes	characteristic
323	#548	—	2	1	no	—
324	#547	C₄×C₆₁	1	1	yes	characteristic
325	#547	C₄×C₆₁	1	2	yes	—
326	#533	C₂²×C₆₁	1	2	yes	—
327	#533	C₂²×C₆₁	1	1	yes	characteristic
328	#529	—	2	1	no	—
329	#529	—	2	2	no	—
330	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
331	C₄×#41	C₂²×C₆₁	1	2	yes	—
332	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
333	C₂×C₄×Q₈	—	2	1	no	—
334	#446	—	2	1	no	—
335	#435	—	2	2	no	—
336	C₈×D₄	—	2	1	no	—
337	#433	—	2	2	no	—
338	#432	C₂²×C₆₁	1	2	yes	—
339	#431	C₂²×C₆₁	1	1	yes	characteristic
340	#431	C₂²×C₆₁	1	2	yes	—
341	#431	C₂²×C₆₁	1	2	yes	—
342	#424	—	2	1	no	—
343	#423	C₂²×C₆₁	1	1	yes	characteristic
344	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
345	#408	—	2	1	no	—
346	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
347	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
348	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
349	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
350	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
351	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
352	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
353	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
354	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
355	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
356	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
357	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
358	C₆₁	#17523	1	1	yes	characteristic, direct product factor, Sylow
359	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
360	C₂×#41	C₂×C₄×C₆₁	1	2	yes	—
361	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
362	C₂²×Q₈	—	2	1	no	—
363	C₂×M₁₆	—	2	1	no	—
364	C₂×M₁₆	—	2	1	no	—
365	C₂²×C₈	—	2	1	no	—
366	C₂²×C₈	—	2	2	no	—
367	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
368	C₄⋊Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
369	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
370	C₄⋊Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
371	#126	—	2	2	no	—
372	#126	—	2	2	no	—
373	#126	—	2	2	no	—
374	#125	—	2	2	no	—
375	#125	—	2	1	no	—
376	#125	—	2	1	no	—
377	#125	—	2	2	no	—
378	#124	—	2	1	no	—
379	#124	—	2	2	no	—
380	#123	—	2	2	no	—
381	#123	—	2	2	no	—
382	#123	—	2	2	no	—
383	C₂²⋊Q₈	—	2	1	no	—
384	C₂²⋊Q₈	—	2	2	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
388	C₂²⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
389	C₂²⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
390	#121	—	2	1	no	—
391	#121	—	2	2	no	—
392	C₄×Q₈	—	2	1	no	—
393	C₄×Q₈	—	2	2	no	—
394	C₄×Q₈	—	2	1	no	—
395	C₄×Q₈	—	2	1	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
398	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
399	C₄×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
400	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
401	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
402	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
408	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
409	C₄×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
410	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
411	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
412	C₄×D₄	D₄×C₆₁	1	2	yes	—
413	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
414	#117	—	2	2	no	—
415	#117	—	2	1	no	—
416	#117	—	2	1	no	—
417	#117	—	2	2	no	—
418	#117	C₂×C₄×C₆₁	1	1	yes	characteristic
419	#117	C₂×C₄×C₆₁	1	2	yes	—
420	#117	D₄×C₆₁	1	1	yes	characteristic
421	#117	D₄×C₆₁	1	2	yes	—
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄²	—	2	1	no	—
425	C₂×C₄²	—	2	2	no	—
426	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
428	C₄⋊C₈	—	2	2	no	—
429	C₄⋊C₈	—	2	1	no	—
430	C₄⋊C₈	—	2	2	no	—
431	C₄⋊C₈	—	2	2	no	—
432	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
433	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
434	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
435	C₂²⋊C₈	—	2	2	no	—
436	C₂²⋊C₈	—	2	2	no	—
437	C₂²⋊C₈	—	2	1	no	—
438	C₂²⋊C₈	—	2	2	no	—
439	#97	—	2	1	no	—
440	#97	—	2	2	no	—
441	C₄×C₈	—	2	1	no	—
442	C₄×C₈	—	2	1	no	—
443	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
444	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
445	C₄×C₈	C₂³×C₆₁	1	2	yes	—
446	#41	—	2	2	no	—
447	#41	—	2	4	no	—
448	#41	—	4	1	no	—
449	#41	—	4	2	no	—
450	#41	—	4	2	no	—
451	C₂×Q₈	—	2	2	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	4	4	no	—
454	C₂×Q₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
455	C₂×Q₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
456	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
460	C₂×D₄	C₂²⋊C₄×C₆₁	1	2	yes	—
461	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
468	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
470	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
473	M₁₆	—	4	2	no	—
474	M₁₆	—	8	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	4	2	no	—
482	C₂×C₈	—	8	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
497	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
498	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
499	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
500	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
501	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
502	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	1	no	—
511	C₂²⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
512	C₂²⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
513	C₂²⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	—	2	1	no	—
518	C₄²	—	2	1	no	—
519	C₄²	—	2	2	no	—
520	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
521	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
522	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
523	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
524	C₄²	C₂×D₄×C₆₁	1	2	yes	—
525	C₄²	C₂×D₄×C₆₁	1	2	yes	—
526	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
527	C₂³	—	2	1	no	—
528	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
529	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
530	Q₈	—	2	2	no	—
531	Q₈	—	2	1	no	—
532	Q₈	—	2	2	no	—
533	Q₈	—	2	1	no	—
534	Q₈	—	2	2	no	—
535	Q₈	—	4	4	no	—
536	Q₈	—	4	2	no	—
537	D₄	—	2	1	no	—
538	D₄	—	2	2	no	—
539	D₄	—	4	2	no	—
540	D₄	—	4	2	no	—
541	D₄	—	4	4	no	—
542	D₄	#131×C₆₁	1	2	yes	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	4	2	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
565	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
566	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
567	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
568	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
569	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
570	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
571	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
572	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	2	no	—
576	C₈	—	4	1	no	—
577	C₈	—	4	2	no	—
578	C₈	—	4	1	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	2	1	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	4	1	no	—
587	C₂²	—	4	1	no	—
588	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
589	C₄	—	2	1	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	1	no	—
600	C₄	C₆₁×#421	1	2	yes	—
601	C₄	C₂×#131×C₆₁	1	1	yes	characteristic
602	C₄	C₆₁×#568	1	1	yes	characteristic
603	C₄	C₆₁×#577	1	1	yes	characteristic
604	C₄	C₆₁×#578	1	1	yes	characteristic
605	C₂	—	4	1	no	—
606	C₂	—	4	2	no	—
607	C₂	—	4	2	no	—
608	C₂	C₆₁×#2880	1	1	yes	characteristic
609	C₂	C₆₁×#2898	1	1	yes	characteristic
610	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	C₆₁×#3482	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₄×#41×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2990	C₂	semidirect
3. C₆₁×#2922	C₂	semidirect
4. C₆₁×#1497	C₂	semidirect
5. C₆₁×#1493	C₂	semidirect
6. C₆₁×#1489	C₂	semidirect
7. C₆₁×#1488	C₂	semidirect
8. C₆₁×#1485	C₂	semidirect
9. C₆₁×#1481	C₂	semidirect
10. C₆₁×#1477	C₂	semidirect
11. C₆₁×#1476	C₂	semidirect
273. #17523	C₆₁	direct (N × H)
301. #2990	C₂×C₆₁	semidirect
302. #2922	C₂×C₆₁	semidirect
303. #1497	C₂×C₆₁	semidirect
304. #1493	C₂×C₆₁	semidirect
305. #1489	C₂×C₆₁	semidirect
306. #1488	C₂×C₆₁	semidirect
307. #1485	C₂×C₆₁	semidirect
308. #1481	C₂×C₆₁	semidirect
309. #1477	C₂×C₆₁	semidirect
310. #1476	C₂×C₆₁	semidirect
358. C₆₁	#17523	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×#3482	2
C₆₁×#2990	2
C₆₁×#2922	2
C₆₁×#1497	2
C₆₁×#1493	2
C₆₁×#1489	2
C₆₁×#1488	2
C₆₁×#1485	2
C₆₁×#1481	2
C₆₁×#1477	2
C₆₁×#1476	2
#17523	61

Subgroups — C61×#17523 #29520956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#17523 #29520956