Subgroups — C₆₁×#24347 #29527780

Order 15616.

1140 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#478	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#442	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2538	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2058	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2025	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2023	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1879	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#1879	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#1864	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1862	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#554	—	2	4	no	—
18	C₆₁×#554	—	2	4	no	—
19	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
23	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
24	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
25	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
26	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₆₁×#478	—	2	4	no	—
33	C₆₁×#442	—	2	4	no	—
34	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
42	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
44	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
45	C₆₁×#396	—	2	1	no	—
46	C₆₁×#396	—	2	1	no	—
47	C₆₁×#390	C₂²	1	1	yes	characteristic
48	C₆₁×#390	C₂²	1	1	yes	characteristic
49	C₆₁×#389	C₂²	1	1	yes	characteristic
50	C₆₁×#389	C₂²	1	1	yes	characteristic
51	C₆₁×#384	C₂²	1	1	yes	characteristic
52	C₆₁×#380	—	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×#336	C₂²	1	1	yes	characteristic
60	C₆₁×#336	C₂²	1	1	yes	characteristic
61	C₆₁×#336	C₂²	1	1	yes	characteristic
62	C₆₁×#336	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₆₁	—	2	1	no	—
64	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
65	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₁	—	2	1	no	—
68	C₂³×C₄×C₆₁	—	2	1	no	—
69	C₂³×C₄×C₆₁	—	2	1	no	—
70	C₂³×C₄×C₆₁	—	2	1	no	—
71	C₂×D₈×C₆₁	—	2	2	no	—
72	C₂×D₈×C₆₁	—	2	2	no	—
73	C₂×D₈×C₆₁	Q₈	1	4	yes	complemented
74	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₆₁	—	2	2	no	—
79	C₄⋊Q₈×C₆₁	—	2	2	no	—
80	#125×C₆₁	—	2	2	no	—
81	#125×C₆₁	—	2	2	no	—
82	#125×C₆₁	—	2	2	no	—
83	#125×C₆₁	D₄	1	4	yes	—
84	C₂²⋊Q₈×C₆₁	—	2	8	no	—
85	C₂²⋊Q₈×C₆₁	—	2	8	no	—
86	C₂²⋊Q₈×C₆₁	—	2	8	no	—
87	C₂²⋊Q₈×C₆₁	—	4	4	no	—
88	C₄×Q₈×C₆₁	—	2	2	no	—
89	C₄×Q₈×C₆₁	—	2	2	no	—
90	C₄×D₄×C₆₁	—	2	2	no	—
91	C₄×D₄×C₆₁	—	2	2	no	—
92	C₄×D₄×C₆₁	—	2	2	no	—
93	C₄×D₄×C₆₁	—	2	2	no	—
94	C₄×D₄×C₆₁	—	2	2	no	—
95	C₄×D₄×C₆₁	—	2	2	no	—
96	C₄×D₄×C₆₁	—	2	2	no	—
97	C₄×D₄×C₆₁	D₄	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
99	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
105	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
107	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
117	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
125	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
128	#107×C₆₁	—	2	2	no	—
129	#106×C₆₁	—	2	2	no	—
130	#106×C₆₁	—	2	2	no	—
131	C₄⋊C₈×C₆₁	—	2	2	no	—
132	#102×C₆₁	—	2	2	no	—
133	#102×C₆₁	—	2	2	no	—
134	#102×C₆₁	—	2	2	no	—
135	#102×C₆₁	—	2	2	no	—
136	#102×C₆₁	—	2	2	no	—
137	#102×C₆₁	—	2	2	no	—
138	#97×C₆₁	—	2	2	no	—
139	#95×C₆₁	—	2	1	no	—
140	#95×C₆₁	—	2	1	no	—
141	C₂⁴×C₆₁	—	2	1	no	—
142	C₂⁴×C₆₁	—	2	1	no	—
143	C₂×Q₈×C₆₁	—	2	4	no	—
144	C₂×Q₈×C₆₁	—	2	2	no	—
145	C₂×Q₈×C₆₁	—	2	2	no	—
146	C₂×Q₈×C₆₁	—	4	1	no	—
147	C₂×Q₈×C₆₁	—	8	1	no	—
148	C₂×Q₈×C₆₁	—	8	1	no	—
149	C₂×D₄×C₆₁	—	2	1	no	—
150	C₂×D₄×C₆₁	—	2	1	no	—
151	C₂×D₄×C₆₁	—	2	1	no	—
152	C₂×D₄×C₆₁	—	2	1	no	—
153	C₂×D₄×C₆₁	—	2	1	no	—
154	C₂×D₄×C₆₁	—	2	1	no	—
155	C₂×D₄×C₆₁	—	2	1	no	—
156	C₂×D₄×C₆₁	—	2	1	no	—
157	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
158	C₂×D₄×C₆₁	C₂×Q₈	1	2	yes	—
159	C₂×D₄×C₆₁	C₂×Q₈	1	2	yes	—
160	C₂×D₄×C₆₁	#41	1	2	yes	—
161	C₂²×C₄×C₆₁	—	2	4	no	—
162	C₂²×C₄×C₆₁	—	2	1	no	—
163	C₂²×C₄×C₆₁	—	2	1	no	—
164	C₂²×C₄×C₆₁	—	2	1	no	—
165	C₂²×C₄×C₆₁	—	2	4	no	—
166	C₂²×C₄×C₆₁	—	2	4	no	—
167	C₂²×C₄×C₆₁	—	2	1	no	—
168	C₂²×C₄×C₆₁	—	2	1	no	—
169	C₂²×C₄×C₆₁	—	2	1	no	—
170	C₂²×C₄×C₆₁	—	4	2	no	—
171	C₂²×C₄×C₆₁	—	4	2	no	—
172	C₂²×C₄×C₆₁	—	4	2	no	—
173	C₂²×C₄×C₆₁	—	4	2	no	—
174	C₂²×C₄×C₆₁	—	4	2	no	—
175	C₂²×C₄×C₆₁	—	8	1	no	—
176	C₂²×C₄×C₆₁	—	8	1	no	—
177	C₂²×C₄×C₆₁	—	8	1	no	—
178	C₂²×C₄×C₆₁	—	8	1	no	—
179	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
186	D₈×C₆₁	—	2	8	no	—
187	C₂×C₈×C₆₁	—	2	1	no	—
188	C₂×C₈×C₆₁	—	2	1	no	—
189	C₂×C₈×C₆₁	—	2	1	no	—
190	C₂×C₈×C₆₁	—	2	1	no	—
191	C₂×C₈×C₆₁	—	2	1	no	—
192	C₂×C₈×C₆₁	—	2	1	no	—
193	C₂×C₈×C₆₁	—	2	1	no	—
194	C₂×C₈×C₆₁	—	2	1	no	—
195	C₂×C₈×C₆₁	—	2	1	no	—
196	C₂×C₈×C₆₁	—	2	1	no	—
197	C₂×C₈×C₆₁	—	2	1	no	—
198	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
199	C₄⋊C₄×C₆₁	—	2	4	no	—
200	C₄⋊C₄×C₆₁	—	2	4	no	—
201	C₄⋊C₄×C₆₁	—	2	4	no	—
202	C₄⋊C₄×C₆₁	—	2	2	no	—
203	C₄⋊C₄×C₆₁	—	2	4	no	—
204	C₄⋊C₄×C₆₁	—	2	4	no	—
205	C₄⋊C₄×C₆₁	—	2	2	no	—
206	C₄⋊C₄×C₆₁	—	2	2	no	—
207	C₄⋊C₄×C₆₁	—	2	1	no	—
208	C₄⋊C₄×C₆₁	—	2	1	no	—
209	C₄⋊C₄×C₆₁	—	2	1	no	—
210	C₄⋊C₄×C₆₁	—	2	1	no	—
211	C₄⋊C₄×C₆₁	—	2	1	no	—
212	C₄⋊C₄×C₆₁	—	2	1	no	—
213	C₄⋊C₄×C₆₁	—	2	1	no	—
214	C₄⋊C₄×C₆₁	—	2	1	no	—
215	C₄⋊C₄×C₆₁	—	2	1	no	—
216	C₄⋊C₄×C₆₁	—	4	2	no	—
217	C₄⋊C₄×C₆₁	—	4	2	no	—
218	C₄⋊C₄×C₆₁	—	4	2	no	—
219	C₄⋊C₄×C₆₁	—	8	1	no	—
220	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
221	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
222	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
223	C₂²⋊C₄×C₆₁	—	2	4	no	—
224	C₂²⋊C₄×C₆₁	—	2	4	no	—
225	C₂²⋊C₄×C₆₁	—	2	4	no	—
226	C₂²⋊C₄×C₆₁	—	4	2	no	—
227	C₂²⋊C₄×C₆₁	—	4	2	no	—
228	C₂²⋊C₄×C₆₁	—	4	2	no	—
229	C₂²⋊C₄×C₆₁	—	8	1	no	—
230	C₂²⋊C₄×C₆₁	—	8	1	no	—
231	C₄²×C₆₁	—	2	1	no	—
232	C₄²×C₆₁	—	2	1	no	—
233	C₄²×C₆₁	—	2	1	no	—
234	C₄²×C₆₁	—	2	1	no	—
235	C₄²×C₆₁	C₂×D₄	1	2	yes	—
236	C₄²×C₆₁	C₂×D₄	1	2	yes	—
237	C₂³×C₆₁	—	2	2	no	—
238	C₂³×C₆₁	—	2	2	no	—
239	C₂³×C₆₁	—	2	2	no	—
240	C₂³×C₆₁	—	2	2	no	—
241	C₂³×C₆₁	—	4	1	no	—
242	C₂³×C₆₁	—	4	1	no	—
243	C₂³×C₆₁	—	4	1	no	—
244	C₂³×C₆₁	—	4	1	no	—
245	C₂³×C₆₁	—	4	1	no	—
246	C₂³×C₆₁	—	4	1	no	—
247	C₂³×C₆₁	—	4	1	no	—
248	C₂³×C₆₁	—	4	1	no	—
249	C₂³×C₆₁	—	4	1	no	—
250	C₂³×C₆₁	—	4	1	no	—
251	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
252	Q₈×C₆₁	—	4	4	no	—
253	Q₈×C₆₁	—	8	4	no	—
254	D₄×C₆₁	—	2	2	no	—
255	D₄×C₆₁	—	2	2	no	—
256	D₄×C₆₁	—	2	2	no	—
257	D₄×C₆₁	—	2	2	no	—
258	D₄×C₆₁	—	2	2	no	—
259	D₄×C₆₁	—	4	1	no	—
260	D₄×C₆₁	—	4	1	no	—
261	D₄×C₆₁	C₂²⋊Q₈	1	4	yes	complemented
262	C₂×C₄×C₆₁	—	2	1	no	—
263	C₂×C₄×C₆₁	—	2	1	no	—
264	C₂×C₄×C₆₁	—	2	1	no	—
265	C₂×C₄×C₆₁	—	2	1	no	—
266	C₂×C₄×C₆₁	—	2	1	no	—
267	C₂×C₄×C₆₁	—	2	1	no	—
268	C₂×C₄×C₆₁	—	2	1	no	—
269	C₂×C₄×C₆₁	—	2	1	no	—
270	C₂×C₄×C₆₁	—	2	1	no	—
271	C₂×C₄×C₆₁	—	2	2	no	—
272	C₂×C₄×C₆₁	—	2	2	no	—
273	C₂×C₄×C₆₁	—	2	2	no	—
274	C₂×C₄×C₆₁	—	2	1	no	—
275	C₂×C₄×C₆₁	—	2	1	no	—
276	C₂×C₄×C₆₁	—	2	1	no	—
277	C₂×C₄×C₆₁	—	2	1	no	—
278	C₂×C₄×C₆₁	—	2	1	no	—
279	C₂×C₄×C₆₁	—	2	1	no	—
280	C₂×C₄×C₆₁	—	4	8	no	—
281	C₂×C₄×C₆₁	—	4	1	no	—
282	C₂×C₄×C₆₁	—	4	1	no	—
283	C₂×C₄×C₆₁	—	4	1	no	—
284	C₂×C₄×C₆₁	—	4	1	no	—
285	C₂×C₄×C₆₁	—	4	1	no	—
286	C₂×C₄×C₆₁	—	4	1	no	—
287	C₂×C₄×C₆₁	—	4	1	no	—
288	C₂×C₄×C₆₁	—	4	1	no	—
289	C₂×C₄×C₆₁	—	4	1	no	—
290	C₂×C₄×C₆₁	—	4	1	no	—
291	C₂×C₄×C₆₁	—	4	1	no	—
292	C₂×C₄×C₆₁	—	4	1	no	—
293	C₂×C₄×C₆₁	—	4	1	no	—
294	C₂×C₄×C₆₁	—	4	1	no	—
295	C₂×C₄×C₆₁	—	4	1	no	—
296	C₂×C₄×C₆₁	—	8	4	no	—
297	C₂×C₄×C₆₁	—	8	4	no	—
298	C₂×C₄×C₆₁	—	8	4	no	—
299	C₂×C₄×C₆₁	#120	1	2	yes	—
300	C₂×C₄×C₆₁	#120	1	2	yes	—
301	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
302	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
303	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
305	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
306	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
307	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
308	C₈×C₆₁	—	2	2	no	—
309	C₈×C₆₁	—	4	1	no	—
310	C₈×C₆₁	—	4	1	no	—
311	C₈×C₆₁	—	4	1	no	—
312	#24347	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
313	C₂²×C₆₁	—	2	4	no	—
314	C₂²×C₆₁	—	4	2	no	—
315	C₂²×C₆₁	—	4	2	no	—
316	C₂²×C₆₁	—	4	2	no	—
317	C₂²×C₆₁	—	4	2	no	—
318	C₂²×C₆₁	—	4	2	no	—
319	C₂²×C₆₁	—	4	2	no	—
320	C₂²×C₆₁	—	4	2	no	—
321	C₂²×C₆₁	—	4	2	no	—
322	C₂²×C₆₁	—	4	2	no	—
323	C₂²×C₆₁	—	8	1	no	—
324	C₂²×C₆₁	—	8	1	no	—
325	C₂²×C₆₁	—	8	1	no	—
326	C₂²×C₆₁	—	8	1	no	—
327	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
328	C₂²×C₆₁	#548	1	1	yes	characteristic
329	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
330	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
331	C₂²×C₆₁	#575	1	1	yes	characteristic
332	C₂²×C₆₁	#576	1	1	yes	characteristic
333	C₂²×C₆₁	#577	1	1	yes	characteristic
334	C₄×C₆₁	—	2	2	no	—
335	C₄×C₆₁	—	2	2	no	—
336	C₄×C₆₁	—	2	2	no	—
337	C₄×C₆₁	—	4	1	no	—
338	C₄×C₆₁	—	4	2	no	—
339	C₄×C₆₁	—	4	1	no	—
340	C₄×C₆₁	—	4	1	no	—
341	C₄×C₆₁	—	4	2	no	—
342	C₄×C₆₁	—	4	2	no	—
343	C₄×C₆₁	—	8	1	no	—
344	C₄×C₆₁	—	8	1	no	—
345	C₄×C₆₁	—	8	1	no	—
346	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
347	C₄×C₆₁	#536	1	1	yes	characteristic
348	C₄×C₆₁	#548	1	1	yes	characteristic
349	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
350	C₂×#554	C₂×C₆₁	1	1	yes	characteristic, complemented
351	C₂×#478	C₂×C₆₁	1	1	yes	characteristic, complemented
352	C₂×#442	C₂×C₆₁	1	1	yes	characteristic
353	#2538	C₂×C₆₁	1	1	yes	characteristic, complemented
354	#2059	C₂×C₆₁	1	1	yes	characteristic, complemented
355	#2058	C₂×C₆₁	1	1	yes	characteristic, complemented
356	#2025	C₂×C₆₁	1	1	yes	characteristic, complemented
357	#2023	C₂×C₆₁	1	1	yes	characteristic, complemented
358	#1992	C₂×C₆₁	1	1	yes	characteristic, complemented
359	#1985	C₂×C₆₁	1	1	yes	characteristic, complemented
360	#1879	C₂×C₆₁	1	1	yes	characteristic
361	#1879	C₂×C₆₁	1	1	yes	characteristic
362	#1864	C₂×C₆₁	1	1	yes	characteristic, complemented
363	#1862	C₂×C₆₁	1	1	yes	characteristic, complemented
364	#1802	C₂×C₆₁	1	1	yes	characteristic, complemented
365	C₂×C₆₁	—	4	4	no	—
366	C₂×C₆₁	—	8	2	no	—
367	C₂×C₆₁	#2409	1	1	yes	characteristic
368	C₂×C₆₁	#3006	1	1	yes	characteristic
369	C₂×C₆₁	#3012	1	1	yes	characteristic
370	C₂×C₆₁	#3366	1	1	yes	characteristic
371	C₂×C₆₁	#3376	1	1	yes	characteristic
372	C₂×C₆₁	#3384	1	1	yes	characteristic
373	C₂×C₆₁	#3389	1	1	yes	characteristic
374	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
375	#554	—	2	4	no	—
376	#554	—	2	4	no	—
377	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
378	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
379	C₂×#125	C₂²×C₆₁	1	1	yes	characteristic
380	C₂×C₂²⋊Q₈	—	2	1	no	—
381	C₂×C₂²⋊Q₈	—	2	1	no	—
382	C₂×C₂²⋊Q₈	—	2	1	no	—
383	C₂×C₂²⋊Q₈	—	2	1	no	—
384	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
385	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
386	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
387	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
388	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
389	C₂²×C₄⋊C₄	—	2	1	no	—
390	#478	—	2	4	no	—
391	#442	—	2	4	no	—
392	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
393	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
394	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
395	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
396	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
397	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
398	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
399	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
400	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
401	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
402	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
403	#396	—	2	1	no	—
404	#396	—	2	1	no	—
405	#390	C₂²×C₆₁	1	1	yes	characteristic
406	#390	C₂²×C₆₁	1	1	yes	characteristic
407	#389	C₂²×C₆₁	1	1	yes	characteristic
408	#389	C₂²×C₆₁	1	1	yes	characteristic
409	#384	C₂²×C₆₁	1	1	yes	characteristic
410	#380	—	2	1	no	—
411	#340	C₂²×C₆₁	1	1	yes	characteristic
412	#340	C₂²×C₆₁	1	1	yes	characteristic
413	#340	C₂²×C₆₁	1	1	yes	characteristic
414	#340	C₂²×C₆₁	1	1	yes	characteristic
415	#340	C₂²×C₆₁	1	1	yes	characteristic
416	#340	C₂²×C₆₁	1	1	yes	characteristic
417	#336	C₂²×C₆₁	1	1	yes	characteristic
418	#336	C₂²×C₆₁	1	1	yes	characteristic
419	#336	C₂²×C₆₁	1	1	yes	characteristic
420	#336	C₂²×C₆₁	1	1	yes	characteristic
421	C₆₁	#24347	1	1	yes	characteristic, direct product factor, Sylow
422	C₂²×Q₈	—	2	1	no	—
423	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
424	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
425	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
426	C₂³×C₄	—	2	1	no	—
427	C₂³×C₄	—	2	1	no	—
428	C₂³×C₄	—	2	1	no	—
429	C₂³×C₄	—	2	1	no	—
430	C₂×D₈	—	2	2	no	—
431	C₂×D₈	—	2	2	no	—
432	C₂×D₈	Q₈×C₆₁	1	4	yes	complemented
433	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
434	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
435	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
436	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
437	C₄⋊Q₈	—	2	2	no	—
438	C₄⋊Q₈	—	2	2	no	—
439	#125	—	2	2	no	—
440	#125	—	2	2	no	—
441	#125	—	2	2	no	—
442	#125	D₄×C₆₁	1	4	yes	—
443	C₂²⋊Q₈	—	2	8	no	—
444	C₂²⋊Q₈	—	2	8	no	—
445	C₂²⋊Q₈	—	2	8	no	—
446	C₂²⋊Q₈	—	4	4	no	—
447	C₄×Q₈	—	2	2	no	—
448	C₄×Q₈	—	2	2	no	—
449	C₄×D₄	—	2	2	no	—
450	C₄×D₄	—	2	2	no	—
451	C₄×D₄	—	2	2	no	—
452	C₄×D₄	—	2	2	no	—
453	C₄×D₄	—	2	2	no	—
454	C₄×D₄	—	2	2	no	—
455	C₄×D₄	—	2	2	no	—
456	C₄×D₄	D₄×C₆₁	1	4	yes	—
457	C₂×C₄⋊C₄	—	2	8	no	—
458	C₂×C₄⋊C₄	—	2	1	no	—
459	C₂×C₄⋊C₄	—	2	1	no	—
460	C₂×C₄⋊C₄	—	2	1	no	—
461	C₂×C₄⋊C₄	—	2	1	no	—
462	C₂×C₄⋊C₄	—	2	1	no	—
463	C₂×C₄⋊C₄	—	2	1	no	—
464	C₂×C₄⋊C₄	—	2	1	no	—
465	C₂×C₄⋊C₄	—	2	1	no	—
466	C₂×C₄⋊C₄	—	2	1	no	—
467	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
468	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
469	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
470	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
471	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
472	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
473	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
474	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
475	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
476	C₂×C₂²⋊C₄	—	2	1	no	—
477	C₂×C₂²⋊C₄	—	2	1	no	—
478	C₂×C₂²⋊C₄	—	2	1	no	—
479	C₂×C₂²⋊C₄	—	2	1	no	—
480	C₂×C₂²⋊C₄	—	2	1	no	—
481	C₂×C₂²⋊C₄	—	2	1	no	—
482	C₂×C₂²⋊C₄	—	2	1	no	—
483	C₂×C₂²⋊C₄	—	2	1	no	—
484	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
486	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
487	#107	—	2	2	no	—
488	#106	—	2	2	no	—
489	#106	—	2	2	no	—
490	C₄⋊C₈	—	2	2	no	—
491	#102	—	2	2	no	—
492	#102	—	2	2	no	—
493	#102	—	2	2	no	—
494	#102	—	2	2	no	—
495	#102	—	2	2	no	—
496	#102	—	2	2	no	—
497	#97	—	2	2	no	—
498	#95	—	2	1	no	—
499	#95	—	2	1	no	—
500	C₂⁴	—	2	1	no	—
501	C₂⁴	—	2	1	no	—
502	C₂×Q₈	—	2	4	no	—
503	C₂×Q₈	—	2	2	no	—
504	C₂×Q₈	—	2	2	no	—
505	C₂×Q₈	—	4	1	no	—
506	C₂×Q₈	—	8	1	no	—
507	C₂×Q₈	—	8	1	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	2	1	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	1	no	—
516	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
517	C₂×D₄	C₂×Q₈×C₆₁	1	2	yes	—
518	C₂×D₄	C₂×Q₈×C₆₁	1	2	yes	—
519	C₂×D₄	#41×C₆₁	1	2	yes	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	—	2	4	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	4	2	no	—
530	C₂²×C₄	—	4	2	no	—
531	C₂²×C₄	—	4	2	no	—
532	C₂²×C₄	—	4	2	no	—
533	C₂²×C₄	—	4	2	no	—
534	C₂²×C₄	—	8	1	no	—
535	C₂²×C₄	—	8	1	no	—
536	C₂²×C₄	—	8	1	no	—
537	C₂²×C₄	—	8	1	no	—
538	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
539	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
540	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
541	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
542	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
543	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
544	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
545	D₈	—	2	8	no	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	C₂×Q₈×C₆₁	1	2	yes	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	4	no	—
563	C₄⋊C₄	—	2	4	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	4	2	no	—
576	C₄⋊C₄	—	4	2	no	—
577	C₄⋊C₄	—	4	2	no	—
578	C₄⋊C₄	—	8	1	no	—
579	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
580	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
581	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
582	C₂²⋊C₄	—	2	4	no	—
583	C₂²⋊C₄	—	2	4	no	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	4	2	no	—
586	C₂²⋊C₄	—	4	2	no	—
587	C₂²⋊C₄	—	4	2	no	—
588	C₂²⋊C₄	—	8	1	no	—
589	C₂²⋊C₄	—	8	1	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	—	2	1	no	—
593	C₄²	—	2	1	no	—
594	C₄²	C₂×D₄×C₆₁	1	2	yes	—
595	C₄²	C₂×D₄×C₆₁	1	2	yes	—
596	C₂³	—	2	2	no	—
597	C₂³	—	2	2	no	—
598	C₂³	—	2	2	no	—
599	C₂³	—	2	2	no	—
600	C₂³	—	4	1	no	—
601	C₂³	—	4	1	no	—
602	C₂³	—	4	1	no	—
603	C₂³	—	4	1	no	—
604	C₂³	—	4	1	no	—
605	C₂³	—	4	1	no	—
606	C₂³	—	4	1	no	—
607	C₂³	—	4	1	no	—
608	C₂³	—	4	1	no	—
609	C₂³	—	4	1	no	—
610	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
611	Q₈	—	4	4	no	—
612	Q₈	—	8	4	no	—
613	D₄	—	2	2	no	—
614	D₄	—	2	2	no	—
615	D₄	—	2	2	no	—
616	D₄	—	2	2	no	—
617	D₄	—	2	2	no	—
618	D₄	—	4	1	no	—
619	D₄	—	4	1	no	—
620	D₄	C₂²⋊Q₈×C₆₁	1	4	yes	complemented
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	4	8	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	8	4	no	—
656	C₂×C₄	—	8	4	no	—
657	C₂×C₄	—	8	4	no	—
658	C₂×C₄	#120×C₆₁	1	2	yes	—
659	C₂×C₄	#120×C₆₁	1	2	yes	—
660	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
661	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
662	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
663	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
664	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
665	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
666	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
667	C₈	—	2	2	no	—
668	C₈	—	4	1	no	—
669	C₈	—	4	1	no	—
670	C₈	—	4	1	no	—
671	C₂²	—	2	4	no	—
672	C₂²	—	4	2	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	4	2	no	—
677	C₂²	—	4	2	no	—
678	C₂²	—	4	2	no	—
679	C₂²	—	4	2	no	—
680	C₂²	—	4	2	no	—
681	C₂²	—	8	1	no	—
682	C₂²	—	8	1	no	—
683	C₂²	—	8	1	no	—
684	C₂²	—	8	1	no	—
685	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
686	C₂²	C₆₁×#548	1	1	yes	characteristic
687	C₂²	D₄×Q₈×C₆₁	1	1	yes	characteristic
688	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
689	C₂²	C₆₁×#575	1	1	yes	characteristic
690	C₂²	C₆₁×#576	1	1	yes	characteristic
691	C₂²	C₆₁×#577	1	1	yes	characteristic
692	C₄	—	2	2	no	—
693	C₄	—	2	2	no	—
694	C₄	—	2	2	no	—
695	C₄	—	4	1	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	8	1	no	—
702	C₄	—	8	1	no	—
703	C₄	—	8	1	no	—
704	C₄	C₂×C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
705	C₄	C₆₁×#536	1	1	yes	characteristic
706	C₄	C₆₁×#548	1	1	yes	characteristic
707	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
708	C₂	—	4	4	no	—
709	C₂	—	8	2	no	—
710	C₂	C₆₁×#2409	1	1	yes	characteristic
711	C₂	C₆₁×#3006	1	1	yes	characteristic
712	C₂	C₆₁×#3012	1	1	yes	characteristic
713	C₂	C₆₁×#3366	1	1	yes	characteristic
714	C₂	C₆₁×#3376	1	1	yes	characteristic
715	C₂	C₆₁×#3384	1	1	yes	characteristic
716	C₂	C₆₁×#3389	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂²×D₈×C₆₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#554	C₂	semidirect
2. C₂×C₆₁×#478	C₂	semidirect
4. C₆₁×#2538	C₂	semidirect
5. C₆₁×#2059	C₂	semidirect
6. C₆₁×#2058	C₂	semidirect
7. C₆₁×#2025	C₂	semidirect
8. C₆₁×#2023	C₂	semidirect
9. C₆₁×#1992	C₂	semidirect
10. C₆₁×#1985	C₂	semidirect
13. C₆₁×#1864	C₂	semidirect
14. C₆₁×#1862	C₂	semidirect
15. C₆₁×#1802	C₂	semidirect
73. C₂×D₈×C₆₁	Q₈	semidirect
261. D₄×C₆₁	C₂²⋊Q₈	semidirect
312. #24347	C₆₁	direct (N × H)
350. C₂×#554	C₂×C₆₁	semidirect
351. C₂×#478	C₂×C₆₁	semidirect
353. #2538	C₂×C₆₁	semidirect
354. #2059	C₂×C₆₁	semidirect
355. #2058	C₂×C₆₁	semidirect
356. #2025	C₂×C₆₁	semidirect
357. #2023	C₂×C₆₁	semidirect
358. #1992	C₂×C₆₁	semidirect
359. #1985	C₂×C₆₁	semidirect
362. #1864	C₂×C₆₁	semidirect
363. #1862	C₂×C₆₁	semidirect
364. #1802	C₂×C₆₁	semidirect
421. C₆₁	#24347	direct (N × H)
432. C₂×D₈	Q₈×C₆₁	semidirect
620. D₄	C₂²⋊Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#554	2
C₂×C₆₁×#478	2
C₂×C₆₁×#442	2
C₆₁×#2538	2
C₆₁×#2059	2
C₆₁×#2058	2
C₆₁×#2025	2
C₆₁×#2023	2
C₆₁×#1992	2
C₆₁×#1985	2
C₆₁×#1879	2
C₆₁×#1864	2
C₆₁×#1862	2
C₆₁×#1802	2
#24347	61

Subgroups — C61×#24347 #29527780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24347 #29527780