Subgroups — C₆₁×#60945 #29564378

Order 15616.

1866 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#529	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3250	C₂	1	4	yes	maximal, complemented
3	C₆₁×#3232	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3211	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3189	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#3187	C₂	1	2	yes	maximal, complemented
7	C₆₁×#3186	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#3168	C₂	1	4	yes	maximal, complemented
9	C₆₁×#3108	C₂	1	4	yes	maximal, complemented
10	C₆₁×#3105	C₂	1	4	yes	maximal, complemented
11	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#529	—	2	2	no	—
13	C₆₁×#529	—	2	2	no	—
14	C₆₁×#529	—	2	2	no	—
15	C₆₁×#529	—	2	2	no	—
16	C₆₁×#529	C₂²	1	4	yes	—
17	C₆₁×#529	C₂²	1	4	yes	—
18	C₆₁×#529	C₂²	1	4	yes	complemented
19	C₆₁×#529	C₂²	1	4	yes	complemented
20	C₂×#126×C₆₁	—	2	2	no	—
21	C₂×#125×C₆₁	C₂²	1	2	yes	complemented
22	C₂×#124×C₆₁	C₂²	1	2	yes	complemented
23	C₂×#123×C₆₁	—	2	2	no	—
24	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
25	C₂×#121×C₆₁	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	complemented
27	C₂×C₄×D₄×C₆₁	—	2	2	no	—
28	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	complemented
29	C₂×#117×C₆₁	—	2	2	no	—
30	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
31	C₆₁×#499	C₂²	1	4	yes	complemented
32	C₆₁×#495	C₂²	1	4	yes	complemented
33	C₆₁×#488	C₂²	1	8	yes	complemented
34	C₆₁×#484	C₂²	1	4	yes	complemented
35	C₆₁×#483	C₂²	1	4	yes	complemented
36	C₆₁×#481	C₂²	1	4	yes	—
37	C₆₁×#480	C₂²	1	4	yes	—
38	C₆₁×#479	C₂²	1	8	yes	complemented
39	C₆₁×#478	C₂²	1	8	yes	complemented
40	C₆₁×#467	C₂²	1	4	yes	complemented
41	C₆₁×#466	C₂²	1	4	yes	—
42	C₆₁×#465	C₂²	1	8	yes	complemented
43	C₆₁×#464	C₂²	1	8	yes	complemented
44	C₆₁×#463	C₂²	1	8	yes	complemented
45	C₆₁×#451	C₂²	1	4	yes	complemented
46	C₆₁×#450	C₂²	1	4	yes	complemented
47	C₆₁×#448	C₂²	1	4	yes	—
48	C₆₁×#447	C₂²	1	4	yes	—
49	C₄×Q₁₆×C₆₁	C₂²	1	4	yes	complemented
50	C₄×SD₁₆×C₆₁	C₂²	1	8	yes	complemented
51	C₄×D₈×C₆₁	C₂²	1	4	yes	complemented
52	C₆₁×#431	C₂²	1	4	yes	complemented
53	C₆₁×#431	C₂²	1	4	yes	complemented
54	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	complemented
55	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
56	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
57	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	complemented
58	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
59	C₂²×D₄×C₆₁	—	2	2	no	—
60	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₆₁	—	2	2	no	—
62	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
66	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
67	C₂×D₈×C₆₁	C₂³	1	4	yes	—
68	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
70	#126×C₆₁	—	2	4	no	—
71	#126×C₆₁	—	2	4	no	—
72	#126×C₆₁	—	2	4	no	—
73	#126×C₆₁	—	2	4	no	—
74	#125×C₆₁	—	2	2	no	—
75	#125×C₆₁	—	2	2	no	—
76	#125×C₆₁	C₂³	1	4	yes	—
77	#125×C₆₁	C₂³	1	4	yes	—
78	#124×C₆₁	—	2	2	no	—
79	#124×C₆₁	—	2	2	no	—
80	#124×C₆₁	C₂³	1	4	yes	—
81	#124×C₆₁	C₂³	1	4	yes	—
82	#123×C₆₁	—	2	4	no	—
83	#123×C₆₁	—	2	4	no	—
84	#123×C₆₁	—	2	4	no	—
85	#123×C₆₁	—	2	4	no	—
86	C₂²⋊Q₈×C₆₁	—	2	2	no	—
87	C₂²⋊Q₈×C₆₁	—	2	2	no	—
88	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
89	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
90	#121×C₆₁	—	2	2	no	—
91	#121×C₆₁	—	2	2	no	—
92	#121×C₆₁	C₂³	1	4	yes	—
93	#121×C₆₁	C₂³	1	4	yes	—
94	C₄×Q₈×C₆₁	—	2	2	no	—
95	C₄×Q₈×C₆₁	—	2	2	no	—
96	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
97	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
98	C₄×D₄×C₆₁	—	2	4	no	—
99	C₄×D₄×C₆₁	—	2	4	no	—
100	C₄×D₄×C₆₁	—	2	4	no	—
101	C₄×D₄×C₆₁	—	2	4	no	—
102	C₄×D₄×C₆₁	—	2	2	no	—
103	C₄×D₄×C₆₁	—	2	2	no	—
104	C₄×D₄×C₆₁	C₂³	1	4	yes	—
105	C₄×D₄×C₆₁	C₂³	1	4	yes	—
106	#117×C₆₁	—	2	4	no	—
107	#117×C₆₁	—	2	4	no	—
108	#117×C₆₁	—	2	4	no	—
109	#117×C₆₁	—	2	4	no	—
110	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
111	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
113	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
114	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
120	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
121	C₂×C₄²×C₆₁	—	2	2	no	—
122	C₂×C₄²×C₆₁	D₄	1	2	yes	—
123	C₂×C₄²×C₆₁	D₄	1	2	yes	—
124	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
130	#107×C₆₁	C₂³	1	4	yes	—
131	#107×C₆₁	C₂³	1	4	yes	—
132	#106×C₆₁	C₂³	1	8	yes	—
133	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
134	#103×C₆₁	C₂³	1	8	yes	—
135	#103×C₆₁	C₂³	1	8	yes	—
136	#102×C₆₁	C₂³	1	8	yes	—
137	#102×C₆₁	C₂³	1	8	yes	—
138	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
139	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
140	C₄×C₈×C₆₁	C₂³	1	4	yes	—
141	C₄×C₈×C₆₁	C₂³	1	4	yes	—
142	C₂⁴×C₆₁	—	2	2	no	—
143	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₆₁	—	2	2	no	—
145	C₂×Q₈×C₆₁	—	4	4	no	—
146	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
147	C₂×D₄×C₆₁	—	2	4	no	—
148	C₂×D₄×C₆₁	—	2	4	no	—
149	C₂×D₄×C₆₁	—	2	2	no	—
150	C₂×D₄×C₆₁	—	4	4	no	—
151	C₂×D₄×C₆₁	—	4	8	no	—
152	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₁	—	2	4	no	—
154	C₂²×C₄×C₆₁	—	2	2	no	—
155	C₂²×C₄×C₆₁	—	2	2	no	—
156	C₂²×C₄×C₆₁	—	2	2	no	—
157	C₂²×C₄×C₆₁	—	2	4	no	—
158	C₂²×C₄×C₆₁	—	2	4	no	—
159	C₂²×C₄×C₆₁	—	4	2	no	—
160	C₂²×C₄×C₆₁	—	4	2	no	—
161	C₂²×C₄×C₆₁	—	4	8	no	—
162	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
174	Q₁₆×C₆₁	—	4	4	no	—
175	SD₁₆×C₆₁	—	4	8	no	—
176	D₈×C₆₁	—	4	4	no	—
177	C₂×C₈×C₆₁	—	4	2	no	—
178	C₂×C₈×C₆₁	—	4	2	no	—
179	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
180	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
181	C₄⋊C₄×C₆₁	—	2	4	no	—
182	C₄⋊C₄×C₆₁	—	2	4	no	—
183	C₄⋊C₄×C₆₁	—	2	4	no	—
184	C₄⋊C₄×C₆₁	—	2	4	no	—
185	C₄⋊C₄×C₆₁	—	2	4	no	—
186	C₄⋊C₄×C₆₁	—	2	4	no	—
187	C₄⋊C₄×C₆₁	—	2	4	no	—
188	C₄⋊C₄×C₆₁	—	2	4	no	—
189	C₄⋊C₄×C₆₁	—	2	2	no	—
190	C₄⋊C₄×C₆₁	—	2	2	no	—
191	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
192	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
193	C₂²⋊C₄×C₆₁	—	2	4	no	—
194	C₂²⋊C₄×C₆₁	—	2	4	no	—
195	C₂²⋊C₄×C₆₁	—	2	4	no	—
196	C₂²⋊C₄×C₆₁	—	2	4	no	—
197	C₂²⋊C₄×C₆₁	—	2	4	no	—
198	C₂²⋊C₄×C₆₁	—	2	4	no	—
199	C₂²⋊C₄×C₆₁	—	2	4	no	—
200	C₂²⋊C₄×C₆₁	—	2	4	no	—
201	C₂²⋊C₄×C₆₁	—	2	4	no	—
202	C₂²⋊C₄×C₆₁	—	2	4	no	—
203	C₄²×C₆₁	—	2	4	no	—
204	C₄²×C₆₁	—	2	4	no	—
205	C₄²×C₆₁	C₂×D₄	1	2	yes	—
206	C₄²×C₆₁	C₂×D₄	1	2	yes	—
207	C₄²×C₆₁	C₂×D₄	1	2	yes	—
208	C₄²×C₆₁	C₂×D₄	1	2	yes	—
209	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₆₁	C₂×D₄	1	2	yes	—
212	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
213	C₄²×C₆₁	C₂⁴	1	2	yes	—
214	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
215	C₂³×C₆₁	—	2	4	no	—
216	C₂³×C₆₁	—	4	1	no	—
217	C₂³×C₆₁	—	4	1	no	—
218	C₂³×C₆₁	—	4	4	no	—
219	C₂³×C₆₁	—	4	4	no	—
220	C₂³×C₆₁	—	4	4	no	—
221	C₂³×C₆₁	#135	1	4	yes	—
222	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
223	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
224	Q₈×C₆₁	—	2	8	no	—
225	Q₈×C₆₁	—	4	4	no	—
226	D₄×C₆₁	—	2	8	no	—
227	D₄×C₆₁	—	4	4	no	—
228	D₄×C₆₁	—	4	8	no	—
229	D₄×C₆₁	—	4	8	no	—
230	C₂×C₄×C₆₁	—	2	4	no	—
231	C₂×C₄×C₆₁	—	2	4	no	—
232	C₂×C₄×C₆₁	—	2	2	no	—
233	C₂×C₄×C₆₁	—	2	2	no	—
234	C₂×C₄×C₆₁	—	2	4	no	—
235	C₂×C₄×C₆₁	—	2	2	no	—
236	C₂×C₄×C₆₁	—	2	2	no	—
237	C₂×C₄×C₆₁	—	2	4	no	—
238	C₂×C₄×C₆₁	—	2	4	no	—
239	C₂×C₄×C₆₁	—	2	4	no	—
240	C₂×C₄×C₆₁	—	2	4	no	—
241	C₂×C₄×C₆₁	—	4	2	no	—
242	C₂×C₄×C₆₁	—	4	1	no	—
243	C₂×C₄×C₆₁	—	4	1	no	—
244	C₂×C₄×C₆₁	—	4	2	no	—
245	C₂×C₄×C₆₁	—	4	8	no	—
246	C₂×C₄×C₆₁	—	4	4	no	—
247	C₂×C₄×C₆₁	—	4	4	no	—
248	C₂×C₄×C₆₁	—	4	4	no	—
249	C₂×C₄×C₆₁	—	4	8	no	—
250	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
259	C₈×C₆₁	—	2	4	no	—
260	C₈×C₆₁	—	2	4	no	—
261	#60945	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂²×C₆₁	—	2	2	no	—
263	C₂²×C₆₁	—	2	1	no	—
264	C₂²×C₆₁	—	2	2	no	—
265	C₂²×C₆₁	—	2	1	no	—
266	C₂²×C₆₁	—	4	4	no	—
267	C₂²×C₆₁	—	4	4	no	—
268	C₂²×C₆₁	—	4	4	no	—
269	C₂²×C₆₁	—	4	4	no	—
270	C₂²×C₆₁	—	8	8	no	—
271	C₂²×C₆₁	C₂×#135	1	4	yes	—
272	C₂²×C₆₁	C₂×#135	1	2	yes	—
273	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
274	C₄×C₆₁	—	2	1	no	—
275	C₄×C₆₁	—	2	2	no	—
276	C₄×C₆₁	—	2	1	no	—
277	C₄×C₆₁	—	2	2	no	—
278	C₄×C₆₁	—	2	1	no	—
279	C₄×C₆₁	—	2	2	no	—
280	C₄×C₆₁	—	2	1	no	—
281	C₄×C₆₁	—	4	4	no	—
282	C₄×C₆₁	—	4	4	no	—
283	C₄×C₆₁	—	4	4	no	—
284	C₄×C₆₁	#575	1	2	yes	—
285	C₄×C₆₁	#585	1	2	yes	—
286	C₂×#529	C₂×C₆₁	1	2	yes	complemented
287	#3250	C₂×C₆₁	1	4	yes	complemented
288	#3232	C₂×C₆₁	1	4	yes	complemented
289	#3211	C₂×C₆₁	1	4	yes	complemented
290	#3189	C₂×C₆₁	1	1	yes	characteristic, complemented
291	#3187	C₂×C₆₁	1	2	yes	complemented
292	#3186	C₂×C₆₁	1	1	yes	characteristic
293	#3168	C₂×C₆₁	1	4	yes	complemented
294	#3108	C₂×C₆₁	1	4	yes	complemented
295	#3105	C₂×C₆₁	1	4	yes	complemented
296	C₂×#431	C₂×C₆₁	1	1	yes	characteristic, complemented
297	C₂×C₆₁	—	2	4	no	—
298	C₂×C₆₁	—	2	2	no	—
299	C₂×C₆₁	—	8	4	no	—
300	C₂×C₆₁	#3449	1	1	yes	characteristic
301	C₂×C₆₁	C₂²×#135	1	1	yes	characteristic
302	C₂×C₆₁	C₂×#575	1	1	yes	characteristic
303	#529	—	2	2	no	—
304	#529	—	2	2	no	—
305	#529	—	2	2	no	—
306	#529	—	2	2	no	—
307	#529	C₂²×C₆₁	1	4	yes	—
308	#529	C₂²×C₆₁	1	4	yes	—
309	#529	C₂²×C₆₁	1	4	yes	complemented
310	#529	C₂²×C₆₁	1	4	yes	complemented
311	C₂×#126	—	2	2	no	—
312	C₂×#125	C₂²×C₆₁	1	2	yes	complemented
313	C₂×#124	C₂²×C₆₁	1	2	yes	complemented
314	C₂×#123	—	2	2	no	—
315	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
316	C₂×#121	C₂²×C₆₁	1	2	yes	—
317	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	complemented
318	C₂×C₄×D₄	—	2	2	no	—
319	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	complemented
320	C₂×#117	—	2	2	no	—
321	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
322	#499	C₂²×C₆₁	1	4	yes	complemented
323	#495	C₂²×C₆₁	1	4	yes	complemented
324	#488	C₂²×C₆₁	1	8	yes	complemented
325	#484	C₂²×C₆₁	1	4	yes	complemented
326	#483	C₂²×C₆₁	1	4	yes	complemented
327	#481	C₂²×C₆₁	1	4	yes	—
328	#480	C₂²×C₆₁	1	4	yes	—
329	#479	C₂²×C₆₁	1	8	yes	complemented
330	#478	C₂²×C₆₁	1	8	yes	complemented
331	#467	C₂²×C₆₁	1	4	yes	complemented
332	#466	C₂²×C₆₁	1	4	yes	—
333	#465	C₂²×C₆₁	1	8	yes	complemented
334	#464	C₂²×C₆₁	1	8	yes	complemented
335	#463	C₂²×C₆₁	1	8	yes	complemented
336	#451	C₂²×C₆₁	1	4	yes	complemented
337	#450	C₂²×C₆₁	1	4	yes	complemented
338	#448	C₂²×C₆₁	1	4	yes	—
339	#447	C₂²×C₆₁	1	4	yes	—
340	C₄×Q₁₆	C₂²×C₆₁	1	4	yes	complemented
341	C₄×SD₁₆	C₂²×C₆₁	1	8	yes	complemented
342	C₄×D₈	C₂²×C₆₁	1	4	yes	complemented
343	#431	C₂²×C₆₁	1	4	yes	complemented
344	#431	C₂²×C₆₁	1	4	yes	complemented
345	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	complemented
346	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
347	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
348	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	complemented
349	C₆₁	#60945	1	1	yes	characteristic, direct product factor, Sylow
350	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
351	C₂²×D₄	—	2	2	no	—
352	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
353	C₂³×C₄	—	2	2	no	—
354	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
355	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
357	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
358	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
359	C₂×D₈	C₂³×C₆₁	1	4	yes	—
360	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
361	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
362	#126	—	2	4	no	—
363	#126	—	2	4	no	—
364	#126	—	2	4	no	—
365	#126	—	2	4	no	—
366	#125	—	2	2	no	—
367	#125	—	2	2	no	—
368	#125	C₂³×C₆₁	1	4	yes	—
369	#125	C₂³×C₆₁	1	4	yes	—
370	#124	—	2	2	no	—
371	#124	—	2	2	no	—
372	#124	C₂³×C₆₁	1	4	yes	—
373	#124	C₂³×C₆₁	1	4	yes	—
374	#123	—	2	4	no	—
375	#123	—	2	4	no	—
376	#123	—	2	4	no	—
377	#123	—	2	4	no	—
378	C₂²⋊Q₈	—	2	2	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
381	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
382	#121	—	2	2	no	—
383	#121	—	2	2	no	—
384	#121	C₂³×C₆₁	1	4	yes	—
385	#121	C₂³×C₆₁	1	4	yes	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	—	2	2	no	—
388	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
389	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
390	C₄×D₄	—	2	4	no	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	—	2	4	no	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	C₂³×C₆₁	1	4	yes	—
397	C₄×D₄	C₂³×C₆₁	1	4	yes	—
398	#117	—	2	4	no	—
399	#117	—	2	4	no	—
400	#117	—	2	4	no	—
401	#117	—	2	4	no	—
402	C₂×C₄⋊C₄	—	2	2	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄⋊C₄	—	2	2	no	—
405	C₂×C₄⋊C₄	—	2	2	no	—
406	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	—	2	2	no	—
413	C₂×C₄²	—	2	2	no	—
414	C₂×C₄²	D₄×C₆₁	1	2	yes	—
415	C₂×C₄²	D₄×C₆₁	1	2	yes	—
416	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
417	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
419	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
420	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
422	#107	C₂³×C₆₁	1	4	yes	—
423	#107	C₂³×C₆₁	1	4	yes	—
424	#106	C₂³×C₆₁	1	8	yes	—
425	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
426	#103	C₂³×C₆₁	1	8	yes	—
427	#103	C₂³×C₆₁	1	8	yes	—
428	#102	C₂³×C₆₁	1	8	yes	—
429	#102	C₂³×C₆₁	1	8	yes	—
430	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
431	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
432	C₄×C₈	C₂³×C₆₁	1	4	yes	—
433	C₄×C₈	C₂³×C₆₁	1	4	yes	—
434	C₂⁴	—	2	2	no	—
435	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	4	4	no	—
438	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	4	4	no	—
443	C₂×D₄	—	4	8	no	—
444	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	8	no	—
454	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
456	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
457	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
461	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
463	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
466	Q₁₆	—	4	4	no	—
467	SD₁₆	—	4	8	no	—
468	D₈	—	4	4	no	—
469	C₂×C₈	—	4	2	no	—
470	C₂×C₈	—	4	2	no	—
471	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
472	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
484	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₄²	—	2	4	no	—
496	C₄²	—	2	4	no	—
497	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
498	C₄²	C₂×D₄×C₆₁	1	2	yes	—
499	C₄²	C₂×D₄×C₆₁	1	2	yes	—
500	C₄²	C₂×D₄×C₆₁	1	2	yes	—
501	C₄²	C₂×D₄×C₆₁	1	2	yes	—
502	C₄²	C₂×D₄×C₆₁	1	2	yes	—
503	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
504	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
505	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
506	C₄²	C₂⁴×C₆₁	1	2	yes	—
507	C₂³	—	2	4	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	4	no	—
511	C₂³	—	4	4	no	—
512	C₂³	—	4	4	no	—
513	C₂³	#135×C₆₁	1	4	yes	—
514	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
515	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
516	Q₈	—	2	8	no	—
517	Q₈	—	4	4	no	—
518	D₄	—	2	8	no	—
519	D₄	—	4	4	no	—
520	D₄	—	4	8	no	—
521	D₄	—	4	8	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	8	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	4	no	—
541	C₂×C₄	—	4	8	no	—
542	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
543	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
544	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
545	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
547	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
550	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
551	C₈	—	2	4	no	—
552	C₈	—	2	4	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	8	8	no	—
562	C₂²	C₂×#135×C₆₁	1	4	yes	—
563	C₂²	C₂×#135×C₆₁	1	2	yes	—
564	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
565	C₄	—	2	1	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	1	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	1	no	—
570	C₄	—	2	1	no	—
571	C₄	—	2	2	no	—
572	C₄	—	4	4	no	—
573	C₄	—	4	4	no	—
574	C₄	—	4	4	no	—
575	C₄	C₆₁×#575	1	2	yes	—
576	C₄	C₆₁×#585	1	2	yes	—
577	C₂	—	2	4	no	—
578	C₂	—	2	2	no	—
579	C₂	—	8	4	no	—
580	C₂	C₆₁×#3449	1	1	yes	characteristic
581	C₂	C₂²×#135×C₆₁	1	1	yes	characteristic
582	C₂	C₂×C₆₁×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
socle	C₂²×C₆₁	C₂³×D₄
prime radical	C₆₁×#3186	C₂
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
1. C₂×C₆₁×#529	C₂	semidirect
2. C₆₁×#3250	C₂	semidirect
3. C₆₁×#3232	C₂	semidirect
4. C₆₁×#3211	C₂	semidirect
5. C₆₁×#3189	C₂	semidirect
6. C₆₁×#3187	C₂	semidirect
8. C₆₁×#3168	C₂	semidirect
9. C₆₁×#3108	C₂	semidirect
10. C₆₁×#3105	C₂	semidirect
11. C₂×C₆₁×#431	C₂	semidirect
18. C₆₁×#529	C₂²	semidirect
19. C₆₁×#529	C₂²	semidirect
21. C₂×#125×C₆₁	C₂²	semidirect
22. C₂×#124×C₆₁	C₂²	semidirect
26. C₂×C₄×Q₈×C₆₁	C₂²	semidirect
28. C₂×C₄×D₄×C₆₁	C₂²	semidirect
31. C₆₁×#499	C₂²	semidirect
32. C₆₁×#495	C₂²	semidirect
33. C₆₁×#488	C₂²	semidirect
34. C₆₁×#484	C₂²	semidirect
35. C₆₁×#483	C₂²	semidirect
38. C₆₁×#479	C₂²	semidirect
39. C₆₁×#478	C₂²	semidirect
40. C₆₁×#467	C₂²	semidirect
42. C₆₁×#465	C₂²	semidirect
43. C₆₁×#464	C₂²	semidirect
44. C₆₁×#463	C₂²	semidirect
45. C₆₁×#451	C₂²	semidirect
46. C₆₁×#450	C₂²	semidirect
49. C₄×Q₁₆×C₆₁	C₂²	semidirect
50. C₄×SD₁₆×C₆₁	C₂²	semidirect
51. C₄×D₈×C₆₁	C₂²	semidirect
52. C₆₁×#431	C₂²	semidirect
53. C₆₁×#431	C₂²	semidirect
54. C₂×C₄⋊C₈×C₆₁	C₂²	semidirect
57. C₂×C₄×C₈×C₆₁	C₂²	semidirect
261. #60945	C₆₁	direct (N × H)
286. C₂×#529	C₂×C₆₁	semidirect
287. #3250	C₂×C₆₁	semidirect
288. #3232	C₂×C₆₁	semidirect
289. #3211	C₂×C₆₁	semidirect
290. #3189	C₂×C₆₁	semidirect
291. #3187	C₂×C₆₁	semidirect
293. #3168	C₂×C₆₁	semidirect
294. #3108	C₂×C₆₁	semidirect
295. #3105	C₂×C₆₁	semidirect
296. C₂×#431	C₂×C₆₁	semidirect
309. #529	C₂²×C₆₁	semidirect
310. #529	C₂²×C₆₁	semidirect
312. C₂×#125	C₂²×C₆₁	semidirect
313. C₂×#124	C₂²×C₆₁	semidirect
317. C₂×C₄×Q₈	C₂²×C₆₁	semidirect
319. C₂×C₄×D₄	C₂²×C₆₁	semidirect
322. #499	C₂²×C₆₁	semidirect
323. #495	C₂²×C₆₁	semidirect
324. #488	C₂²×C₆₁	semidirect
325. #484	C₂²×C₆₁	semidirect
326. #483	C₂²×C₆₁	semidirect
329. #479	C₂²×C₆₁	semidirect
330. #478	C₂²×C₆₁	semidirect
331. #467	C₂²×C₆₁	semidirect
333. #465	C₂²×C₆₁	semidirect
334. #464	C₂²×C₆₁	semidirect
335. #463	C₂²×C₆₁	semidirect
336. #451	C₂²×C₆₁	semidirect
337. #450	C₂²×C₆₁	semidirect
340. C₄×Q₁₆	C₂²×C₆₁	semidirect
341. C₄×SD₁₆	C₂²×C₆₁	semidirect
342. C₄×D₈	C₂²×C₆₁	semidirect
343. #431	C₂²×C₆₁	semidirect
344. #431	C₂²×C₆₁	semidirect
345. C₂×C₄⋊C₈	C₂²×C₆₁	semidirect
348. C₂×C₄×C₈	C₂²×C₆₁	semidirect
349. C₆₁	#60945	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#529	2
C₆₁×#3250	2
C₆₁×#3232	2
C₆₁×#3211	2
C₆₁×#3189	2
C₆₁×#3187	2
C₆₁×#3186	2
C₆₁×#3168	2
C₆₁×#3108	2
C₆₁×#3105	2
C₂×C₆₁×#431	2
#60945	61

Subgroups — C61×#60945 #29564378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60945 #29564378