Subgroups — C₆₁×#24840 #29528273

Order 15616.

860 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#479	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2851	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂²⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2088	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2086	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2068	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2066	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1989	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1949	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1892	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1892	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1808	C₂	1	1	yes	maximal, characteristic
14	C₆₁×#1807	C₂	1	1	yes	maximal, characteristic
15	C₆₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#479	—	2	4	no	—
23	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₆₁	—	2	4	no	—
28	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#391	C₂²	1	1	yes	characteristic
37	C₆₁×#389	C₂²	1	1	yes	characteristic
38	C₆₁×#389	C₂²	1	1	yes	characteristic
39	C₆₁×#389	C₂²	1	1	yes	characteristic
40	C₆₁×#387	—	2	1	no	—
41	C₆₁×#387	—	2	1	no	—
42	C₆₁×#387	—	2	1	no	—
43	C₆₁×#385	—	2	1	no	—
44	C₆₁×#385	—	2	1	no	—
45	C₆₁×#385	—	2	1	no	—
46	C₆₁×#382	C₂²	1	1	yes	characteristic
47	C₆₁×#380	C₂²	1	1	yes	characteristic
48	C₆₁×#380	C₂²	1	1	yes	characteristic
49	C₆₁×#380	C₂²	1	1	yes	characteristic
50	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
51	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄×C₆₁	—	2	1	no	—
53	C₆₁×#340	C₂²	1	1	yes	characteristic
54	C₆₁×#340	C₂²	1	1	yes	characteristic
55	C₆₁×#340	C₂²	1	1	yes	characteristic
56	C₆₁×#340	C₂²	1	1	yes	characteristic
57	C₆₁×#340	C₂²	1	1	yes	characteristic
58	C₆₁×#340	C₂²	1	1	yes	characteristic
59	C₆₁×#340	C₂²	1	1	yes	characteristic
60	C₆₁×#340	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
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	C₂²×C₈×C₆₁	—	2	2	no	—
66	C₂²×C₈×C₆₁	—	2	2	no	—
67	C₂²×C₈×C₆₁	—	2	2	no	—
68	C₂²×C₈×C₆₁	—	2	2	no	—
69	C₂²×C₈×C₆₁	—	4	1	no	—
70	C₂²×C₈×C₆₁	—	4	1	no	—
71	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	#125×C₆₁	—	2	2	no	—
76	#125×C₆₁	—	2	2	no	—
77	C₂²⋊Q₈×C₆₁	—	2	2	no	—
78	C₂²⋊Q₈×C₆₁	Q₈	1	4	yes	—
79	C₄×Q₈×C₆₁	—	2	2	no	—
80	C₄×Q₈×C₆₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
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	2	no	—
100	C₂×C₄²×C₆₁	—	2	2	no	—
101	C₂×C₄²×C₆₁	—	2	1	no	—
102	C₂×C₄²×C₆₁	—	2	1	no	—
103	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
104	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
107	#107×C₆₁	—	2	2	no	—
108	#107×C₆₁	—	4	1	no	—
109	#106×C₆₁	—	2	2	no	—
110	#106×C₆₁	—	4	1	no	—
111	C₄⋊C₈×C₆₁	—	2	2	no	—
112	C₄⋊C₈×C₆₁	—	2	2	no	—
113	C₄⋊C₈×C₆₁	—	2	2	no	—
114	C₄⋊C₈×C₆₁	D₄	1	4	yes	—
115	#103×C₆₁	—	2	2	no	—
116	#103×C₆₁	—	2	2	no	—
117	#103×C₆₁	—	4	1	no	—
118	#103×C₆₁	—	4	1	no	—
119	#95×C₆₁	—	2	1	no	—
120	#95×C₆₁	—	2	1	no	—
121	#95×C₆₁	—	2	1	no	—
122	#95×C₆₁	—	2	1	no	—
123	#95×C₆₁	—	2	1	no	—
124	#95×C₆₁	—	2	1	no	—
125	#95×C₆₁	—	2	1	no	—
126	#95×C₆₁	—	2	1	no	—
127	#95×C₆₁	—	2	1	no	—
128	#95×C₆₁	—	2	1	no	—
129	#95×C₆₁	—	2	1	no	—
130	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Q₈×C₆₁	—	2	1	no	—
132	C₂×Q₈×C₆₁	—	2	1	no	—
133	C₂×Q₈×C₆₁	—	2	1	no	—
134	C₂×Q₈×C₆₁	—	2	1	no	—
135	C₂×Q₈×C₆₁	—	2	1	no	—
136	C₂×Q₈×C₆₁	C₂×Q₈	1	2	yes	—
137	C₂²×C₄×C₆₁	—	2	2	no	—
138	C₂²×C₄×C₆₁	—	2	2	no	—
139	C₂²×C₄×C₆₁	—	2	2	no	—
140	C₂²×C₄×C₆₁	—	2	1	no	—
141	C₂²×C₄×C₆₁	—	2	1	no	—
142	C₂²×C₄×C₆₁	—	2	1	no	—
143	C₂²×C₄×C₆₁	—	2	1	no	—
144	C₂²×C₄×C₆₁	—	2	2	no	—
145	C₂²×C₄×C₆₁	—	2	1	no	—
146	C₂²×C₄×C₆₁	—	2	1	no	—
147	C₂²×C₄×C₆₁	—	2	2	no	—
148	C₂²×C₄×C₆₁	—	2	1	no	—
149	C₂²×C₄×C₆₁	—	2	1	no	—
150	C₂²×C₄×C₆₁	—	2	1	no	—
151	C₂²×C₄×C₆₁	—	2	1	no	—
152	C₂²×C₄×C₆₁	—	2	1	no	—
153	C₂²×C₄×C₆₁	—	2	1	no	—
154	C₂²×C₄×C₆₁	—	4	1	no	—
155	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
161	C₂²×C₄×C₆₁	#41	1	2	yes	—
162	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
163	C₂×C₈×C₆₁	—	2	1	no	—
164	C₂×C₈×C₆₁	—	2	1	no	—
165	C₂×C₈×C₆₁	—	2	1	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	—	2	1	no	—
173	C₂×C₈×C₆₁	—	4	4	no	—
174	C₂×C₈×C₆₁	—	4	4	no	—
175	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
176	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
177	C₄⋊C₄×C₆₁	—	2	4	no	—
178	C₄⋊C₄×C₆₁	—	2	1	no	—
179	C₄⋊C₄×C₆₁	—	2	1	no	—
180	C₄⋊C₄×C₆₁	—	2	1	no	—
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	—	2	1	no	—
191	C₄⋊C₄×C₆₁	—	4	2	no	—
192	C₄⋊C₄×C₆₁	—	4	2	no	—
193	C₄⋊C₄×C₆₁	—	4	2	no	—
194	C₄⋊C₄×C₆₁	—	4	2	no	—
195	C₄⋊C₄×C₆₁	—	4	2	no	—
196	C₄⋊C₄×C₆₁	—	4	2	no	—
197	C₄⋊C₄×C₆₁	C₂×Q₈	1	2	yes	—
198	C₂²⋊C₄×C₆₁	—	2	4	no	—
199	C₂²⋊C₄×C₆₁	—	4	2	no	—
200	C₂²⋊C₄×C₆₁	—	4	2	no	—
201	C₂²⋊C₄×C₆₁	—	4	2	no	—
202	C₄²×C₆₁	—	2	2	no	—
203	C₄²×C₆₁	—	2	2	no	—
204	C₄²×C₆₁	—	2	1	no	—
205	C₄²×C₆₁	—	2	1	no	—
206	C₄²×C₆₁	—	4	2	no	—
207	C₄²×C₆₁	—	4	2	no	—
208	C₄²×C₆₁	C₂×D₄	1	2	yes	—
209	C₄²×C₆₁	#41	1	2	yes	—
210	C₂³×C₆₁	—	2	1	no	—
211	C₂³×C₆₁	—	2	1	no	—
212	C₂³×C₆₁	—	2	1	no	—
213	C₂³×C₆₁	—	2	1	no	—
214	C₂³×C₆₁	C₂²⋊Q₈	1	2	yes	—
215	C₂³×C₆₁	#135	1	2	yes	—
216	C₂³×C₆₁	#137	1	2	yes	—
217	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
218	Q₈×C₆₁	—	2	2	no	—
219	Q₈×C₆₁	—	2	2	no	—
220	Q₈×C₆₁	—	4	1	no	—
221	Q₈×C₆₁	—	4	1	no	—
222	C₂×C₄×C₆₁	—	2	8	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	—	2	2	no	—
232	C₂×C₄×C₆₁	—	2	2	no	—
233	C₂×C₄×C₆₁	—	2	2	no	—
234	C₂×C₄×C₆₁	—	2	2	no	—
235	C₂×C₄×C₆₁	—	2	1	no	—
236	C₂×C₄×C₆₁	—	2	2	no	—
237	C₂×C₄×C₆₁	—	2	2	no	—
238	C₂×C₄×C₆₁	—	2	1	no	—
239	C₂×C₄×C₆₁	—	2	1	no	—
240	C₂×C₄×C₆₁	—	4	4	no	—
241	C₂×C₄×C₆₁	—	4	1	no	—
242	C₂×C₄×C₆₁	—	4	1	no	—
243	C₂×C₄×C₆₁	—	4	1	no	—
244	C₂×C₄×C₆₁	—	4	1	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	—	4	1	no	—
252	C₂×C₄×C₆₁	—	4	1	no	—
253	C₂×C₄×C₆₁	—	4	1	no	—
254	C₂×C₄×C₆₁	—	4	1	no	—
255	C₂×C₄×C₆₁	—	4	1	no	—
256	C₂×C₄×C₆₁	—	4	1	no	—
257	C₂×C₄×C₆₁	—	4	1	no	—
258	C₂×C₄×C₆₁	—	4	1	no	—
259	C₂×C₄×C₆₁	—	4	1	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	#121	1	2	yes	—
264	C₂×C₄×C₆₁	#121	1	2	yes	—
265	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
266	C₂×C₄×C₆₁	#135	1	2	yes	—
267	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
269	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
272	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
273	C₈×C₆₁	—	2	2	no	—
274	C₈×C₆₁	—	2	2	no	—
275	C₈×C₆₁	—	4	1	no	—
276	C₈×C₆₁	—	4	1	no	—
277	#24840	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂²×C₆₁	—	2	2	no	—
279	C₂²×C₆₁	—	2	2	no	—
280	C₂²×C₆₁	—	2	2	no	—
281	C₂²×C₆₁	—	2	2	no	—
282	C₂²×C₆₁	—	2	2	no	—
283	C₂²×C₆₁	—	2	2	no	—
284	C₂²×C₆₁	#479	1	4	yes	complemented
285	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
286	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
287	C₂²×C₆₁	#537	1	1	yes	characteristic
288	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
289	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
290	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
291	C₂²×C₆₁	#575	1	1	yes	characteristic
292	C₄×C₆₁	—	2	2	no	—
293	C₄×C₆₁	—	2	2	no	—
294	C₄×C₆₁	—	2	2	no	—
295	C₄×C₆₁	—	2	2	no	—
296	C₄×C₆₁	—	4	2	no	—
297	C₄×C₆₁	—	4	2	no	—
298	C₄×C₆₁	—	4	1	no	—
299	C₄×C₆₁	—	8	1	no	—
300	C₄×C₆₁	—	8	1	no	—
301	C₄×C₆₁	—	8	1	no	—
302	C₄×C₆₁	—	8	1	no	—
303	C₄×C₆₁	—	8	1	no	—
304	C₄×C₆₁	—	8	1	no	—
305	C₄×C₆₁	#546	1	1	yes	characteristic
306	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
307	C₄×C₆₁	#552	1	1	yes	characteristic
308	C₄×C₆₁	#554	1	1	yes	characteristic
309	C₂×#479	C₂×C₆₁	1	1	yes	characteristic, complemented
310	C₂²×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
311	#2851	C₂×C₆₁	1	1	yes	characteristic
312	C₄×C₂²⋊Q₈	C₂×C₆₁	1	1	yes	characteristic
313	#2088	C₂×C₆₁	1	1	yes	characteristic, complemented
314	#2086	C₂×C₆₁	1	1	yes	characteristic, complemented
315	#2068	C₂×C₆₁	1	1	yes	characteristic, complemented
316	#2066	C₂×C₆₁	1	1	yes	characteristic, complemented
317	#1989	C₂×C₆₁	1	1	yes	characteristic, complemented
318	#1949	C₂×C₆₁	1	1	yes	characteristic, complemented
319	#1892	C₂×C₆₁	1	1	yes	characteristic
320	#1892	C₂×C₆₁	1	1	yes	characteristic
321	#1808	C₂×C₆₁	1	1	yes	characteristic
322	#1807	C₂×C₆₁	1	1	yes	characteristic
323	#1805	C₂×C₆₁	1	1	yes	characteristic, complemented
324	C₂×C₆₁	—	2	4	no	—
325	C₂×C₆₁	#2417	1	1	yes	characteristic
326	C₂×C₆₁	#3050	1	1	yes	characteristic
327	C₂×C₆₁	#3056	1	1	yes	characteristic
328	C₂×C₆₁	C₂×#479	1	1	yes	characteristic
329	C₂×C₆₁	#3076	1	1	yes	characteristic
330	C₂×C₆₁	#3248	1	1	yes	characteristic
331	C₂×C₆₁	#3250	1	1	yes	characteristic
332	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
333	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
334	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
335	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
336	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
337	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
338	#479	—	2	4	no	—
339	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
340	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
341	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
342	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
343	C₂×C₄⋊C₈	—	2	4	no	—
344	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
345	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
346	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
347	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
348	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
349	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
350	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
351	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
352	#391	C₂²×C₆₁	1	1	yes	characteristic
353	#389	C₂²×C₆₁	1	1	yes	characteristic
354	#389	C₂²×C₆₁	1	1	yes	characteristic
355	#389	C₂²×C₆₁	1	1	yes	characteristic
356	#387	—	2	1	no	—
357	#387	—	2	1	no	—
358	#387	—	2	1	no	—
359	#385	—	2	1	no	—
360	#385	—	2	1	no	—
361	#385	—	2	1	no	—
362	#382	C₂²×C₆₁	1	1	yes	characteristic
363	#380	C₂²×C₆₁	1	1	yes	characteristic
364	#380	C₂²×C₆₁	1	1	yes	characteristic
365	#380	C₂²×C₆₁	1	1	yes	characteristic
366	C₄×C₄⋊C₄	—	2	1	no	—
367	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
368	C₄×C₂²⋊C₄	—	2	1	no	—
369	#340	C₂²×C₆₁	1	1	yes	characteristic
370	#340	C₂²×C₆₁	1	1	yes	characteristic
371	#340	C₂²×C₆₁	1	1	yes	characteristic
372	#340	C₂²×C₆₁	1	1	yes	characteristic
373	#340	C₂²×C₆₁	1	1	yes	characteristic
374	#340	C₂²×C₆₁	1	1	yes	characteristic
375	#340	C₂²×C₆₁	1	1	yes	characteristic
376	#340	C₂²×C₆₁	1	1	yes	characteristic
377	C₆₁	#24840	1	1	yes	characteristic, direct product factor, Sylow
378	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
379	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
380	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
382	C₂²×C₈	—	2	2	no	—
383	C₂²×C₈	—	2	2	no	—
384	C₂²×C₈	—	2	2	no	—
385	C₂²×C₈	—	2	2	no	—
386	C₂²×C₈	—	4	1	no	—
387	C₂²×C₈	—	4	1	no	—
388	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
390	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
391	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
392	#125	—	2	2	no	—
393	#125	—	2	2	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	Q₈×C₆₁	1	4	yes	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×Q₈	—	2	2	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₄²	—	2	2	no	—
417	C₂×C₄²	—	2	2	no	—
418	C₂×C₄²	—	2	1	no	—
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
421	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
424	#107	—	2	2	no	—
425	#107	—	4	1	no	—
426	#106	—	2	2	no	—
427	#106	—	4	1	no	—
428	C₄⋊C₈	—	2	2	no	—
429	C₄⋊C₈	—	2	2	no	—
430	C₄⋊C₈	—	2	2	no	—
431	C₄⋊C₈	D₄×C₆₁	1	4	yes	—
432	#103	—	2	2	no	—
433	#103	—	2	2	no	—
434	#103	—	4	1	no	—
435	#103	—	4	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	—	2	1	no	—
439	#95	—	2	1	no	—
440	#95	—	2	1	no	—
441	#95	—	2	1	no	—
442	#95	—	2	1	no	—
443	#95	—	2	1	no	—
444	#95	—	2	1	no	—
445	#95	—	2	1	no	—
446	#95	—	2	1	no	—
447	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	C₂×Q₈×C₆₁	1	2	yes	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	4	1	no	—
472	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
478	C₂²×C₄	#41×C₆₁	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
480	C₂×C₈	—	2	1	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	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	4	4	no	—
491	C₂×C₈	—	4	4	no	—
492	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
493	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	4	2	no	—
509	C₄⋊C₄	—	4	2	no	—
510	C₄⋊C₄	—	4	2	no	—
511	C₄⋊C₄	—	4	2	no	—
512	C₄⋊C₄	—	4	2	no	—
513	C₄⋊C₄	—	4	2	no	—
514	C₄⋊C₄	C₂×Q₈×C₆₁	1	2	yes	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	4	2	no	—
517	C₂²⋊C₄	—	4	2	no	—
518	C₂²⋊C₄	—	4	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	4	2	no	—
524	C₄²	—	4	2	no	—
525	C₄²	C₂×D₄×C₆₁	1	2	yes	—
526	C₄²	#41×C₆₁	1	2	yes	—
527	C₂³	—	2	1	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	1	no	—
530	C₂³	—	2	1	no	—
531	C₂³	C₂²⋊Q₈×C₆₁	1	2	yes	—
532	C₂³	#135×C₆₁	1	2	yes	—
533	C₂³	#137×C₆₁	1	2	yes	—
534	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
535	Q₈	—	2	2	no	—
536	Q₈	—	2	2	no	—
537	Q₈	—	4	1	no	—
538	Q₈	—	4	1	no	—
539	C₂×C₄	—	2	8	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	—	2	1	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	#121×C₆₁	1	2	yes	—
581	C₂×C₄	#121×C₆₁	1	2	yes	—
582	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
583	C₂×C₄	#135×C₆₁	1	2	yes	—
584	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
585	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
586	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
587	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
588	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
589	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
590	C₈	—	2	2	no	—
591	C₈	—	2	2	no	—
592	C₈	—	4	1	no	—
593	C₈	—	4	1	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	2	no	—
600	C₂²	C₆₁×#479	1	4	yes	complemented
601	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
602	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
603	C₂²	C₆₁×#537	1	1	yes	characteristic
604	C₂²	C₂×#135×C₆₁	1	1	yes	characteristic
605	C₂²	C₂×#135×C₆₁	1	1	yes	characteristic
606	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
607	C₂²	C₆₁×#575	1	1	yes	characteristic
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	2	2	no	—
611	C₄	—	2	2	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	2	no	—
615	C₄	—	8	1	no	—
616	C₄	—	8	1	no	—
617	C₄	—	8	1	no	—
618	C₄	—	8	1	no	—
619	C₄	—	8	1	no	—
620	C₄	—	8	1	no	—
621	C₄	C₆₁×#546	1	1	yes	characteristic
622	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
623	C₄	C₆₁×#552	1	1	yes	characteristic
624	C₄	C₆₁×#554	1	1	yes	characteristic
625	C₂	—	2	4	no	—
626	C₂	C₆₁×#2417	1	1	yes	characteristic
627	C₂	C₆₁×#3050	1	1	yes	characteristic
628	C₂	C₆₁×#3056	1	1	yes	characteristic
629	C₂	C₂×C₆₁×#479	1	1	yes	characteristic
630	C₂	C₆₁×#3076	1	1	yes	characteristic
631	C₂	C₆₁×#3248	1	1	yes	characteristic
632	C₂	C₆₁×#3250	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#479	C₂	semidirect
5. C₆₁×#2088	C₂	semidirect
6. C₆₁×#2086	C₂	semidirect
7. C₆₁×#2068	C₂	semidirect
8. C₆₁×#2066	C₂	semidirect
9. C₆₁×#1989	C₂	semidirect
10. C₆₁×#1949	C₂	semidirect
15. C₆₁×#1805	C₂	semidirect
277. #24840	C₆₁	direct (N × H)
284. C₂²×C₆₁	#479	semidirect
309. C₂×#479	C₂×C₆₁	semidirect
313. #2088	C₂×C₆₁	semidirect
314. #2086	C₂×C₆₁	semidirect
315. #2068	C₂×C₆₁	semidirect
316. #2066	C₂×C₆₁	semidirect
317. #1989	C₂×C₆₁	semidirect
318. #1949	C₂×C₆₁	semidirect
323. #1805	C₂×C₆₁	semidirect
377. C₆₁	#24840	direct (N × H)
600. C₂²	C₆₁×#479	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#479	2
C₂²×C₄⋊C₈×C₆₁	2
C₆₁×#2851	2
C₄×C₂²⋊Q₈×C₆₁	2
C₆₁×#2088	2
C₆₁×#2086	2
C₆₁×#2068	2
C₆₁×#2066	2
C₆₁×#1989	2
C₆₁×#1949	2
C₆₁×#1892	2
C₆₁×#1808	2
C₆₁×#1807	2
C₆₁×#1805	2
#24840	61

Subgroups — C61×#24840 #29528273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24840 #29528273