Subgroups — C₆₁×#26071 #29529504

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2442	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2440	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2085	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2059	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1944	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1853	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1852	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1847	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1807	C₂	1	1	yes	maximal, characteristic
15	C₆₁×#1752	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
20	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#501	—	2	4	no	—
23	C₂×#107×C₆₁	—	2	4	no	—
24	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#400	—	2	1	no	—
38	C₆₁×#398	—	2	1	no	—
39	C₆₁×#398	—	2	1	no	—
40	C₆₁×#395	—	2	1	no	—
41	C₆₁×#389	C₂²	1	1	yes	characteristic
42	C₆₁×#389	C₂²	1	1	yes	characteristic
43	C₆₁×#389	C₂²	1	1	yes	characteristic
44	C₆₁×#389	C₂²	1	1	yes	characteristic
45	C₆₁×#385	—	2	1	no	—
46	C₆₁×#385	—	2	1	no	—
47	C₆₁×#384	C₂²	1	1	yes	characteristic
48	C₆₁×#382	C₂²	1	1	yes	characteristic
49	C₆₁×#380	—	2	1	no	—
50	C₆₁×#380	C₂²	1	1	yes	characteristic
51	C₆₁×#380	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
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×#336	C₂²	1	1	yes	characteristic
60	C₆₁×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₆₁	—	2	1	no	—
62	C₂³×C₄×C₆₁	—	2	1	no	—
63	C₂³×C₄×C₆₁	—	2	1	no	—
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	Q₈	1	4	yes	complemented
68	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₆₁	—	2	2	no	—
73	C₄⋊Q₈×C₆₁	—	4	1	no	—
74	#125×C₆₁	—	2	2	no	—
75	#125×C₆₁	—	4	1	no	—
76	C₂²⋊Q₈×C₆₁	—	4	4	no	—
77	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₁	—	2	1	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	—	2	1	no	—
89	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
91	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
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×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
104	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
106	#107×C₆₁	—	2	2	no	—
107	#107×C₆₁	—	2	2	no	—
108	#107×C₆₁	—	2	2	no	—
109	#107×C₆₁	—	2	2	no	—
110	#107×C₆₁	—	4	1	no	—
111	#107×C₆₁	D₄	1	4	yes	—
112	#106×C₆₁	—	2	2	no	—
113	#106×C₆₁	—	2	2	no	—
114	#106×C₆₁	—	2	2	no	—
115	#106×C₆₁	—	4	1	no	—
116	C₂²⋊C₈×C₆₁	Q₈	1	4	yes	complemented
117	#97×C₆₁	—	2	2	no	—
118	#97×C₆₁	—	2	2	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	D₄	1	1	yes	characteristic
128	#95×C₆₁	D₄	1	1	yes	characteristic
129	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Q₈×C₆₁	—	4	2	no	—
131	C₂×Q₈×C₆₁	—	4	2	no	—
132	C₂×Q₈×C₆₁	—	8	1	no	—
133	C₂²×C₄×C₆₁	—	2	1	no	—
134	C₂²×C₄×C₆₁	—	2	1	no	—
135	C₂²×C₄×C₆₁	—	2	1	no	—
136	C₂²×C₄×C₆₁	—	2	1	no	—
137	C₂²×C₄×C₆₁	—	2	1	no	—
138	C₂²×C₄×C₆₁	—	2	1	no	—
139	C₂²×C₄×C₆₁	—	2	1	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	1	no	—
145	C₂²×C₄×C₆₁	—	4	2	no	—
146	C₂²×C₄×C₆₁	—	4	2	no	—
147	C₂²×C₄×C₆₁	—	4	2	no	—
148	C₂²×C₄×C₆₁	—	4	2	no	—
149	C₂²×C₄×C₆₁	—	8	1	no	—
150	C₂²×C₄×C₆₁	—	8	1	no	—
151	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
157	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
158	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
159	C₂×C₈×C₆₁	—	2	1	no	—
160	C₂×C₈×C₆₁	—	2	1	no	—
161	C₂×C₈×C₆₁	—	2	1	no	—
162	C₂×C₈×C₆₁	—	2	1	no	—
163	C₂×C₈×C₆₁	—	2	1	no	—
164	C₂×C₈×C₆₁	—	2	1	no	—
165	C₂×C₈×C₆₁	—	2	8	no	—
166	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
167	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
168	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
169	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
170	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
171	C₂×C₈×C₆₁	#41	1	2	yes	—
172	C₄⋊C₄×C₆₁	—	2	1	no	—
173	C₄⋊C₄×C₆₁	—	2	1	no	—
174	C₄⋊C₄×C₆₁	—	2	1	no	—
175	C₄⋊C₄×C₆₁	—	2	1	no	—
176	C₄⋊C₄×C₆₁	—	2	1	no	—
177	C₄⋊C₄×C₆₁	—	2	1	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	—	4	2	no	—
187	C₄⋊C₄×C₆₁	—	4	2	no	—
188	C₄⋊C₄×C₆₁	—	4	2	no	—
189	C₄⋊C₄×C₆₁	—	4	2	no	—
190	C₄⋊C₄×C₆₁	—	4	2	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	—	8	1	no	—
195	C₄⋊C₄×C₆₁	—	8	1	no	—
196	C₄⋊C₄×C₆₁	—	8	1	no	—
197	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
198	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
199	C₂²⋊C₄×C₆₁	—	2	2	no	—
200	C₂²⋊C₄×C₆₁	—	2	2	no	—
201	C₂²⋊C₄×C₆₁	—	4	2	no	—
202	C₂²⋊C₄×C₆₁	—	4	2	no	—
203	C₄²×C₆₁	—	2	1	no	—
204	C₄²×C₆₁	—	2	1	no	—
205	C₄²×C₆₁	—	2	1	no	—
206	C₄²×C₆₁	—	2	1	no	—
207	C₂³×C₆₁	—	2	1	no	—
208	C₂³×C₆₁	—	2	1	no	—
209	C₂³×C₆₁	—	2	1	no	—
210	C₂³×C₆₁	—	2	1	no	—
211	C₂³×C₆₁	C₄⋊Q₈	1	2	yes	—
212	C₂³×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
213	C₂³×C₆₁	#137	1	2	yes	—
214	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
215	Q₈×C₆₁	—	8	4	no	—
216	C₂×C₄×C₆₁	—	2	1	no	—
217	C₂×C₄×C₆₁	—	2	1	no	—
218	C₂×C₄×C₆₁	—	2	2	no	—
219	C₂×C₄×C₆₁	—	2	2	no	—
220	C₂×C₄×C₆₁	—	2	2	no	—
221	C₂×C₄×C₆₁	—	2	2	no	—
222	C₂×C₄×C₆₁	—	2	1	no	—
223	C₂×C₄×C₆₁	—	2	1	no	—
224	C₂×C₄×C₆₁	—	2	2	no	—
225	C₂×C₄×C₆₁	—	2	2	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	1	no	—
232	C₂×C₄×C₆₁	—	2	1	no	—
233	C₂×C₄×C₆₁	—	2	1	no	—
234	C₂×C₄×C₆₁	—	2	1	no	—
235	C₂×C₄×C₆₁	—	2	1	no	—
236	C₂×C₄×C₆₁	—	4	1	no	—
237	C₂×C₄×C₆₁	—	4	1	no	—
238	C₂×C₄×C₆₁	—	4	1	no	—
239	C₂×C₄×C₆₁	—	4	1	no	—
240	C₂×C₄×C₆₁	—	4	1	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	—	8	4	no	—
259	C₂×C₄×C₆₁	—	8	4	no	—
260	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
261	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
263	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
264	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
266	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
267	C₈×C₆₁	—	2	2	no	—
268	C₈×C₆₁	—	2	2	no	—
269	C₈×C₆₁	—	2	2	no	—
270	C₈×C₆₁	C₂²⋊Q₈	1	4	yes	complemented
271	#26071	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂²×C₆₁	—	2	2	no	—
273	C₂²×C₆₁	—	2	2	no	—
274	C₂²×C₆₁	—	2	2	no	—
275	C₂²×C₆₁	—	2	2	no	—
276	C₂²×C₆₁	—	2	2	no	—
277	C₂²×C₆₁	—	2	2	no	—
278	C₂²×C₆₁	#501	1	4	yes	complemented
279	C₂²×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
280	C₂²×C₆₁	D₄²	1	1	yes	characteristic
281	C₂²×C₆₁	#547	1	1	yes	characteristic
282	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
284	C₂²×C₆₁	#576	1	1	yes	characteristic
285	C₂²×C₆₁	#578	1	1	yes	characteristic
286	C₄×C₆₁	—	2	2	no	—
287	C₄×C₆₁	—	4	1	no	—
288	C₄×C₆₁	—	4	1	no	—
289	C₄×C₆₁	—	4	1	no	—
290	C₄×C₆₁	—	4	2	no	—
291	C₄×C₆₁	—	4	2	no	—
292	C₄×C₆₁	—	4	1	no	—
293	C₄×C₆₁	—	8	1	no	—
294	C₄×C₆₁	—	8	1	no	—
295	C₄×C₆₁	—	8	1	no	—
296	C₄×C₆₁	—	8	1	no	—
297	C₄×C₆₁	—	8	1	no	—
298	C₄×C₆₁	—	8	1	no	—
299	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
300	C₄×C₆₁	#544	1	1	yes	characteristic
301	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
302	C₄×C₆₁	#554	1	1	yes	characteristic
303	C₂×#501	C₂×C₆₁	1	1	yes	characteristic, complemented
304	C₂²×#107	C₂×C₆₁	1	1	yes	characteristic
305	#2442	C₂×C₆₁	1	1	yes	characteristic
306	#2440	C₂×C₆₁	1	1	yes	characteristic
307	#2085	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#2083	C₂×C₆₁	1	1	yes	characteristic, complemented
309	#2059	C₂×C₆₁	1	1	yes	characteristic, complemented
310	#2057	C₂×C₆₁	1	1	yes	characteristic, complemented
311	#1944	C₂×C₆₁	1	1	yes	characteristic, complemented
312	#1943	C₂×C₆₁	1	1	yes	characteristic, complemented
313	#1853	C₂×C₆₁	1	1	yes	characteristic
314	#1852	C₂×C₆₁	1	1	yes	characteristic
315	#1847	C₂×C₆₁	1	1	yes	characteristic, complemented
316	#1807	C₂×C₆₁	1	1	yes	characteristic
317	#1752	C₂×C₆₁	1	1	yes	characteristic
318	C₂×C₆₁	—	2	4	no	—
319	C₂×C₆₁	#2440	1	1	yes	characteristic
320	C₂×C₆₁	C₂×#501	1	1	yes	characteristic
321	C₂×C₆₁	#3164	1	1	yes	characteristic
322	C₂×C₆₁	#3264	1	1	yes	characteristic
323	C₂×C₆₁	#3270	1	1	yes	characteristic
324	C₂×C₆₁	#3343	1	1	yes	characteristic
325	C₂×C₆₁	#3346	1	1	yes	characteristic
326	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
327	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
328	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
329	C₂×C₂²⋊Q₈	—	2	1	no	—
330	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
331	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
332	#501	—	2	4	no	—
333	C₂×#107	—	2	4	no	—
334	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
335	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
336	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
337	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
338	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
339	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
340	C₂×#106	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×#106	C₂²×C₆₁	1	1	yes	characteristic
344	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
345	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
346	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
347	#400	—	2	1	no	—
348	#398	—	2	1	no	—
349	#398	—	2	1	no	—
350	#395	—	2	1	no	—
351	#389	C₂²×C₆₁	1	1	yes	characteristic
352	#389	C₂²×C₆₁	1	1	yes	characteristic
353	#389	C₂²×C₆₁	1	1	yes	characteristic
354	#389	C₂²×C₆₁	1	1	yes	characteristic
355	#385	—	2	1	no	—
356	#385	—	2	1	no	—
357	#384	C₂²×C₆₁	1	1	yes	characteristic
358	#382	C₂²×C₆₁	1	1	yes	characteristic
359	#380	—	2	1	no	—
360	#380	C₂²×C₆₁	1	1	yes	characteristic
361	#380	C₂²×C₆₁	1	1	yes	characteristic
362	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
363	#340	C₂²×C₆₁	1	1	yes	characteristic
364	#340	C₂²×C₆₁	1	1	yes	characteristic
365	#340	C₂²×C₆₁	1	1	yes	characteristic
366	#340	C₂²×C₆₁	1	1	yes	characteristic
367	#340	C₂²×C₆₁	1	1	yes	characteristic
368	#340	C₂²×C₆₁	1	1	yes	characteristic
369	#336	C₂²×C₆₁	1	1	yes	characteristic
370	#336	C₂²×C₆₁	1	1	yes	characteristic
371	C₆₁	#26071	1	1	yes	characteristic, direct product factor, Sylow
372	C₂²×Q₈	—	2	1	no	—
373	C₂³×C₄	—	2	1	no	—
374	C₂³×C₄	—	2	1	no	—
375	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
376	C₂²×C₈	—	2	2	no	—
377	C₂²×C₈	—	2	2	no	—
378	C₂²×C₈	Q₈×C₆₁	1	4	yes	complemented
379	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
380	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
382	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
383	C₄⋊Q₈	—	2	2	no	—
384	C₄⋊Q₈	—	4	1	no	—
385	#125	—	2	2	no	—
386	#125	—	4	1	no	—
387	C₂²⋊Q₈	—	4	4	no	—
388	C₂×C₄⋊C₄	—	2	2	no	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	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	—	4	1	no	—
402	C₂×C₄⋊C₄	—	4	1	no	—
403	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
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×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
414	C₂×C₂²⋊C₄	D₄×C₆₁	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
417	#107	—	2	2	no	—
418	#107	—	2	2	no	—
419	#107	—	2	2	no	—
420	#107	—	2	2	no	—
421	#107	—	4	1	no	—
422	#107	D₄×C₆₁	1	4	yes	—
423	#106	—	2	2	no	—
424	#106	—	2	2	no	—
425	#106	—	2	2	no	—
426	#106	—	4	1	no	—
427	C₂²⋊C₈	Q₈×C₆₁	1	4	yes	complemented
428	#97	—	2	2	no	—
429	#97	—	2	2	no	—
430	#95	—	2	1	no	—
431	#95	—	2	1	no	—
432	#95	—	2	1	no	—
433	#95	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	D₄×C₆₁	1	1	yes	characteristic
439	#95	D₄×C₆₁	1	1	yes	characteristic
440	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
441	C₂×Q₈	—	4	2	no	—
442	C₂×Q₈	—	4	2	no	—
443	C₂×Q₈	—	8	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	—	4	2	no	—
460	C₂²×C₄	—	8	1	no	—
461	C₂²×C₄	—	8	1	no	—
462	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
467	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
468	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	8	no	—
477	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
478	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
479	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
480	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
481	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
482	C₂×C₈	#41×C₆₁	1	2	yes	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	4	2	no	—
498	C₄⋊C₄	—	4	2	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	—	4	2	no	—
501	C₄⋊C₄	—	4	2	no	—
502	C₄⋊C₄	—	4	2	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₄⋊C₄	—	4	2	no	—
505	C₄⋊C₄	—	8	1	no	—
506	C₄⋊C₄	—	8	1	no	—
507	C₄⋊C₄	—	8	1	no	—
508	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
509	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	4	2	no	—
513	C₂²⋊C₄	—	4	2	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	2	1	no	—
517	C₄²	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	—	2	1	no	—
522	C₂³	C₄⋊Q₈×C₆₁	1	2	yes	—
523	C₂³	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
524	C₂³	#137×C₆₁	1	2	yes	—
525	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
526	Q₈	—	8	4	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	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	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	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	—	8	4	no	—
570	C₂×C₄	—	8	4	no	—
571	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
572	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
573	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
574	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
575	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
576	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
577	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
578	C₈	—	2	2	no	—
579	C₈	—	2	2	no	—
580	C₈	—	2	2	no	—
581	C₈	C₂²⋊Q₈×C₆₁	1	4	yes	complemented
582	C₂²	—	2	2	no	—
583	C₂²	—	2	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	C₆₁×#501	1	4	yes	complemented
589	C₂²	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
590	C₂²	D₄²×C₆₁	1	1	yes	characteristic
591	C₂²	C₆₁×#547	1	1	yes	characteristic
592	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
593	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
594	C₂²	C₆₁×#576	1	1	yes	characteristic
595	C₂²	C₆₁×#578	1	1	yes	characteristic
596	C₄	—	2	2	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	2	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	1	no	—
603	C₄	—	8	1	no	—
604	C₄	—	8	1	no	—
605	C₄	—	8	1	no	—
606	C₄	—	8	1	no	—
607	C₄	—	8	1	no	—
608	C₄	—	8	1	no	—
609	C₄	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
610	C₄	C₆₁×#544	1	1	yes	characteristic
611	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
612	C₄	C₆₁×#554	1	1	yes	characteristic
613	C₂	—	2	4	no	—
614	C₂	C₆₁×#2440	1	1	yes	characteristic
615	C₂	C₂×C₆₁×#501	1	1	yes	characteristic
616	C₂	C₆₁×#3164	1	1	yes	characteristic
617	C₂	C₆₁×#3264	1	1	yes	characteristic
618	C₂	C₆₁×#3270	1	1	yes	characteristic
619	C₂	C₆₁×#3343	1	1	yes	characteristic
620	C₂	C₆₁×#3346	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 ◁ C₄×C₂²⋊C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#501	C₂	semidirect
5. C₆₁×#2085	C₂	semidirect
6. C₆₁×#2083	C₂	semidirect
7. C₆₁×#2059	C₂	semidirect
8. C₆₁×#2057	C₂	semidirect
9. C₆₁×#1944	C₂	semidirect
10. C₆₁×#1943	C₂	semidirect
13. C₆₁×#1847	C₂	semidirect
67. C₂²×C₈×C₆₁	Q₈	semidirect
116. C₂²⋊C₈×C₆₁	Q₈	semidirect
270. C₈×C₆₁	C₂²⋊Q₈	semidirect
271. #26071	C₆₁	direct (N × H)
278. C₂²×C₆₁	#501	semidirect
303. C₂×#501	C₂×C₆₁	semidirect
307. #2085	C₂×C₆₁	semidirect
308. #2083	C₂×C₆₁	semidirect
309. #2059	C₂×C₆₁	semidirect
310. #2057	C₂×C₆₁	semidirect
311. #1944	C₂×C₆₁	semidirect
312. #1943	C₂×C₆₁	semidirect
315. #1847	C₂×C₆₁	semidirect
371. C₆₁	#26071	direct (N × H)
378. C₂²×C₈	Q₈×C₆₁	semidirect
427. C₂²⋊C₈	Q₈×C₆₁	semidirect
581. C₈	C₂²⋊Q₈×C₆₁	semidirect
588. C₂²	C₆₁×#501	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#501	2
C₂²×#107×C₆₁	2
C₆₁×#2442	2
C₆₁×#2440	2
C₆₁×#2085	2
C₆₁×#2083	2
C₆₁×#2059	2
C₆₁×#2057	2
C₆₁×#1944	2
C₆₁×#1943	2
C₆₁×#1853	2
C₆₁×#1852	2
C₆₁×#1847	2
C₆₁×#1807	2
C₆₁×#1752	2
#26071	61

Subgroups — C61×#26071 #29529504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#26071 #29529504