Subgroups — C₆₁×#22431 #29525864

Order 15616.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#547	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#417	C₂	1	2	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1894	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1872	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1788	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₁	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#547	—	2	4	no	—
15	C₆₁×#547	C₄	1	8	yes	complemented
16	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#123×C₆₁	—	2	2	no	—
18	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
19	C₂×#121×C₆₁	—	2	1	no	—
20	C₂×C₄×D₄×C₆₁	—	2	1	no	—
21	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
24	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
25	C₆₁×#436	—	2	4	no	—
26	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#417	—	2	8	no	—
28	C₂×#103×C₆₁	C₂²	1	2	yes	—
29	C₂×#103×C₆₁	C₂²	1	2	yes	—
30	C₂×#103×C₆₁	C₂²	1	2	yes	—
31	C₂×#103×C₆₁	C₂²	1	2	yes	—
32	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#102×C₆₁	C₂²	1	2	yes	—
34	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#389	—	2	1	no	—
38	C₆₁×#385	—	2	2	no	—
39	C₆₁×#384	C₂²	1	1	yes	characteristic
40	C₆₁×#380	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#340	C₂²	1	1	yes	characteristic
43	C₆₁×#340	C₂²	1	1	yes	characteristic
44	C₆₁×#340	C₂²	1	2	yes	—
45	C₂×#41×C₆₁	—	2	4	no	—
46	C₂×#41×C₆₁	—	2	4	no	—
47	C₂×#41×C₆₁	—	2	4	no	—
48	C₂×#41×C₆₁	—	4	2	no	—
49	C₂×#41×C₆₁	—	4	2	no	—
50	C₂×#41×C₆₁	C₂×C₄	1	8	yes	complemented
51	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
52	C₂²×D₄×C₆₁	—	2	2	no	—
53	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
54	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₁	—	2	2	no	—
56	C₂³×C₄×C₆₁	—	2	1	no	—
57	C₂³×C₄×C₆₁	—	2	2	no	—
58	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
59	C₂×M₁₆×C₆₁	—	2	4	no	—
60	C₂×M₁₆×C₆₁	—	2	4	no	—
61	C₂×M₁₆×C₆₁	—	4	2	no	—
62	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₁	—	2	2	no	—
66	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	complemented
67	#123×C₆₁	—	2	16	no	—
68	C₂²⋊Q₈×C₆₁	—	2	4	no	—
69	C₂²⋊Q₈×C₆₁	C₂×C₄	1	8	yes	complemented
70	#121×C₆₁	—	2	8	no	—
71	C₄×D₄×C₆₁	—	2	2	no	—
72	C₄×D₄×C₆₁	—	4	4	no	—
73	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
74	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	complemented
75	C₄×D₄×C₆₁	D₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
80	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
81	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
83	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
85	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
94	C₂×C₄²×C₆₁	—	2	1	no	—
95	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
96	C₄⋊C₈×C₆₁	—	2	2	no	—
97	#103×C₆₁	—	2	4	no	—
98	#103×C₆₁	—	2	4	no	—
99	#103×C₆₁	—	4	2	no	—
100	#103×C₆₁	—	4	2	no	—
101	#102×C₆₁	—	2	4	no	—
102	#102×C₆₁	—	2	2	no	—
103	#102×C₆₁	D₄	1	4	yes	complemented
104	C₂²⋊C₈×C₆₁	—	2	4	no	—
105	C₄×C₈×C₆₁	—	2	2	no	—
106	#95×C₆₁	—	2	1	no	—
107	#95×C₆₁	—	2	2	no	—
108	C₂⁴×C₆₁	—	2	1	no	—
109	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
110	#41×C₆₁	—	2	16	no	—
111	#41×C₆₁	—	4	8	no	—
112	#41×C₆₁	—	4	8	no	—
113	#41×C₆₁	—	4	8	no	—
114	C₂×Q₈×C₆₁	—	2	2	no	—
115	C₂×Q₈×C₆₁	—	2	2	no	—
116	C₂×Q₈×C₆₁	—	2	2	no	—
117	C₂×Q₈×C₆₁	—	2	2	no	—
118	C₂×Q₈×C₆₁	—	2	2	no	—
119	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
120	C₂×D₄×C₆₁	—	2	8	no	—
121	C₂×D₄×C₆₁	—	2	2	no	—
122	C₂×D₄×C₆₁	—	2	2	no	—
123	C₂×D₄×C₆₁	—	2	1	no	—
124	C₂×D₄×C₆₁	—	2	1	no	—
125	C₂×D₄×C₆₁	—	2	1	no	—
126	C₂×D₄×C₆₁	—	4	4	no	—
127	C₂×D₄×C₆₁	—	4	4	no	—
128	C₂×D₄×C₆₁	C₂²⋊C₄	1	4	yes	—
129	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
131	C₂×D₄×C₆₁	#41	1	2	yes	—
132	C₂²×C₄×C₆₁	—	2	2	no	—
133	C₂²×C₄×C₆₁	—	2	2	no	—
134	C₂²×C₄×C₆₁	—	2	2	no	—
135	C₂²×C₄×C₆₁	—	2	2	no	—
136	C₂²×C₄×C₆₁	—	2	1	no	—
137	C₂²×C₄×C₆₁	—	2	1	no	—
138	C₂²×C₄×C₆₁	—	2	2	no	—
139	C₂²×C₄×C₆₁	—	2	4	no	—
140	C₂²×C₄×C₆₁	—	2	8	no	—
141	C₂²×C₄×C₆₁	—	2	1	no	—
142	C₂²×C₄×C₆₁	—	2	2	no	—
143	C₂²×C₄×C₆₁	—	4	4	no	—
144	C₂²×C₄×C₆₁	—	4	4	no	—
145	C₂²×C₄×C₆₁	—	4	4	no	—
146	C₂²×C₄×C₆₁	—	4	2	no	—
147	C₂²×C₄×C₆₁	—	4	2	no	—
148	C₂²×C₄×C₆₁	—	4	4	no	—
149	C₂²×C₄×C₆₁	—	8	2	no	—
150	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
156	M₁₆×C₆₁	—	4	8	no	—
157	C₂×C₈×C₆₁	—	2	1	no	—
158	C₂×C₈×C₆₁	—	2	1	no	—
159	C₂×C₈×C₆₁	—	2	1	no	—
160	C₂×C₈×C₆₁	—	2	2	no	—
161	C₂×C₈×C₆₁	—	2	2	no	—
162	C₂×C₈×C₆₁	—	2	1	no	—
163	C₂×C₈×C₆₁	—	2	1	no	—
164	C₂×C₈×C₆₁	—	2	2	no	—
165	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
166	C₄⋊C₄×C₆₁	—	2	4	no	—
167	C₄⋊C₄×C₆₁	—	2	8	no	—
168	C₄⋊C₄×C₆₁	—	2	4	no	—
169	C₄⋊C₄×C₆₁	—	2	1	no	—
170	C₄⋊C₄×C₆₁	—	2	1	no	—
171	C₄⋊C₄×C₆₁	—	2	2	no	—
172	C₄⋊C₄×C₆₁	—	2	1	no	—
173	C₄⋊C₄×C₆₁	—	2	1	no	—
174	C₄⋊C₄×C₆₁	—	2	2	no	—
175	C₄⋊C₄×C₆₁	—	4	4	no	—
176	C₄⋊C₄×C₆₁	—	4	2	no	—
177	C₄⋊C₄×C₆₁	—	8	1	no	—
178	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
179	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
180	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
181	C₂²⋊C₄×C₆₁	—	2	8	no	—
182	C₂²⋊C₄×C₆₁	—	2	4	no	—
183	C₂²⋊C₄×C₆₁	—	4	4	no	—
184	C₂²⋊C₄×C₆₁	C₂²⋊C₄	1	8	yes	—
185	C₄²×C₆₁	—	4	2	no	—
186	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
187	C₄²×C₆₁	C₂×D₄	1	2	yes	—
188	C₂³×C₆₁	—	2	2	no	—
189	C₂³×C₆₁	—	2	2	no	—
190	C₂³×C₆₁	—	2	2	no	—
191	C₂³×C₆₁	—	2	2	no	—
192	C₂³×C₆₁	—	2	2	no	—
193	C₂³×C₆₁	—	2	2	no	—
194	C₂³×C₆₁	—	2	2	no	—
195	C₂³×C₆₁	—	2	2	no	—
196	C₂³×C₆₁	—	2	2	no	—
197	C₂³×C₆₁	—	4	1	no	—
198	C₂³×C₆₁	—	4	1	no	—
199	C₂³×C₆₁	—	4	1	no	—
200	C₂³×C₆₁	—	4	1	no	—
201	C₂³×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
202	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₆₁	—	2	4	no	—
204	Q₈×C₆₁	—	2	4	no	—
205	Q₈×C₆₁	—	4	2	no	—
206	Q₈×C₆₁	—	4	2	no	—
207	D₄×C₆₁	—	2	2	no	—
208	D₄×C₆₁	—	2	2	no	—
209	D₄×C₆₁	—	2	2	no	—
210	D₄×C₆₁	—	2	8	no	—
211	D₄×C₆₁	—	4	16	no	—
212	D₄×C₆₁	C₄×D₄	1	4	yes	complemented
213	C₂×C₄×C₆₁	—	2	2	no	—
214	C₂×C₄×C₆₁	—	2	2	no	—
215	C₂×C₄×C₆₁	—	2	1	no	—
216	C₂×C₄×C₆₁	—	2	1	no	—
217	C₂×C₄×C₆₁	—	2	4	no	—
218	C₂×C₄×C₆₁	—	2	4	no	—
219	C₂×C₄×C₆₁	—	2	4	no	—
220	C₂×C₄×C₆₁	—	2	4	no	—
221	C₂×C₄×C₆₁	—	2	2	no	—
222	C₂×C₄×C₆₁	—	2	4	no	—
223	C₂×C₄×C₆₁	—	2	2	no	—
224	C₂×C₄×C₆₁	—	4	16	no	—
225	C₂×C₄×C₆₁	—	4	2	no	—
226	C₂×C₄×C₆₁	—	4	2	no	—
227	C₂×C₄×C₆₁	—	4	1	no	—
228	C₂×C₄×C₆₁	—	4	1	no	—
229	C₂×C₄×C₆₁	—	4	1	no	—
230	C₂×C₄×C₆₁	—	4	1	no	—
231	C₂×C₄×C₆₁	—	4	8	no	—
232	C₂×C₄×C₆₁	—	4	2	no	—
233	C₂×C₄×C₆₁	—	4	1	no	—
234	C₂×C₄×C₆₁	—	4	1	no	—
235	C₂×C₄×C₆₁	—	4	2	no	—
236	C₂×C₄×C₆₁	—	4	2	no	—
237	C₂×C₄×C₆₁	—	4	1	no	—
238	C₂×C₄×C₆₁	—	8	8	no	—
239	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
240	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
241	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
242	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
243	C₂×C₄×C₆₁	#137	1	2	yes	—
244	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
245	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
249	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
250	C₈×C₆₁	—	2	2	no	—
251	C₈×C₆₁	—	4	2	no	—
252	C₈×C₆₁	—	4	1	no	—
253	#22431	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂²×C₆₁	—	2	4	no	—
255	C₂²×C₆₁	—	2	4	no	—
256	C₂²×C₆₁	—	2	4	no	—
257	C₂²×C₆₁	—	2	4	no	—
258	C₂²×C₆₁	—	4	2	no	—
259	C₂²×C₆₁	—	4	2	no	—
260	C₂²×C₆₁	—	4	2	no	—
261	C₂²×C₆₁	—	4	2	no	—
262	C₂²×C₆₁	—	4	2	no	—
263	C₂²×C₆₁	—	4	2	no	—
264	C₂²×C₆₁	—	4	2	no	—
265	C₂²×C₆₁	—	4	2	no	—
266	C₂²×C₆₁	—	4	2	no	—
267	C₂²×C₆₁	—	4	2	no	—
268	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
269	C₂²×C₆₁	D₄²	1	1	yes	characteristic
270	C₂²×C₆₁	#546	1	1	yes	characteristic
271	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
273	C₂²×C₆₁	#577	1	1	yes	characteristic
274	C₂²×C₆₁	#578	1	1	yes	characteristic
275	C₄×C₆₁	—	2	4	no	—
276	C₄×C₆₁	—	2	2	no	—
277	C₄×C₆₁	—	4	4	no	—
278	C₄×C₆₁	—	4	2	no	—
279	C₄×C₆₁	—	4	2	no	—
280	C₄×C₆₁	—	8	1	no	—
281	C₄×C₆₁	—	8	2	no	—
282	C₄×C₆₁	#417	1	4	yes	complemented
283	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
284	C₄×C₆₁	#518	1	1	yes	characteristic
285	C₄×C₆₁	D₄²	1	1	yes	characteristic
286	C₄×C₆₁	#546	1	1	yes	characteristic
287	C₂×#547	C₂×C₆₁	1	1	yes	characteristic
288	C₂×#436	C₂×C₆₁	1	1	yes	characteristic, complemented
289	C₂×#417	C₂×C₆₁	1	2	yes	complemented
290	D₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
291	#1981	C₂×C₆₁	1	1	yes	characteristic, complemented
292	#1956	C₂×C₆₁	1	1	yes	characteristic, complemented
293	#1894	C₂×C₆₁	1	2	yes	complemented
294	#1872	C₂×C₆₁	1	2	yes	complemented
295	#1802	C₂×C₆₁	1	1	yes	characteristic, complemented
296	#1788	C₂×C₆₁	1	2	yes	complemented
297	C₄×#102	C₂×C₆₁	1	1	yes	characteristic, complemented
298	C₂×C₆₁	—	4	4	no	—
299	C₂×C₆₁	—	4	4	no	—
300	C₂×C₆₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
301	C₂×C₆₁	C₂×#417	1	1	yes	characteristic
302	C₂×C₆₁	#2898	1	1	yes	characteristic
303	C₂×C₆₁	#3272	1	1	yes	characteristic
304	C₂×C₆₁	#3276	1	1	yes	characteristic
305	C₂×C₆₁	#3330	1	1	yes	characteristic
306	C₂×C₆₁	#3331	1	1	yes	characteristic
307	C₂²×#41	C₂²×C₆₁	1	2	yes	—
308	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
309	#547	—	2	4	no	—
310	#547	C₄×C₆₁	1	8	yes	complemented
311	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
312	C₂×#123	—	2	2	no	—
313	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
314	C₂×#121	—	2	1	no	—
315	C₂×C₄×D₄	—	2	1	no	—
316	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
317	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
318	C₂²×C₄⋊C₄	—	2	1	no	—
319	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
320	#436	—	2	4	no	—
321	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
322	#417	—	2	8	no	—
323	C₂×#103	C₂²×C₆₁	1	2	yes	—
324	C₂×#103	C₂²×C₆₁	1	2	yes	—
325	C₂×#103	C₂²×C₆₁	1	2	yes	—
326	C₂×#103	C₂²×C₆₁	1	2	yes	—
327	C₂×#102	C₂²×C₆₁	1	2	yes	—
328	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
329	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
330	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
331	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
332	#389	—	2	1	no	—
333	#385	—	2	2	no	—
334	#384	C₂²×C₆₁	1	1	yes	characteristic
335	#380	C₂²×C₆₁	1	2	yes	—
336	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
337	#340	C₂²×C₆₁	1	2	yes	—
338	#340	C₂²×C₆₁	1	1	yes	characteristic
339	#340	C₂²×C₆₁	1	1	yes	characteristic
340	C₆₁	#22431	1	1	yes	characteristic, direct product factor, Sylow
341	C₂×#41	—	2	4	no	—
342	C₂×#41	—	2	4	no	—
343	C₂×#41	—	2	4	no	—
344	C₂×#41	—	4	2	no	—
345	C₂×#41	—	4	2	no	—
346	C₂×#41	C₂×C₄×C₆₁	1	8	yes	complemented
347	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	D₄×C₆₁	1	1	yes	characteristic
350	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
351	C₂³×C₄	—	2	1	no	—
352	C₂³×C₄	—	2	2	no	—
353	C₂³×C₄	—	2	2	no	—
354	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
355	C₂×M₁₆	—	2	4	no	—
356	C₂×M₁₆	—	2	4	no	—
357	C₂×M₁₆	—	4	2	no	—
358	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
359	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
360	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
361	C₄⋊Q₈	—	2	2	no	—
362	C₄⋊Q₈	C₂×C₄×C₆₁	1	4	yes	complemented
363	#123	—	2	16	no	—
364	C₂²⋊Q₈	—	2	4	no	—
365	C₂²⋊Q₈	C₂×C₄×C₆₁	1	8	yes	complemented
366	#121	—	2	8	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	4	4	no	—
369	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
370	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	complemented
371	C₄×D₄	D₄×C₆₁	1	4	yes	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	2	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	8	no	—
376	C₂×C₄⋊C₄	—	2	4	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	4	2	no	—
381	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
392	C₄⋊C₈	—	2	2	no	—
393	#103	—	2	4	no	—
394	#103	—	2	4	no	—
395	#103	—	4	2	no	—
396	#103	—	4	2	no	—
397	#102	—	2	4	no	—
398	#102	—	2	2	no	—
399	#102	D₄×C₆₁	1	4	yes	complemented
400	C₂²⋊C₈	—	2	4	no	—
401	C₄×C₈	—	2	2	no	—
402	#95	—	2	1	no	—
403	#95	—	2	2	no	—
404	C₂⁴	—	2	1	no	—
405	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
406	#41	—	2	16	no	—
407	#41	—	4	8	no	—
408	#41	—	4	8	no	—
409	#41	—	4	8	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	C₂²×C₄×C₆₁	1	4	yes	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	C₂²⋊C₄×C₆₁	1	4	yes	—
425	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
426	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
427	C₂×D₄	#41×C₆₁	1	2	yes	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	4	4	no	—
440	C₂²×C₄	—	4	4	no	—
441	C₂²×C₄	—	4	4	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	4	no	—
445	C₂²×C₄	—	8	2	no	—
446	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
447	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
449	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
450	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
452	M₁₆	—	4	8	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	8	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	4	4	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	—	8	1	no	—
474	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
475	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
476	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
477	C₂²⋊C₄	—	2	8	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	4	4	no	—
480	C₂²⋊C₄	C₂²⋊C₄×C₆₁	1	8	yes	—
481	C₄²	—	4	2	no	—
482	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
483	C₄²	C₂×D₄×C₆₁	1	2	yes	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	1	no	—
495	C₂³	—	4	1	no	—
496	C₂³	—	4	1	no	—
497	C₂³	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
498	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
499	Q₈	—	2	4	no	—
500	Q₈	—	2	4	no	—
501	Q₈	—	4	2	no	—
502	Q₈	—	4	2	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	—	2	8	no	—
506	D₄	—	2	2	no	—
507	D₄	—	4	16	no	—
508	D₄	C₄×D₄×C₆₁	1	4	yes	complemented
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	4	16	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	8	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	8	8	no	—
535	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
536	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
537	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
538	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
539	C₂×C₄	#137×C₆₁	1	2	yes	—
540	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
543	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
544	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
545	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
546	C₈	—	2	2	no	—
547	C₈	—	4	2	no	—
548	C₈	—	4	1	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	2	4	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	2	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	2	no	—
562	C₂²	—	4	2	no	—
563	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
564	C₂²	D₄²×C₆₁	1	1	yes	characteristic
565	C₂²	C₆₁×#546	1	1	yes	characteristic
566	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
567	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
568	C₂²	C₆₁×#577	1	1	yes	characteristic
569	C₂²	C₆₁×#578	1	1	yes	characteristic
570	C₄	—	2	4	no	—
571	C₄	—	2	2	no	—
572	C₄	—	4	4	no	—
573	C₄	—	4	2	no	—
574	C₄	—	4	2	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	2	no	—
577	C₄	C₆₁×#417	1	4	yes	complemented
578	C₄	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
579	C₄	C₆₁×#518	1	1	yes	characteristic
580	C₄	D₄²×C₆₁	1	1	yes	characteristic
581	C₄	C₆₁×#546	1	1	yes	characteristic
582	C₂	—	4	4	no	—
583	C₂	—	4	4	no	—
584	C₂	D₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
585	C₂	C₂×C₆₁×#417	1	1	yes	characteristic
586	C₂	C₆₁×#2898	1	1	yes	characteristic
587	C₂	C₆₁×#3272	1	1	yes	characteristic
588	C₂	C₆₁×#3276	1	1	yes	characteristic
589	C₂	C₆₁×#3330	1	1	yes	characteristic
590	C₂	C₆₁×#3331	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×C₆₁×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄×D₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#436	C₂	semidirect
3. C₂×C₆₁×#417	C₂	semidirect
4. D₄×C₄⋊C₄×C₆₁	C₂	semidirect
5. C₆₁×#1981	C₂	semidirect
6. C₆₁×#1956	C₂	semidirect
7. C₆₁×#1894	C₂	semidirect
8. C₆₁×#1872	C₂	semidirect
9. C₆₁×#1802	C₂	semidirect
10. C₆₁×#1788	C₂	semidirect
11. C₄×#102×C₆₁	C₂	semidirect
15. C₆₁×#547	C₄	semidirect
50. C₂×#41×C₆₁	C₂×C₄	semidirect
66. C₄⋊Q₈×C₆₁	C₂×C₄	semidirect
69. C₂²⋊Q₈×C₆₁	C₂×C₄	semidirect
74. C₄×D₄×C₆₁	C₂×C₄	semidirect
103. #102×C₆₁	D₄	semidirect
212. D₄×C₆₁	C₄×D₄	semidirect
253. #22431	C₆₁	direct (N × H)
282. C₄×C₆₁	#417	semidirect
288. C₂×#436	C₂×C₆₁	semidirect
289. C₂×#417	C₂×C₆₁	semidirect
290. D₄×C₄⋊C₄	C₂×C₆₁	semidirect
291. #1981	C₂×C₆₁	semidirect
292. #1956	C₂×C₆₁	semidirect
293. #1894	C₂×C₆₁	semidirect
294. #1872	C₂×C₆₁	semidirect
295. #1802	C₂×C₆₁	semidirect
296. #1788	C₂×C₆₁	semidirect
297. C₄×#102	C₂×C₆₁	semidirect
310. #547	C₄×C₆₁	semidirect
340. C₆₁	#22431	direct (N × H)
346. C₂×#41	C₂×C₄×C₆₁	semidirect
362. C₄⋊Q₈	C₂×C₄×C₆₁	semidirect
365. C₂²⋊Q₈	C₂×C₄×C₆₁	semidirect
370. C₄×D₄	C₂×C₄×C₆₁	semidirect
399. #102	D₄×C₆₁	semidirect
508. D₄	C₄×D₄×C₆₁	semidirect
577. C₄	C₆₁×#417	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#547	2
C₂×C₆₁×#436	2
C₂×C₆₁×#417	2
D₄×C₄⋊C₄×C₆₁	2
C₆₁×#1981	2
C₆₁×#1956	2
C₆₁×#1894	2
C₆₁×#1872	2
C₆₁×#1802	2
C₆₁×#1788	2
C₄×#102×C₆₁	2
#22431	61

Subgroups — C61×#22431 #29525864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22431 #29525864