Subgroups — C₆₁×#26169 #29529602

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#464	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2448	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2093	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#2086	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2042	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#1934	C₂	1	1	yes	maximal, characteristic
10	C₆₁×#1916	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1880	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1880	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1878	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1863	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#1863	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#464	—	2	4	no	—
24	C₄×Q₁₆×C₆₁	—	2	4	no	—
25	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#399	—	2	1	no	—
38	C₆₁×#397	—	2	1	no	—
39	C₆₁×#391	C₂²	1	1	yes	characteristic
40	C₆₁×#390	C₂²	1	1	yes	characteristic
41	C₆₁×#390	C₂²	1	1	yes	characteristic
42	C₆₁×#389	—	2	1	no	—
43	C₆₁×#389	—	2	1	no	—
44	C₆₁×#389	C₂²	1	1	yes	characteristic
45	C₆₁×#389	C₂²	1	1	yes	characteristic
46	C₆₁×#389	C₂²	1	1	yes	characteristic
47	C₆₁×#388	—	2	1	no	—
48	C₆₁×#388	—	2	1	no	—
49	C₆₁×#384	—	2	1	no	—
50	C₆₁×#384	C₂²	1	1	yes	characteristic
51	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
52	C₄×C₄⋊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×#336	C₂²	1	1	yes	characteristic
58	C₆₁×#336	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	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆×C₆₁	—	2	2	no	—
65	C₂×Q₁₆×C₆₁	—	2	2	no	—
66	C₂×Q₁₆×C₆₁	D₄	1	4	yes	—
67	C₂×SD₁₆×C₆₁	—	2	2	no	—
68	C₂×SD₁₆×C₆₁	—	2	2	no	—
69	C₂×SD₁₆×C₆₁	—	2	2	no	—
70	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
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	#124×C₆₁	—	2	2	no	—
75	#124×C₆₁	—	4	1	no	—
76	C₄×Q₈×C₆₁	—	2	2	no	—
77	C₄×Q₈×C₆₁	—	2	2	no	—
78	C₄×Q₈×C₆₁	—	2	2	no	—
79	C₄×Q₈×C₆₁	—	2	2	no	—
80	C₄×Q₈×C₆₁	—	2	2	no	—
81	C₄×Q₈×C₆₁	—	4	1	no	—
82	C₄×Q₈×C₆₁	—	4	1	no	—
83	C₄×Q₈×C₆₁	—	4	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	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₁	D₄	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	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
106	C₂×C₄²×C₆₁	—	2	1	no	—
107	C₂×C₄²×C₆₁	—	2	1	no	—
108	C₂×C₄²×C₆₁	—	2	1	no	—
109	C₂×C₄²×C₆₁	—	2	1	no	—
110	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
113	#107×C₆₁	—	2	2	no	—
114	#107×C₆₁	—	4	1	no	—
115	C₄⋊C₈×C₆₁	—	2	2	no	—
116	#103×C₆₁	—	2	2	no	—
117	#103×C₆₁	—	2	2	no	—
118	#103×C₆₁	—	2	2	no	—
119	#103×C₆₁	—	2	2	no	—
120	#103×C₆₁	—	2	2	no	—
121	#102×C₆₁	—	2	2	no	—
122	#102×C₆₁	—	2	2	no	—
123	#102×C₆₁	—	4	1	no	—
124	C₄×C₈×C₆₁	—	2	2	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	#95×C₆₁	D₄	1	1	yes	characteristic
131	#95×C₆₁	D₄	1	1	yes	characteristic
132	C₂⁴×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	—	2	1	no	—
137	C₂×Q₈×C₆₁	—	2	1	no	—
138	C₂×Q₈×C₆₁	—	2	1	no	—
139	C₂×Q₈×C₆₁	—	2	1	no	—
140	C₂×Q₈×C₆₁	—	2	1	no	—
141	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
142	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₆₁	#41	1	2	yes	—
144	C₂×Q₈×C₆₁	#41	1	2	yes	—
145	C₂×D₄×C₆₁	—	2	1	no	—
146	C₂×D₄×C₆₁	—	2	1	no	—
147	C₂×D₄×C₆₁	—	2	1	no	—
148	C₂×D₄×C₆₁	—	2	1	no	—
149	C₂×D₄×C₆₁	—	2	1	no	—
150	C₂×D₄×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	—	2	1	no	—
155	C₂²×C₄×C₆₁	—	2	1	no	—
156	C₂²×C₄×C₆₁	—	2	1	no	—
157	C₂²×C₄×C₆₁	—	2	1	no	—
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×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
165	Q₁₆×C₆₁	—	2	8	no	—
166	SD₁₆×C₆₁	—	4	4	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	—	2	1	no	—
174	C₂×C₈×C₆₁	—	2	1	no	—
175	C₂×C₈×C₆₁	—	2	1	no	—
176	C₂×C₈×C₆₁	—	2	1	no	—
177	C₂×C₈×C₆₁	—	2	1	no	—
178	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
179	C₄⋊C₄×C₆₁	—	2	4	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	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	—	8	1	no	—
196	C₄⋊C₄×C₆₁	—	8	1	no	—
197	C₄⋊C₄×C₆₁	—	8	1	no	—
198	C₄⋊C₄×C₆₁	—	8	1	no	—
199	C₄⋊C₄×C₆₁	#41	1	2	yes	—
200	C₂²⋊C₄×C₆₁	—	4	2	no	—
201	C₂²⋊C₄×C₆₁	—	4	2	no	—
202	C₂²⋊C₄×C₆₁	—	4	2	no	—
203	C₂²⋊C₄×C₆₁	—	4	2	no	—
204	C₂²⋊C₄×C₆₁	—	8	1	no	—
205	C₂²⋊C₄×C₆₁	—	8	1	no	—
206	C₄²×C₆₁	—	2	4	no	—
207	C₄²×C₆₁	—	2	1	no	—
208	C₄²×C₆₁	—	2	1	no	—
209	C₄²×C₆₁	—	2	1	no	—
210	C₄²×C₆₁	—	2	1	no	—
211	C₄²×C₆₁	—	2	1	no	—
212	C₄²×C₆₁	—	2	1	no	—
213	C₄²×C₆₁	—	4	2	no	—
214	C₄²×C₆₁	—	4	2	no	—
215	C₄²×C₆₁	—	4	2	no	—
216	C₂³×C₆₁	—	4	1	no	—
217	C₂³×C₆₁	—	4	1	no	—
218	C₂³×C₆₁	—	4	1	no	—
219	C₂³×C₆₁	—	4	1	no	—
220	C₂³×C₆₁	—	4	1	no	—
221	C₂³×C₆₁	—	4	1	no	—
222	C₂³×C₆₁	—	4	1	no	—
223	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₁	—	2	2	no	—
225	Q₈×C₆₁	—	2	2	no	—
226	Q₈×C₆₁	—	2	2	no	—
227	Q₈×C₆₁	—	2	2	no	—
228	Q₈×C₆₁	—	2	2	no	—
229	Q₈×C₆₁	—	4	1	no	—
230	Q₈×C₆₁	—	4	1	no	—
231	Q₈×C₆₁	#123	1	4	yes	—
232	D₄×C₆₁	—	4	1	no	—
233	D₄×C₆₁	—	4	1	no	—
234	D₄×C₆₁	—	4	1	no	—
235	D₄×C₆₁	—	4	1	no	—
236	C₂×C₄×C₆₁	—	2	1	no	—
237	C₂×C₄×C₆₁	—	2	2	no	—
238	C₂×C₄×C₆₁	—	2	1	no	—
239	C₂×C₄×C₆₁	—	2	2	no	—
240	C₂×C₄×C₆₁	—	2	2	no	—
241	C₂×C₄×C₆₁	—	2	1	no	—
242	C₂×C₄×C₆₁	—	2	1	no	—
243	C₂×C₄×C₆₁	—	2	1	no	—
244	C₂×C₄×C₆₁	—	2	2	no	—
245	C₂×C₄×C₆₁	—	2	1	no	—
246	C₂×C₄×C₆₁	—	2	1	no	—
247	C₂×C₄×C₆₁	—	2	1	no	—
248	C₂×C₄×C₆₁	—	2	2	no	—
249	C₂×C₄×C₆₁	—	2	1	no	—
250	C₂×C₄×C₆₁	—	2	1	no	—
251	C₂×C₄×C₆₁	—	2	1	no	—
252	C₂×C₄×C₆₁	—	2	1	no	—
253	C₂×C₄×C₆₁	—	2	1	no	—
254	C₂×C₄×C₆₁	—	2	1	no	—
255	C₂×C₄×C₆₁	—	2	1	no	—
256	C₂×C₄×C₆₁	—	2	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	—	4	1	no	—
264	C₂×C₄×C₆₁	—	4	1	no	—
265	C₂×C₄×C₆₁	—	4	1	no	—
266	C₂×C₄×C₆₁	—	4	1	no	—
267	C₂×C₄×C₆₁	—	4	1	no	—
268	C₂×C₄×C₆₁	—	4	1	no	—
269	C₂×C₄×C₆₁	—	4	1	no	—
270	C₂×C₄×C₆₁	—	4	1	no	—
271	C₂×C₄×C₆₁	—	4	1	no	—
272	C₂×C₄×C₆₁	—	4	1	no	—
273	C₂×C₄×C₆₁	#135	1	2	yes	—
274	C₂×C₄×C₆₁	#137	1	2	yes	—
275	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
281	C₈×C₆₁	—	2	2	no	—
282	C₈×C₆₁	—	4	1	no	—
283	C₈×C₆₁	—	4	1	no	—
284	C₈×C₆₁	—	4	1	no	—
285	#26169	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
286	C₂²×C₆₁	—	8	1	no	—
287	C₂²×C₆₁	—	8	1	no	—
288	C₂²×C₆₁	—	8	1	no	—
289	C₂²×C₆₁	—	8	1	no	—
290	C₂²×C₆₁	—	8	1	no	—
291	C₂²×C₆₁	—	8	1	no	—
292	C₂²×C₆₁	—	8	1	no	—
293	C₂²×C₆₁	#525	1	1	yes	characteristic
294	C₂²×C₆₁	D₄²	1	1	yes	characteristic
295	C₂²×C₆₁	#548	1	1	yes	characteristic
296	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
297	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
298	C₂²×C₆₁	#576	1	1	yes	characteristic
299	C₂²×C₆₁	#578	1	1	yes	characteristic
300	C₄×C₆₁	—	2	2	no	—
301	C₄×C₆₁	—	2	4	no	—
302	C₄×C₆₁	—	2	2	no	—
303	C₄×C₆₁	—	4	1	no	—
304	C₄×C₆₁	—	4	2	no	—
305	C₄×C₆₁	—	4	2	no	—
306	C₄×C₆₁	—	4	1	no	—
307	C₄×C₆₁	—	4	1	no	—
308	C₄×C₆₁	—	4	1	no	—
309	C₄×C₆₁	—	8	1	no	—
310	C₄×C₆₁	—	8	1	no	—
311	C₄×C₆₁	—	8	1	no	—
312	C₄×C₆₁	—	8	1	no	—
313	C₄×C₆₁	C₂×#123	1	1	yes	characteristic
314	C₄×C₆₁	#539	1	1	yes	characteristic
315	C₄×C₆₁	#547	1	1	yes	characteristic
316	C₄×C₆₁	#553	1	1	yes	characteristic
317	C₂×#464	C₂×C₆₁	1	1	yes	characteristic
318	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
319	#2448	C₂×C₆₁	1	1	yes	characteristic
320	Q₈×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
321	#2093	C₂×C₆₁	1	1	yes	characteristic
322	#2086	C₂×C₆₁	1	1	yes	characteristic, complemented
323	#2072	C₂×C₆₁	1	1	yes	characteristic, complemented
324	#2042	C₂×C₆₁	1	1	yes	characteristic
325	#1934	C₂×C₆₁	1	1	yes	characteristic
326	#1916	C₂×C₆₁	1	1	yes	characteristic, complemented
327	#1880	C₂×C₆₁	1	1	yes	characteristic
328	#1880	C₂×C₆₁	1	1	yes	characteristic
329	#1878	C₂×C₆₁	1	1	yes	characteristic, complemented
330	#1863	C₂×C₆₁	1	1	yes	characteristic, complemented
331	#1863	C₂×C₆₁	1	1	yes	characteristic, complemented
332	C₂×C₆₁	—	16	1	no	—
333	C₂×C₆₁	#2443	1	1	yes	characteristic
334	C₂×C₆₁	#3108	1	1	yes	characteristic
335	C₂×C₆₁	#3124	1	1	yes	characteristic
336	C₂×C₆₁	#3270	1	1	yes	characteristic
337	C₂×C₆₁	#3287	1	1	yes	characteristic
338	C₂×C₆₁	#3364	1	1	yes	characteristic
339	C₂×C₆₁	#3375	1	1	yes	characteristic
340	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
341	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
342	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
343	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
345	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
346	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
347	#464	—	2	4	no	—
348	C₄×Q₁₆	—	2	4	no	—
349	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
350	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
351	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
352	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
353	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
354	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
355	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
356	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
357	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
358	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
359	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
360	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
361	#399	—	2	1	no	—
362	#397	—	2	1	no	—
363	#391	C₂²×C₆₁	1	1	yes	characteristic
364	#390	C₂²×C₆₁	1	1	yes	characteristic
365	#390	C₂²×C₆₁	1	1	yes	characteristic
366	#389	—	2	1	no	—
367	#389	—	2	1	no	—
368	#389	C₂²×C₆₁	1	1	yes	characteristic
369	#389	C₂²×C₆₁	1	1	yes	characteristic
370	#389	C₂²×C₆₁	1	1	yes	characteristic
371	#388	—	2	1	no	—
372	#388	—	2	1	no	—
373	#384	—	2	1	no	—
374	#384	C₂²×C₆₁	1	1	yes	characteristic
375	C₄×C₄⋊C₄	—	2	1	no	—
376	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
377	#340	C₂²×C₆₁	1	1	yes	characteristic
378	#340	C₂²×C₆₁	1	1	yes	characteristic
379	#340	C₂²×C₆₁	1	1	yes	characteristic
380	#340	C₂²×C₆₁	1	1	yes	characteristic
381	#336	C₂²×C₆₁	1	1	yes	characteristic
382	#336	C₂²×C₆₁	1	1	yes	characteristic
383	#336	C₂²×C₆₁	1	1	yes	characteristic
384	#336	C₂²×C₆₁	1	1	yes	characteristic
385	C₆₁	#26169	1	1	yes	characteristic, direct product factor, Sylow
386	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
387	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
388	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
389	C₂×Q₁₆	—	2	2	no	—
390	C₂×Q₁₆	—	2	2	no	—
391	C₂×Q₁₆	D₄×C₆₁	1	4	yes	—
392	C₂×SD₁₆	—	2	2	no	—
393	C₂×SD₁₆	—	2	2	no	—
394	C₂×SD₁₆	—	2	2	no	—
395	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
396	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
399	#124	—	2	2	no	—
400	#124	—	4	1	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×Q₈	—	2	2	no	—
405	C₄×Q₈	—	2	2	no	—
406	C₄×Q₈	—	4	1	no	—
407	C₄×Q₈	—	4	1	no	—
408	C₄×Q₈	—	4	1	no	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
419	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₄²	—	2	1	no	—
432	C₂×C₄²	—	2	1	no	—
433	C₂×C₄²	—	2	1	no	—
434	C₂×C₄²	—	2	1	no	—
435	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
438	#107	—	2	2	no	—
439	#107	—	4	1	no	—
440	C₄⋊C₈	—	2	2	no	—
441	#103	—	2	2	no	—
442	#103	—	2	2	no	—
443	#103	—	2	2	no	—
444	#103	—	2	2	no	—
445	#103	—	2	2	no	—
446	#102	—	2	2	no	—
447	#102	—	2	2	no	—
448	#102	—	4	1	no	—
449	C₄×C₈	—	2	2	no	—
450	#95	—	2	1	no	—
451	#95	—	2	1	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	D₄×C₆₁	1	1	yes	characteristic
456	#95	D₄×C₆₁	1	1	yes	characteristic
457	C₂⁴	—	2	1	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
467	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
468	C₂×Q₈	#41×C₆₁	1	2	yes	—
469	C₂×Q₈	#41×C₆₁	1	2	yes	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
490	Q₁₆	—	2	8	no	—
491	SD₁₆	—	4	4	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	4	2	no	—
517	C₄⋊C₄	—	4	2	no	—
518	C₄⋊C₄	—	4	2	no	—
519	C₄⋊C₄	—	4	2	no	—
520	C₄⋊C₄	—	8	1	no	—
521	C₄⋊C₄	—	8	1	no	—
522	C₄⋊C₄	—	8	1	no	—
523	C₄⋊C₄	—	8	1	no	—
524	C₄⋊C₄	#41×C₆₁	1	2	yes	—
525	C₂²⋊C₄	—	4	2	no	—
526	C₂²⋊C₄	—	4	2	no	—
527	C₂²⋊C₄	—	4	2	no	—
528	C₂²⋊C₄	—	4	2	no	—
529	C₂²⋊C₄	—	8	1	no	—
530	C₂²⋊C₄	—	8	1	no	—
531	C₄²	—	2	4	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	2	1	no	—
534	C₄²	—	2	1	no	—
535	C₄²	—	2	1	no	—
536	C₄²	—	2	1	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	4	2	no	—
539	C₄²	—	4	2	no	—
540	C₄²	—	4	2	no	—
541	C₂³	—	4	1	no	—
542	C₂³	—	4	1	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	1	no	—
545	C₂³	—	4	1	no	—
546	C₂³	—	4	1	no	—
547	C₂³	—	4	1	no	—
548	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
549	Q₈	—	2	2	no	—
550	Q₈	—	2	2	no	—
551	Q₈	—	2	2	no	—
552	Q₈	—	2	2	no	—
553	Q₈	—	2	2	no	—
554	Q₈	—	4	1	no	—
555	Q₈	—	4	1	no	—
556	Q₈	#123×C₆₁	1	4	yes	—
557	D₄	—	4	1	no	—
558	D₄	—	4	1	no	—
559	D₄	—	4	1	no	—
560	D₄	—	4	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	#135×C₆₁	1	2	yes	—
599	C₂×C₄	#137×C₆₁	1	2	yes	—
600	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
601	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
602	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
603	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
604	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
605	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
606	C₈	—	2	2	no	—
607	C₈	—	4	1	no	—
608	C₈	—	4	1	no	—
609	C₈	—	4	1	no	—
610	C₂²	—	8	1	no	—
611	C₂²	—	8	1	no	—
612	C₂²	—	8	1	no	—
613	C₂²	—	8	1	no	—
614	C₂²	—	8	1	no	—
615	C₂²	—	8	1	no	—
616	C₂²	—	8	1	no	—
617	C₂²	C₆₁×#525	1	1	yes	characteristic
618	C₂²	D₄²×C₆₁	1	1	yes	characteristic
619	C₂²	C₆₁×#548	1	1	yes	characteristic
620	C₂²	C₂×#135×C₆₁	1	1	yes	characteristic
621	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
622	C₂²	C₆₁×#576	1	1	yes	characteristic
623	C₂²	C₆₁×#578	1	1	yes	characteristic
624	C₄	—	2	2	no	—
625	C₄	—	2	4	no	—
626	C₄	—	2	2	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	1	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	2	no	—
632	C₄	—	4	1	no	—
633	C₄	—	8	1	no	—
634	C₄	—	8	1	no	—
635	C₄	—	8	1	no	—
636	C₄	—	8	1	no	—
637	C₄	C₂×#123×C₆₁	1	1	yes	characteristic
638	C₄	C₆₁×#539	1	1	yes	characteristic
639	C₄	C₆₁×#547	1	1	yes	characteristic
640	C₄	C₆₁×#553	1	1	yes	characteristic
641	C₂	—	16	1	no	—
642	C₂	C₆₁×#2443	1	1	yes	characteristic
643	C₂	C₆₁×#3108	1	1	yes	characteristic
644	C₂	C₆₁×#3124	1	1	yes	characteristic
645	C₂	C₆₁×#3270	1	1	yes	characteristic
646	C₂	C₆₁×#3287	1	1	yes	characteristic
647	C₂	C₆₁×#3364	1	1	yes	characteristic
648	C₂	C₆₁×#3375	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₆₁	C₂	semidirect
4. Q₈×C₄⋊C₄×C₆₁	C₂	semidirect
6. C₆₁×#2086	C₂	semidirect
7. C₆₁×#2072	C₂	semidirect
10. C₆₁×#1916	C₂	semidirect
13. C₆₁×#1878	C₂	semidirect
14. C₆₁×#1863	C₂	semidirect
15. C₆₁×#1863	C₂	semidirect
285. #26169	C₆₁	direct (N × H)
318. C₂×C₄×Q₁₆	C₂×C₆₁	semidirect
320. Q₈×C₄⋊C₄	C₂×C₆₁	semidirect
322. #2086	C₂×C₆₁	semidirect
323. #2072	C₂×C₆₁	semidirect
326. #1916	C₂×C₆₁	semidirect
329. #1878	C₂×C₆₁	semidirect
330. #1863	C₂×C₆₁	semidirect
331. #1863	C₂×C₆₁	semidirect
385. C₆₁	#26169	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×#464	2
C₂×C₄×Q₁₆×C₆₁	2
C₆₁×#2448	2
Q₈×C₄⋊C₄×C₆₁	2
C₆₁×#2093	2
C₆₁×#2086	2
C₆₁×#2072	2
C₆₁×#2042	2
C₆₁×#1934	2
C₆₁×#1916	2
C₆₁×#1880	2
C₆₁×#1878	2
C₆₁×#1863	2
#26169	61

Subgroups — C61×#26169 #29529602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#26169 #29529602