Subgroups — C₆₁×#24231 #29527664

Order 15616.

1368 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3036	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#454	C₂	1	2	yes	maximal, complemented
4	C₄×AGL(1,ℤ/8)×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2547	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2038	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2003	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1980	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1793	C₂	1	2	yes	maximal, complemented
12	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	2	yes	—
13	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	1	yes	characteristic
15	D₄×Q₈×C₆₁	—	2	1	no	—
16	D₄×Q₈×C₆₁	—	2	1	no	—
17	C₆₁×#547	—	2	1	no	—
18	C₆₁×#547	—	2	1	no	—
19	C₆₁×#547	—	2	2	no	—
20	C₆₁×#547	—	2	2	no	—
21	C₆₁×#536	—	2	2	no	—
22	C₆₁×#536	C₂²	1	2	yes	—
23	C₆₁×#532	—	2	1	no	—
24	C₆₁×#532	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
26	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#124×C₆₁	C₂²	1	2	yes	—
28	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#463	—	2	2	no	—
31	C₆₁×#461	—	2	1	no	—
32	C₆₁×#459	—	2	1	no	—
33	C₆₁×#454	—	2	8	no	—
34	C₆₁×#442	—	2	1	no	—
35	C₆₁×#440	—	2	1	no	—
36	C₄×SD₁₆×C₆₁	—	2	1	no	—
37	C₄×D₈×C₆₁	—	2	1	no	—
38	C₆₁×#428	C₂²	1	1	yes	characteristic
39	C₆₁×#423	C₂²	1	1	yes	characteristic
40	C₂×#104×C₆₁	C₂²	1	2	yes	—
41	C₂×#104×C₆₁	C₂²	1	2	yes	—
42	C₂×#104×C₆₁	C₂²	1	2	yes	—
43	C₆₁×#418	C₂²	1	1	yes	characteristic
44	C₆₁×#417	C₂²	1	2	yes	—
45	C₆₁×#417	C₂²	1	1	yes	characteristic
46	C₂×#101×C₆₁	C₂²	1	2	yes	—
47	C₂×#101×C₆₁	C₂²	1	2	yes	—
48	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
49	C₆₁×#397	—	2	2	no	—
50	C₆₁×#391	C₂²	1	2	yes	—
51	C₆₁×#380	—	2	2	no	—
52	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
53	C₆₁×#339	C₂²	1	2	yes	—
54	2₋¹⁺⁴×C₆₁	—	2	4	no	—
55	2₋¹⁺⁴×C₆₁	—	2	4	no	—
56	2₋¹⁺⁴×C₆₁	—	4	2	no	—
57	2₋¹⁺⁴×C₆₁	—	4	2	no	—
58	C₂×#41×C₆₁	—	2	2	no	—
59	C₂×#41×C₆₁	—	2	2	no	—
60	C₂×#41×C₆₁	—	2	4	no	—
61	C₂×#41×C₆₁	C₂³	1	2	yes	—
62	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₆₁	—	2	2	no	—
64	C₂²×Q₈×C₆₁	—	2	2	no	—
65	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₁	—	2	1	no	—
68	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
69	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
70	AGL(1,ℤ/8)×C₆₁	—	4	2	no	—
71	AGL(1,ℤ/8)×C₆₁	D₄	1	4	yes	complemented
72	C₂×SD₁₆×C₆₁	—	2	1	no	—
73	C₂×SD₁₆×C₆₁	—	2	2	no	—
74	C₂×D₈×C₆₁	—	2	1	no	—
75	C₂×D₈×C₆₁	—	2	2	no	—
76	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
77	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
78	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
79	C₄⋊Q₈×C₆₁	—	2	2	no	—
80	C₄⋊Q₈×C₆₁	—	2	2	no	—
81	C₄⋊Q₈×C₆₁	—	2	2	no	—
82	C₄⋊Q₈×C₆₁	—	2	1	no	—
83	C₄⋊Q₈×C₆₁	—	2	1	no	—
84	C₄⋊Q₈×C₆₁	—	2	1	no	—
85	C₄⋊Q₈×C₆₁	—	2	1	no	—
86	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
87	#127×C₆₁	D₄	1	1	yes	characteristic
88	#124×C₆₁	—	2	2	no	—
89	#124×C₆₁	—	2	4	no	—
90	#124×C₆₁	—	4	2	no	—
91	#124×C₆₁	D₄	1	2	yes	—
92	#123×C₆₁	—	2	4	no	—
93	#123×C₆₁	—	2	4	no	—
94	#123×C₆₁	—	2	4	no	—
95	C₂²⋊Q₈×C₆₁	—	2	4	no	—
96	C₂²⋊Q₈×C₆₁	—	2	4	no	—
97	C₂²⋊Q₈×C₆₁	—	2	2	no	—
98	C₂²⋊Q₈×C₆₁	—	2	2	no	—
99	#121×C₆₁	—	2	2	no	—
100	#121×C₆₁	—	2	2	no	—
101	#121×C₆₁	—	2	1	no	—
102	#121×C₆₁	—	2	1	no	—
103	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
104	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
105	C₄×D₄×C₆₁	—	2	2	no	—
106	C₄×D₄×C₆₁	—	2	2	no	—
107	C₄×D₄×C₆₁	—	2	1	no	—
108	C₄×D₄×C₆₁	—	2	1	no	—
109	C₄×D₄×C₆₁	—	2	1	no	—
110	C₄×D₄×C₆₁	—	2	1	no	—
111	C₄×D₄×C₆₁	—	2	1	no	—
112	C₄×D₄×C₆₁	—	2	1	no	—
113	C₄×D₄×C₆₁	—	2	1	no	—
114	C₄×D₄×C₆₁	—	2	1	no	—
115	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
116	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
117	#117×C₆₁	—	2	1	no	—
118	#117×C₆₁	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
121	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
122	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
123	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
124	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
125	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
126	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
127	C₂×C₄²×C₆₁	—	2	1	no	—
128	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
130	#107×C₆₁	—	2	1	no	—
131	#106×C₆₁	—	2	1	no	—
132	C₄⋊C₈×C₆₁	—	2	1	no	—
133	C₄⋊C₈×C₆₁	—	2	1	no	—
134	#104×C₆₁	—	2	4	no	—
135	#104×C₆₁	—	2	4	no	—
136	#104×C₆₁	—	4	2	no	—
137	#103×C₆₁	—	2	1	no	—
138	#103×C₆₁	—	2	2	no	—
139	#102×C₆₁	—	2	1	no	—
140	#102×C₆₁	—	2	2	no	—
141	#102×C₆₁	—	2	1	no	—
142	#102×C₆₁	—	2	1	no	—
143	#101×C₆₁	—	2	4	no	—
144	#101×C₆₁	—	4	2	no	—
145	#97×C₆₁	—	2	1	no	—
146	C₄×C₈×C₆₁	—	2	1	no	—
147	#95×C₆₁	—	2	1	no	—
148	#95×C₆₁	—	2	4	no	—
149	C₂⁴×C₆₁	—	2	1	no	—
150	#41×C₆₁	—	2	1	no	—
151	#41×C₆₁	—	2	1	no	—
152	#41×C₆₁	—	2	2	no	—
153	#41×C₆₁	—	2	2	no	—
154	#41×C₆₁	—	2	2	no	—
155	#41×C₆₁	—	2	2	no	—
156	#41×C₆₁	—	4	4	no	—
157	#41×C₆₁	—	4	4	no	—
158	#41×C₆₁	—	4	8	no	—
159	#41×C₆₁	—	4	8	no	—
160	#41×C₆₁	—	4	4	no	—
161	#41×C₆₁	—	4	4	no	—
162	#41×C₆₁	C₂×D₄	1	2	yes	—
163	#41×C₆₁	#41	1	2	yes	—
164	C₂×Q₈×C₆₁	—	2	4	no	—
165	C₂×Q₈×C₆₁	—	2	2	no	—
166	C₂×Q₈×C₆₁	—	2	2	no	—
167	C₂×Q₈×C₆₁	—	2	2	no	—
168	C₂×Q₈×C₆₁	—	2	2	no	—
169	C₂×Q₈×C₆₁	—	2	2	no	—
170	C₂×Q₈×C₆₁	—	2	2	no	—
171	C₂×Q₈×C₆₁	—	4	4	no	—
172	C₂×Q₈×C₆₁	—	4	4	no	—
173	C₂×Q₈×C₆₁	—	4	4	no	—
174	C₂×Q₈×C₆₁	—	4	4	no	—
175	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
176	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
177	C₂×D₄×C₆₁	—	2	1	no	—
178	C₂×D₄×C₆₁	—	2	4	no	—
179	C₂×D₄×C₆₁	—	2	2	no	—
180	C₂×D₄×C₆₁	—	2	1	no	—
181	C₂×D₄×C₆₁	—	2	1	no	—
182	C₂×D₄×C₆₁	—	2	1	no	—
183	C₂×D₄×C₆₁	—	2	1	no	—
184	C₂×D₄×C₆₁	—	2	1	no	—
185	C₂×D₄×C₆₁	—	2	1	no	—
186	C₂×D₄×C₆₁	—	2	1	no	—
187	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
188	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
189	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₆₁	—	2	1	no	—
191	C₂²×C₄×C₆₁	—	2	4	no	—
192	C₂²×C₄×C₆₁	—	2	4	no	—
193	C₂²×C₄×C₆₁	—	2	1	no	—
194	C₂²×C₄×C₆₁	—	2	1	no	—
195	C₂²×C₄×C₆₁	—	2	2	no	—
196	C₂²×C₄×C₆₁	—	2	1	no	—
197	C₂²×C₄×C₆₁	—	2	2	no	—
198	C₂²×C₄×C₆₁	—	4	4	no	—
199	C₂²×C₄×C₆₁	—	4	1	no	—
200	C₂²×C₄×C₆₁	—	4	1	no	—
201	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
202	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
205	SD₁₆×C₆₁	—	2	4	no	—
206	SD₁₆×C₆₁	—	4	4	no	—
207	D₈×C₆₁	—	2	4	no	—
208	D₈×C₆₁	—	4	4	no	—
209	M₁₆×C₆₁	—	2	1	no	—
210	M₁₆×C₆₁	—	2	2	no	—
211	M₁₆×C₆₁	—	2	1	no	—
212	M₁₆×C₆₁	—	2	2	no	—
213	M₁₆×C₆₁	—	2	1	no	—
214	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
215	C₂×C₈×C₆₁	—	2	1	no	—
216	C₂×C₈×C₆₁	—	2	1	no	—
217	C₂×C₈×C₆₁	—	2	2	no	—
218	C₄⋊C₄×C₆₁	—	2	4	no	—
219	C₄⋊C₄×C₆₁	—	2	4	no	—
220	C₄⋊C₄×C₆₁	—	2	1	no	—
221	C₄⋊C₄×C₆₁	—	2	1	no	—
222	C₄⋊C₄×C₆₁	—	2	2	no	—
223	C₄⋊C₄×C₆₁	—	2	2	no	—
224	C₄⋊C₄×C₆₁	—	2	2	no	—
225	C₄⋊C₄×C₆₁	—	2	2	no	—
226	C₄⋊C₄×C₆₁	—	2	1	no	—
227	C₄⋊C₄×C₆₁	—	2	1	no	—
228	C₄⋊C₄×C₆₁	—	2	1	no	—
229	C₄⋊C₄×C₆₁	—	2	1	no	—
230	C₄⋊C₄×C₆₁	—	4	2	no	—
231	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
232	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
233	C₂²⋊C₄×C₆₁	—	2	4	no	—
234	C₂²⋊C₄×C₆₁	—	2	4	no	—
235	C₂²⋊C₄×C₆₁	—	2	1	no	—
236	C₂²⋊C₄×C₆₁	—	2	1	no	—
237	C₂²⋊C₄×C₆₁	—	2	1	no	—
238	C₂²⋊C₄×C₆₁	—	2	2	no	—
239	C₂²⋊C₄×C₆₁	—	2	2	no	—
240	C₂²⋊C₄×C₆₁	—	4	8	no	—
241	C₂²⋊C₄×C₆₁	—	4	1	no	—
242	C₂²⋊C₄×C₆₁	—	4	1	no	—
243	C₄²×C₆₁	—	2	1	no	—
244	C₄²×C₆₁	—	2	1	no	—
245	C₄²×C₆₁	—	2	1	no	—
246	C₄²×C₆₁	—	2	2	no	—
247	C₄²×C₆₁	—	2	2	no	—
248	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
249	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
250	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
251	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
252	C₂³×C₆₁	—	2	1	no	—
253	C₂³×C₆₁	—	2	2	no	—
254	C₂³×C₆₁	—	2	2	no	—
255	C₂³×C₆₁	—	4	2	no	—
256	C₂³×C₆₁	—	4	2	no	—
257	C₂³×C₆₁	—	4	1	no	—
258	C₂³×C₆₁	—	4	1	no	—
259	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
260	Q₈×C₆₁	—	2	1	no	—
261	Q₈×C₆₁	—	2	2	no	—
262	Q₈×C₆₁	—	2	2	no	—
263	Q₈×C₆₁	—	4	8	no	—
264	Q₈×C₆₁	—	4	2	no	—
265	Q₈×C₆₁	—	4	2	no	—
266	Q₈×C₆₁	—	4	2	no	—
267	Q₈×C₆₁	—	4	2	no	—
268	Q₈×C₆₁	#121	1	2	yes	—
269	D₄×C₆₁	—	2	2	no	—
270	D₄×C₆₁	—	2	2	no	—
271	D₄×C₆₁	—	2	2	no	—
272	D₄×C₆₁	—	2	1	no	—
273	D₄×C₆₁	—	2	2	no	—
274	D₄×C₆₁	—	2	2	no	—
275	D₄×C₆₁	—	2	2	no	—
276	D₄×C₆₁	—	4	2	no	—
277	D₄×C₆₁	—	4	8	no	—
278	D₄×C₆₁	—	4	2	no	—
279	D₄×C₆₁	—	4	2	no	—
280	D₄×C₆₁	—	4	1	no	—
281	D₄×C₆₁	—	4	1	no	—
282	D₄×C₆₁	—	4	2	no	—
283	D₄×C₆₁	#121	1	2	yes	—
284	C₂×C₄×C₆₁	—	2	1	no	—
285	C₂×C₄×C₆₁	—	2	2	no	—
286	C₂×C₄×C₆₁	—	2	2	no	—
287	C₂×C₄×C₆₁	—	2	2	no	—
288	C₂×C₄×C₆₁	—	2	1	no	—
289	C₂×C₄×C₆₁	—	2	1	no	—
290	C₂×C₄×C₆₁	—	2	2	no	—
291	C₂×C₄×C₆₁	—	2	4	no	—
292	C₂×C₄×C₆₁	—	2	4	no	—
293	C₂×C₄×C₆₁	—	2	4	no	—
294	C₂×C₄×C₆₁	—	4	1	no	—
295	C₂×C₄×C₆₁	—	4	2	no	—
296	C₂×C₄×C₆₁	—	4	8	no	—
297	C₂×C₄×C₆₁	—	4	8	no	—
298	C₂×C₄×C₆₁	—	4	2	no	—
299	C₂×C₄×C₆₁	—	4	2	no	—
300	C₂×C₄×C₆₁	—	4	2	no	—
301	C₂×C₄×C₆₁	—	4	2	no	—
302	C₂×C₄×C₆₁	—	4	2	no	—
303	C₂×C₄×C₆₁	—	4	2	no	—
304	C₂×C₄×C₆₁	—	4	1	no	—
305	C₂×C₄×C₆₁	—	4	2	no	—
306	C₂×C₄×C₆₁	—	8	2	no	—
307	C₂×C₄×C₆₁	—	8	2	no	—
308	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
309	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
310	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
311	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
312	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
313	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
316	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
317	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
318	C₈×C₆₁	—	2	2	no	—
319	C₈×C₆₁	—	4	1	no	—
320	C₈×C₆₁	—	4	2	no	—
321	#24231	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
322	C₂²×C₆₁	—	2	1	no	—
323	C₂²×C₆₁	—	2	1	no	—
324	C₂²×C₆₁	—	2	2	no	—
325	C₂²×C₆₁	—	4	4	no	—
326	C₂²×C₆₁	—	4	1	no	—
327	C₂²×C₆₁	—	4	1	no	—
328	C₂²×C₆₁	—	4	2	no	—
329	C₂²×C₆₁	—	4	2	no	—
330	C₂²×C₆₁	—	4	2	no	—
331	C₂²×C₆₁	—	4	2	no	—
332	C₂²×C₆₁	—	8	2	no	—
333	C₂²×C₆₁	—	8	1	no	—
334	C₂²×C₆₁	—	8	1	no	—
335	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
336	C₂²×C₆₁	D₄²	1	1	yes	characteristic
337	C₂²×C₆₁	#546	1	1	yes	characteristic
338	C₄×C₆₁	—	2	2	no	—
339	C₄×C₆₁	—	2	1	no	—
340	C₄×C₆₁	—	2	1	no	—
341	C₄×C₆₁	—	2	1	no	—
342	C₄×C₆₁	—	4	4	no	—
343	C₄×C₆₁	—	4	2	no	—
344	C₄×C₆₁	—	4	1	no	—
345	C₄×C₆₁	—	4	1	no	—
346	C₄×C₆₁	—	8	2	no	—
347	C₄×C₆₁	—	8	1	no	—
348	C₄×C₆₁	#454	1	2	yes	complemented
349	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
350	C₄×C₆₁	#535	1	1	yes	characteristic
351	C₄×C₆₁	D₄²	1	1	yes	characteristic
352	C₄×C₆₁	#546	1	1	yes	characteristic
353	#3499	C₂×C₆₁	1	1	yes	characteristic, complemented
354	#3036	C₂×C₆₁	1	1	yes	characteristic, complemented
355	C₂×#454	C₂×C₆₁	1	2	yes	complemented
356	C₄×AGL(1,ℤ/8)	C₂×C₆₁	1	1	yes	characteristic, complemented
357	#2547	C₂×C₆₁	1	1	yes	characteristic, complemented
358	#2038	C₂×C₆₁	1	2	yes	complemented
359	#2003	C₂×C₆₁	1	2	yes	complemented
360	#1980	C₂×C₆₁	1	1	yes	characteristic, complemented
361	#1966	C₂×C₆₁	1	1	yes	characteristic, complemented
362	#1806	C₂×C₆₁	1	1	yes	characteristic, complemented
363	#1793	C₂×C₆₁	1	2	yes	complemented
364	C₂×C₆₁	—	2	1	no	—
365	C₂×C₆₁	—	2	1	no	—
366	C₂×C₆₁	—	4	2	no	—
367	C₂×C₆₁	—	8	2	no	—
368	C₂×C₆₁	—	8	2	no	—
369	C₂×C₆₁	#2408	1	1	yes	characteristic
370	C₂×C₆₁	C₂×#454	1	1	yes	characteristic
371	C₂×C₆₁	#3020	1	1	yes	characteristic
372	C₂×2₋¹⁺⁴	C₂²×C₆₁	1	2	yes	—
373	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
374	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	1	yes	characteristic
375	D₄×Q₈	—	2	1	no	—
376	D₄×Q₈	—	2	1	no	—
377	#547	—	2	1	no	—
378	#547	—	2	1	no	—
379	#547	—	2	2	no	—
380	#547	—	2	2	no	—
381	#536	—	2	2	no	—
382	#536	C₂²×C₆₁	1	2	yes	—
383	#532	—	2	1	no	—
384	#532	C₂²×C₆₁	1	1	yes	characteristic
385	C₂×C₄⋊Q₈	—	2	1	no	—
386	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
387	C₂×#124	C₂²×C₆₁	1	2	yes	—
388	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
389	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
390	#463	—	2	2	no	—
391	#461	—	2	1	no	—
392	#459	—	2	1	no	—
393	#454	—	2	8	no	—
394	#442	—	2	1	no	—
395	#440	—	2	1	no	—
396	C₄×SD₁₆	—	2	1	no	—
397	C₄×D₈	—	2	1	no	—
398	#428	C₂²×C₆₁	1	1	yes	characteristic
399	#423	C₂²×C₆₁	1	1	yes	characteristic
400	C₂×#104	C₂²×C₆₁	1	2	yes	—
401	C₂×#104	C₂²×C₆₁	1	2	yes	—
402	C₂×#104	C₂²×C₆₁	1	2	yes	—
403	#418	C₂²×C₆₁	1	1	yes	characteristic
404	#417	C₂²×C₆₁	1	1	yes	characteristic
405	#417	C₂²×C₆₁	1	2	yes	—
406	C₂×#101	C₂²×C₆₁	1	2	yes	—
407	C₂×#101	C₂²×C₆₁	1	2	yes	—
408	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
409	#397	—	2	2	no	—
410	#391	C₂²×C₆₁	1	2	yes	—
411	#380	—	2	2	no	—
412	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
413	#339	C₂²×C₆₁	1	2	yes	—
414	C₆₁	#24231	1	1	yes	characteristic, direct product factor, Sylow
415	2₋¹⁺⁴	—	2	4	no	—
416	2₋¹⁺⁴	—	2	4	no	—
417	2₋¹⁺⁴	—	4	2	no	—
418	2₋¹⁺⁴	—	4	2	no	—
419	C₂×#41	—	2	2	no	—
420	C₂×#41	—	2	2	no	—
421	C₂×#41	—	2	4	no	—
422	C₂×#41	C₂³×C₆₁	1	2	yes	—
423	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
424	C₂²×Q₈	—	2	2	no	—
425	C₂²×Q₈	—	2	2	no	—
426	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
427	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
428	C₂³×C₄	—	2	1	no	—
429	AGL(1,ℤ/8)	—	2	2	no	—
430	AGL(1,ℤ/8)	—	2	4	no	—
431	AGL(1,ℤ/8)	—	4	2	no	—
432	AGL(1,ℤ/8)	D₄×C₆₁	1	4	yes	complemented
433	C₂×SD₁₆	—	2	1	no	—
434	C₂×SD₁₆	—	2	2	no	—
435	C₂×D₈	—	2	1	no	—
436	C₂×D₈	—	2	2	no	—
437	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
438	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
439	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
440	C₄⋊Q₈	—	2	2	no	—
441	C₄⋊Q₈	—	2	2	no	—
442	C₄⋊Q₈	—	2	2	no	—
443	C₄⋊Q₈	—	2	1	no	—
444	C₄⋊Q₈	—	2	1	no	—
445	C₄⋊Q₈	—	2	1	no	—
446	C₄⋊Q₈	—	2	1	no	—
447	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
448	#127	D₄×C₆₁	1	1	yes	characteristic
449	#124	—	2	2	no	—
450	#124	—	2	4	no	—
451	#124	—	4	2	no	—
452	#124	D₄×C₆₁	1	2	yes	—
453	#123	—	2	4	no	—
454	#123	—	2	4	no	—
455	#123	—	2	4	no	—
456	C₂²⋊Q₈	—	2	4	no	—
457	C₂²⋊Q₈	—	2	4	no	—
458	C₂²⋊Q₈	—	2	2	no	—
459	C₂²⋊Q₈	—	2	2	no	—
460	#121	—	2	2	no	—
461	#121	—	2	2	no	—
462	#121	—	2	1	no	—
463	#121	—	2	1	no	—
464	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
465	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
466	C₄×D₄	—	2	2	no	—
467	C₄×D₄	—	2	2	no	—
468	C₄×D₄	—	2	1	no	—
469	C₄×D₄	—	2	1	no	—
470	C₄×D₄	—	2	1	no	—
471	C₄×D₄	—	2	1	no	—
472	C₄×D₄	—	2	1	no	—
473	C₄×D₄	—	2	1	no	—
474	C₄×D₄	—	2	1	no	—
475	C₄×D₄	—	2	1	no	—
476	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
477	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
478	#117	—	2	1	no	—
479	#117	C₂³×C₆₁	1	1	yes	characteristic
480	C₂×C₄⋊C₄	—	2	2	no	—
481	C₂×C₄⋊C₄	—	2	2	no	—
482	C₂×C₄⋊C₄	—	2	1	no	—
483	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
484	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
485	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
486	C₂×C₂²⋊C₄	—	2	1	no	—
487	C₂×C₂²⋊C₄	—	2	4	no	—
488	C₂×C₄²	—	2	1	no	—
489	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
490	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
491	#107	—	2	1	no	—
492	#106	—	2	1	no	—
493	C₄⋊C₈	—	2	1	no	—
494	C₄⋊C₈	—	2	1	no	—
495	#104	—	2	4	no	—
496	#104	—	2	4	no	—
497	#104	—	4	2	no	—
498	#103	—	2	1	no	—
499	#103	—	2	2	no	—
500	#102	—	2	1	no	—
501	#102	—	2	2	no	—
502	#102	—	2	1	no	—
503	#102	—	2	1	no	—
504	#101	—	2	4	no	—
505	#101	—	4	2	no	—
506	#97	—	2	1	no	—
507	C₄×C₈	—	2	1	no	—
508	#95	—	2	1	no	—
509	#95	—	2	4	no	—
510	C₂⁴	—	2	1	no	—
511	#41	—	2	1	no	—
512	#41	—	2	1	no	—
513	#41	—	2	2	no	—
514	#41	—	2	2	no	—
515	#41	—	2	2	no	—
516	#41	—	2	2	no	—
517	#41	—	4	4	no	—
518	#41	—	4	4	no	—
519	#41	—	4	8	no	—
520	#41	—	4	8	no	—
521	#41	—	4	4	no	—
522	#41	—	4	4	no	—
523	#41	C₂×D₄×C₆₁	1	2	yes	—
524	#41	#41×C₆₁	1	2	yes	—
525	C₂×Q₈	—	2	4	no	—
526	C₂×Q₈	—	2	2	no	—
527	C₂×Q₈	—	2	2	no	—
528	C₂×Q₈	—	2	2	no	—
529	C₂×Q₈	—	2	2	no	—
530	C₂×Q₈	—	2	2	no	—
531	C₂×Q₈	—	2	2	no	—
532	C₂×Q₈	—	4	4	no	—
533	C₂×Q₈	—	4	4	no	—
534	C₂×Q₈	—	4	4	no	—
535	C₂×Q₈	—	4	4	no	—
536	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
537	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
538	C₂×D₄	—	2	1	no	—
539	C₂×D₄	—	2	4	no	—
540	C₂×D₄	—	2	2	no	—
541	C₂×D₄	—	2	1	no	—
542	C₂×D₄	—	2	1	no	—
543	C₂×D₄	—	2	1	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	1	no	—
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	—	2	1	no	—
548	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
549	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
550	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
551	C₂²×C₄	—	2	1	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	—	2	4	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	2	2	no	—
557	C₂²×C₄	—	2	1	no	—
558	C₂²×C₄	—	2	2	no	—
559	C₂²×C₄	—	4	4	no	—
560	C₂²×C₄	—	4	1	no	—
561	C₂²×C₄	—	4	1	no	—
562	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
563	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
564	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
565	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
566	SD₁₆	—	2	4	no	—
567	SD₁₆	—	4	4	no	—
568	D₈	—	2	4	no	—
569	D₈	—	4	4	no	—
570	M₁₆	—	2	1	no	—
571	M₁₆	—	2	2	no	—
572	M₁₆	—	2	1	no	—
573	M₁₆	—	2	2	no	—
574	M₁₆	—	2	1	no	—
575	M₁₆	C₂×D₄×C₆₁	1	2	yes	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	2	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	4	2	no	—
592	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
593	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
594	C₂²⋊C₄	—	2	4	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	1	no	—
597	C₂²⋊C₄	—	2	1	no	—
598	C₂²⋊C₄	—	2	1	no	—
599	C₂²⋊C₄	—	2	2	no	—
600	C₂²⋊C₄	—	2	2	no	—
601	C₂²⋊C₄	—	4	8	no	—
602	C₂²⋊C₄	—	4	1	no	—
603	C₂²⋊C₄	—	4	1	no	—
604	C₄²	—	2	1	no	—
605	C₄²	—	2	1	no	—
606	C₄²	—	2	1	no	—
607	C₄²	—	2	2	no	—
608	C₄²	—	2	2	no	—
609	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
610	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
611	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
612	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
613	C₂³	—	2	1	no	—
614	C₂³	—	2	2	no	—
615	C₂³	—	2	2	no	—
616	C₂³	—	4	2	no	—
617	C₂³	—	4	2	no	—
618	C₂³	—	4	1	no	—
619	C₂³	—	4	1	no	—
620	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
621	Q₈	—	2	1	no	—
622	Q₈	—	2	2	no	—
623	Q₈	—	2	2	no	—
624	Q₈	—	4	8	no	—
625	Q₈	—	4	2	no	—
626	Q₈	—	4	2	no	—
627	Q₈	—	4	2	no	—
628	Q₈	—	4	2	no	—
629	Q₈	#121×C₆₁	1	2	yes	—
630	D₄	—	2	2	no	—
631	D₄	—	2	2	no	—
632	D₄	—	2	1	no	—
633	D₄	—	2	2	no	—
634	D₄	—	2	2	no	—
635	D₄	—	2	2	no	—
636	D₄	—	2	2	no	—
637	D₄	—	4	2	no	—
638	D₄	—	4	8	no	—
639	D₄	—	4	2	no	—
640	D₄	—	4	2	no	—
641	D₄	—	4	1	no	—
642	D₄	—	4	1	no	—
643	D₄	—	4	2	no	—
644	D₄	#121×C₆₁	1	2	yes	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	4	no	—
650	C₂×C₄	—	2	4	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	4	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	2	no	—
657	C₂×C₄	—	4	8	no	—
658	C₂×C₄	—	4	8	no	—
659	C₂×C₄	—	4	2	no	—
660	C₂×C₄	—	4	2	no	—
661	C₂×C₄	—	4	2	no	—
662	C₂×C₄	—	4	2	no	—
663	C₂×C₄	—	4	2	no	—
664	C₂×C₄	—	4	2	no	—
665	C₂×C₄	—	4	2	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	8	2	no	—
668	C₂×C₄	—	8	2	no	—
669	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
670	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
671	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
672	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
673	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
674	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
675	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
676	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
677	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
678	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
679	C₈	—	2	2	no	—
680	C₈	—	4	1	no	—
681	C₈	—	4	2	no	—
682	C₂²	—	2	1	no	—
683	C₂²	—	2	1	no	—
684	C₂²	—	2	2	no	—
685	C₂²	—	4	4	no	—
686	C₂²	—	4	1	no	—
687	C₂²	—	4	1	no	—
688	C₂²	—	4	2	no	—
689	C₂²	—	4	2	no	—
690	C₂²	—	4	2	no	—
691	C₂²	—	4	2	no	—
692	C₂²	—	8	2	no	—
693	C₂²	—	8	1	no	—
694	C₂²	—	8	1	no	—
695	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
696	C₂²	D₄²×C₆₁	1	1	yes	characteristic
697	C₂²	C₆₁×#546	1	1	yes	characteristic
698	C₄	—	2	2	no	—
699	C₄	—	2	1	no	—
700	C₄	—	2	1	no	—
701	C₄	—	2	1	no	—
702	C₄	—	4	4	no	—
703	C₄	—	4	2	no	—
704	C₄	—	4	1	no	—
705	C₄	—	4	1	no	—
706	C₄	—	8	2	no	—
707	C₄	—	8	1	no	—
708	C₄	C₆₁×#454	1	2	yes	complemented
709	C₄	C₂×#121×C₆₁	1	1	yes	characteristic
710	C₄	C₆₁×#535	1	1	yes	characteristic
711	C₄	D₄²×C₆₁	1	1	yes	characteristic
712	C₄	C₆₁×#546	1	1	yes	characteristic
713	C₂	—	2	1	no	—
714	C₂	—	2	1	no	—
715	C₂	—	4	2	no	—
716	C₂	—	8	2	no	—
717	C₂	—	8	2	no	—
718	C₂	C₆₁×#2408	1	1	yes	characteristic
719	C₂	C₂×C₆₁×#454	1	1	yes	characteristic
720	C₂	C₆₁×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3499	C₂	semidirect
2. C₆₁×#3036	C₂	semidirect
3. C₂×C₆₁×#454	C₂	semidirect
4. C₄×AGL(1,ℤ/8)×C₆₁	C₂	semidirect
5. C₆₁×#2547	C₂	semidirect
6. C₆₁×#2038	C₂	semidirect
7. C₆₁×#2003	C₂	semidirect
8. C₆₁×#1980	C₂	semidirect
9. C₆₁×#1966	C₂	semidirect
10. C₆₁×#1806	C₂	semidirect
11. C₆₁×#1793	C₂	semidirect
71. AGL(1,ℤ/8)×C₆₁	D₄	semidirect
321. #24231	C₆₁	direct (N × H)
348. C₄×C₆₁	#454	semidirect
353. #3499	C₂×C₆₁	semidirect
354. #3036	C₂×C₆₁	semidirect
355. C₂×#454	C₂×C₆₁	semidirect
356. C₄×AGL(1,ℤ/8)	C₂×C₆₁	semidirect
357. #2547	C₂×C₆₁	semidirect
358. #2038	C₂×C₆₁	semidirect
359. #2003	C₂×C₆₁	semidirect
360. #1980	C₂×C₆₁	semidirect
361. #1966	C₂×C₆₁	semidirect
362. #1806	C₂×C₆₁	semidirect
363. #1793	C₂×C₆₁	semidirect
414. C₆₁	#24231	direct (N × H)
432. AGL(1,ℤ/8)	D₄×C₆₁	semidirect
708. C₄	C₆₁×#454	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3499	2
C₆₁×#3036	2
C₂×C₆₁×#454	2
C₄×AGL(1,ℤ/8)×C₆₁	2
C₆₁×#2547	2
C₆₁×#2038	2
C₆₁×#2003	2
C₆₁×#1980	2
C₆₁×#1966	2
C₆₁×#1806	2
C₆₁×#1793	2
#24231	61

Subgroups — C61×#24231 #29527664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24231 #29527664