Subgroups — C₆₁×#60237 #29563670

Order 15616.

2168 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2981	C₂	1	4	yes	maximal, complemented
3	C₆₁×#2980	C₂	1	4	yes	maximal, complemented
4	C₆₁×#2974	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2973	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2935	C₂	1	4	yes	maximal, complemented
7	C₆₁×#2932	C₂	1	4	yes	maximal, complemented
8	C₆₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₆₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#2880	C₂	1	4	yes	maximal, complemented
11	C₂²×#41×C₆₁	C₂²	1	2	yes	—
12	C₂×#131×C₆₁	C₂²	1	8	yes	complemented
13	C₆₁×#532	C₄	1	2	yes	—
14	C₆₁×#532	C₄	1	2	yes	—
15	C₆₁×#532	C₄	1	2	yes	—
16	C₆₁×#532	C₄	1	2	yes	—
17	C₆₁×#532	C₂²	1	2	yes	—
18	C₆₁×#532	C₂²	1	2	yes	—
19	C₆₁×#532	C₂²	1	2	yes	—
20	C₆₁×#532	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
22	C₂×#127×C₆₁	C₄	1	1	yes	characteristic
23	C₂×#124×C₆₁	C₄	1	2	yes	—
24	C₂×#121×C₆₁	C₄	1	2	yes	—
25	C₂×#121×C₆₁	C₄	1	2	yes	—
26	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
28	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#436	C₂²	1	8	yes	complemented
30	C₆₁×#435	C₂²	1	16	yes	complemented
31	C₆₁×#435	C₂²	1	16	yes	complemented
32	C₆₁×#435	C₂²	1	16	yes	complemented
33	C₈×D₄×C₆₁	C₂²	1	8	yes	complemented
34	C₆₁×#432	C₂²	1	2	yes	—
35	C₆₁×#432	C₂²	1	2	yes	—
36	C₆₁×#432	C₂²	1	2	yes	—
37	C₆₁×#432	C₂²	1	2	yes	complemented
38	C₆₁×#432	C₂²	1	2	yes	complemented
39	C₆₁×#432	C₂²	1	2	yes	complemented
40	C₆₁×#431	C₂²	1	2	yes	—
41	C₆₁×#431	C₂²	1	2	yes	complemented
42	C₆₁×#423	C₂²	1	2	yes	complemented
43	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	complemented
44	C₆₁×#408	C₂²	1	8	yes	complemented
45	C₆₁×#408	C₂²	1	16	yes	complemented
46	C₆₁×#407	C₂²	1	2	yes	—
47	C₆₁×#407	C₂²	1	8	yes	—
48	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
49	C₂×C₂²⋊C₈×C₆₁	C₂²	1	8	yes	—
50	C₄×M₁₆×C₆₁	C₂²	1	2	yes	complemented
51	C₂×#97×C₆₁	C₂²	1	2	yes	complemented
52	C₂×#41×C₆₁	—	4	8	no	—
53	C₂×#41×C₆₁	C₂×C₄	1	4	yes	—
54	C₂×#41×C₆₁	C₂×C₄	1	4	yes	—
55	C₂×#41×C₆₁	C₂³	1	4	yes	—
56	C₂×#41×C₆₁	C₂³	1	4	yes	—
57	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
58	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
59	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
60	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
61	C₂²×D₄×C₆₁	C₂×C₄	1	4	yes	—
62	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
66	#131×C₆₁	—	4	16	no	—
67	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
68	C₂×M₁₆×C₆₁	C₂³	1	16	yes	—
69	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
70	C₂²×C₈×C₆₁	C₂³	1	16	yes	—
71	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
72	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
73	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
74	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
75	#127×C₆₁	C₂×C₄	1	2	yes	—
76	#127×C₆₁	C₂×C₄	1	2	yes	—
77	#127×C₆₁	C₂×C₄	1	2	yes	—
78	#127×C₆₁	C₂×C₄	1	2	yes	—
79	#124×C₆₁	C₂×C₄	1	4	yes	—
80	#124×C₆₁	C₂×C₄	1	4	yes	—
81	#124×C₆₁	C₂×C₄	1	4	yes	—
82	#124×C₆₁	C₂×C₄	1	4	yes	—
83	#121×C₆₁	C₂×C₄	1	4	yes	—
84	#121×C₆₁	C₂×C₄	1	4	yes	—
85	#121×C₆₁	C₂×C₄	1	4	yes	—
86	#121×C₆₁	C₂×C₄	1	4	yes	—
87	#121×C₆₁	C₂×C₄	1	4	yes	—
88	#121×C₆₁	C₂×C₄	1	4	yes	—
89	#121×C₆₁	C₂×C₄	1	4	yes	—
90	#121×C₆₁	C₂×C₄	1	4	yes	—
91	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
92	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
93	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
94	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
95	C₄×D₄×C₆₁	C₂³	1	4	yes	—
96	C₄×D₄×C₆₁	C₂³	1	4	yes	—
97	C₄×D₄×C₆₁	C₂³	1	4	yes	—
98	C₄×D₄×C₆₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
101	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
103	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
104	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
107	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
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	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
115	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
116	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
117	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
118	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
119	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
120	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
121	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
122	C₂²⋊C₈×C₆₁	C₂³	1	16	yes	—
123	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
124	C₂²⋊C₈×C₆₁	C₂³	1	16	yes	—
125	#97×C₆₁	C₂³	1	4	yes	—
126	#97×C₆₁	C₂³	1	4	yes	—
127	#97×C₆₁	C₂³	1	4	yes	—
128	C₄×C₈×C₆₁	C₂³	1	4	yes	—
129	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
130	C₂⁴×C₆₁	C₂²×C₄	1	4	yes	—
131	#41×C₆₁	—	2	16	no	—
132	#41×C₆₁	—	4	8	no	—
133	#41×C₆₁	—	4	8	no	—
134	#41×C₆₁	—	4	8	no	—
135	C₂×Q₈×C₆₁	—	4	4	no	—
136	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
137	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
138	C₂×D₄×C₆₁	—	4	4	no	—
139	C₂×D₄×C₆₁	—	4	4	no	—
140	C₂×D₄×C₆₁	—	4	4	no	—
141	C₂×D₄×C₆₁	—	4	8	no	—
142	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
143	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
144	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
145	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
146	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
147	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
148	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
149	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
150	C₂²×C₄×C₆₁	—	2	4	no	—
151	C₂²×C₄×C₆₁	—	4	2	no	—
152	C₂²×C₄×C₆₁	—	4	8	no	—
153	C₂²×C₄×C₆₁	—	4	2	no	—
154	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
171	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
175	M₁₆×C₆₁	—	2	8	no	—
176	M₁₆×C₆₁	—	4	16	no	—
177	C₂×C₈×C₆₁	—	2	8	no	—
178	C₂×C₈×C₆₁	—	4	16	no	—
179	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
180	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
181	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
182	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
183	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
184	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
185	C₂²⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
186	C₂²⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
187	C₂²⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
188	C₂²⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
189	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
190	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
191	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
192	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
193	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
194	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
195	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
197	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
198	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
199	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
200	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
201	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
202	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
203	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
204	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
205	C₂³×C₆₁	—	2	2	no	—
206	C₂³×C₆₁	—	2	2	no	—
207	C₂³×C₆₁	—	2	8	no	—
208	C₂³×C₆₁	—	4	1	no	—
209	C₂³×C₆₁	—	4	4	no	—
210	C₂³×C₆₁	—	4	4	no	—
211	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
212	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
213	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
214	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
215	Q₈×C₆₁	—	2	8	no	—
216	Q₈×C₆₁	—	4	4	no	—
217	D₄×C₆₁	—	2	16	no	—
218	D₄×C₆₁	—	2	8	no	—
219	D₄×C₆₁	—	2	8	no	—
220	D₄×C₆₁	—	4	4	no	—
221	D₄×C₆₁	—	4	4	no	—
222	D₄×C₆₁	—	4	4	no	—
223	D₄×C₆₁	—	4	4	no	—
224	D₄×C₆₁	—	4	8	no	—
225	C₂×C₄×C₆₁	—	2	2	no	—
226	C₂×C₄×C₆₁	—	2	2	no	—
227	C₂×C₄×C₆₁	—	2	2	no	—
228	C₂×C₄×C₆₁	—	2	2	no	—
229	C₂×C₄×C₆₁	—	2	2	no	—
230	C₂×C₄×C₆₁	—	2	2	no	—
231	C₂×C₄×C₆₁	—	2	2	no	—
232	C₂×C₄×C₆₁	—	2	2	no	—
233	C₂×C₄×C₆₁	—	2	16	no	—
234	C₂×C₄×C₆₁	—	2	2	no	—
235	C₂×C₄×C₆₁	—	2	2	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	4	no	—
244	C₂×C₄×C₆₁	—	4	8	no	—
245	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
246	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
249	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
250	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
252	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
253	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂³×C₄	1	8	yes	—
255	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₆₁	H(5,𝔽₂)	1	4	yes	—
258	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
259	C₈×C₆₁	—	2	8	no	—
260	C₈×C₆₁	—	2	8	no	—
261	#60237	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂²×C₆₁	—	2	1	no	—
263	C₂²×C₆₁	—	2	1	no	—
264	C₂²×C₆₁	—	2	1	no	—
265	C₂²×C₆₁	—	2	1	no	—
266	C₂²×C₆₁	—	2	1	no	—
267	C₂²×C₆₁	—	2	8	no	—
268	C₂²×C₆₁	—	2	1	no	—
269	C₂²×C₆₁	—	2	1	no	—
270	C₂²×C₆₁	—	2	8	no	—
271	C₂²×C₆₁	—	2	8	no	—
272	C₂²×C₆₁	—	4	4	no	—
273	C₂²×C₆₁	—	8	8	no	—
274	C₂²×C₆₁	#518	1	2	yes	—
275	C₂²×C₆₁	#568	1	2	yes	—
276	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
277	C₄×C₆₁	—	2	1	no	—
278	C₄×C₆₁	—	2	1	no	—
279	C₄×C₆₁	—	2	8	no	—
280	C₄×C₆₁	—	2	1	no	—
281	C₄×C₆₁	—	2	1	no	—
282	C₄×C₆₁	—	2	1	no	—
283	C₄×C₆₁	—	2	1	no	—
284	C₄×C₆₁	—	2	1	no	—
285	C₄×C₆₁	—	2	1	no	—
286	C₄×C₆₁	—	2	1	no	—
287	C₄×C₆₁	#518	1	2	yes	—
288	C₄×C₆₁	#568	1	2	yes	—
289	C₄×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
290	C₂×#532	C₂×C₆₁	1	1	yes	characteristic
291	#2981	C₂×C₆₁	1	4	yes	complemented
292	#2980	C₂×C₆₁	1	4	yes	complemented
293	#2974	C₂×C₆₁	1	4	yes	complemented
294	#2973	C₂×C₆₁	1	4	yes	complemented
295	#2935	C₂×C₆₁	1	4	yes	complemented
296	#2932	C₂×C₆₁	1	4	yes	complemented
297	#2920	C₂×C₆₁	1	1	yes	characteristic, complemented
298	C₂×#432	C₂×C₆₁	1	1	yes	characteristic, complemented
299	#2880	C₂×C₆₁	1	4	yes	complemented
300	C₂×C₆₁	—	2	2	no	—
301	C₂×C₆₁	—	2	2	no	—
302	C₂×C₆₁	—	4	1	no	—
303	C₂×C₆₁	—	4	8	no	—
304	C₂×C₆₁	C₂×#518	1	1	yes	characteristic
305	C₂×C₆₁	C₂×#568	1	1	yes	characteristic
306	C₂×C₆₁	C₂×#568	1	1	yes	characteristic
307	C₂²×#41	C₂²×C₆₁	1	2	yes	—
308	C₂×#131	C₂²×C₆₁	1	8	yes	complemented
309	#532	C₄×C₆₁	1	2	yes	—
310	#532	C₄×C₆₁	1	2	yes	—
311	#532	C₄×C₆₁	1	2	yes	—
312	#532	C₄×C₆₁	1	2	yes	—
313	#532	C₂²×C₆₁	1	2	yes	—
314	#532	C₂²×C₆₁	1	2	yes	—
315	#532	C₂²×C₆₁	1	2	yes	—
316	#532	C₂²×C₆₁	1	2	yes	—
317	C₂×C₄⋊Q₈	C₄×C₆₁	1	1	yes	characteristic
318	C₂×#127	C₄×C₆₁	1	1	yes	characteristic
319	C₂×#124	C₄×C₆₁	1	2	yes	—
320	C₂×#121	C₄×C₆₁	1	2	yes	—
321	C₂×#121	C₄×C₆₁	1	2	yes	—
322	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
323	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
324	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
325	#436	C₂²×C₆₁	1	8	yes	complemented
326	#435	C₂²×C₆₁	1	16	yes	complemented
327	#435	C₂²×C₆₁	1	16	yes	complemented
328	#435	C₂²×C₆₁	1	16	yes	complemented
329	C₈×D₄	C₂²×C₆₁	1	8	yes	complemented
330	#432	C₂²×C₆₁	1	2	yes	—
331	#432	C₂²×C₆₁	1	2	yes	—
332	#432	C₂²×C₆₁	1	2	yes	—
333	#432	C₂²×C₆₁	1	2	yes	complemented
334	#432	C₂²×C₆₁	1	2	yes	complemented
335	#432	C₂²×C₆₁	1	2	yes	complemented
336	#431	C₂²×C₆₁	1	2	yes	—
337	#431	C₂²×C₆₁	1	2	yes	complemented
338	#423	C₂²×C₆₁	1	2	yes	complemented
339	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	complemented
340	#408	C₂²×C₆₁	1	8	yes	complemented
341	#408	C₂²×C₆₁	1	16	yes	complemented
342	#407	C₂²×C₆₁	1	2	yes	—
343	#407	C₂²×C₆₁	1	8	yes	—
344	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
345	C₂×C₂²⋊C₈	C₂²×C₆₁	1	8	yes	—
346	C₄×M₁₆	C₂²×C₆₁	1	2	yes	complemented
347	C₂×#97	C₂²×C₆₁	1	2	yes	complemented
348	C₆₁	#60237	1	1	yes	characteristic, direct product factor, Sylow
349	C₂×#41	—	4	8	no	—
350	C₂×#41	C₂×C₄×C₆₁	1	4	yes	—
351	C₂×#41	C₂×C₄×C₆₁	1	4	yes	—
352	C₂×#41	C₂³×C₆₁	1	4	yes	—
353	C₂×#41	C₂³×C₆₁	1	4	yes	—
354	C₂²×Q₈	C₂×C₄×C₆₁	1	2	yes	—
355	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
356	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
357	C₂²×D₄	C₂×C₄×C₆₁	1	4	yes	—
358	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
359	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
360	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
362	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
363	#131	—	4	16	no	—
364	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
365	C₂×M₁₆	C₂³×C₆₁	1	16	yes	—
366	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
367	C₂²×C₈	C₂³×C₆₁	1	16	yes	—
368	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
369	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
370	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
371	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
372	#127	C₂×C₄×C₆₁	1	2	yes	—
373	#127	C₂×C₄×C₆₁	1	2	yes	—
374	#127	C₂×C₄×C₆₁	1	2	yes	—
375	#127	C₂×C₄×C₆₁	1	2	yes	—
376	#124	C₂×C₄×C₆₁	1	4	yes	—
377	#124	C₂×C₄×C₆₁	1	4	yes	—
378	#124	C₂×C₄×C₆₁	1	4	yes	—
379	#124	C₂×C₄×C₆₁	1	4	yes	—
380	#121	C₂×C₄×C₆₁	1	4	yes	—
381	#121	C₂×C₄×C₆₁	1	4	yes	—
382	#121	C₂×C₄×C₆₁	1	4	yes	—
383	#121	C₂×C₄×C₆₁	1	4	yes	—
384	#121	C₂×C₄×C₆₁	1	4	yes	—
385	#121	C₂×C₄×C₆₁	1	4	yes	—
386	#121	C₂×C₄×C₆₁	1	4	yes	—
387	#121	C₂×C₄×C₆₁	1	4	yes	—
388	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
389	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
390	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
391	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
392	C₄×D₄	C₂³×C₆₁	1	4	yes	—
393	C₄×D₄	C₂³×C₆₁	1	4	yes	—
394	C₄×D₄	C₂³×C₆₁	1	4	yes	—
395	C₄×D₄	C₂³×C₆₁	1	4	yes	—
396	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
397	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
400	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
401	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
402	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
403	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
404	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
405	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
411	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
412	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
413	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
414	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
415	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
416	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
417	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
418	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
419	C₂²⋊C₈	C₂³×C₆₁	1	16	yes	—
420	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
421	C₂²⋊C₈	C₂³×C₆₁	1	16	yes	—
422	#97	C₂³×C₆₁	1	4	yes	—
423	#97	C₂³×C₆₁	1	4	yes	—
424	#97	C₂³×C₆₁	1	4	yes	—
425	C₄×C₈	C₂³×C₆₁	1	4	yes	—
426	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
427	C₂⁴	C₂²×C₄×C₆₁	1	4	yes	—
428	#41	—	2	16	no	—
429	#41	—	4	8	no	—
430	#41	—	4	8	no	—
431	#41	—	4	8	no	—
432	C₂×Q₈	—	4	4	no	—
433	C₂×Q₈	C₂²×C₄×C₆₁	1	4	yes	—
434	C₂×Q₈	C₂²×C₄×C₆₁	1	4	yes	—
435	C₂×D₄	—	4	4	no	—
436	C₂×D₄	—	4	4	no	—
437	C₂×D₄	—	4	4	no	—
438	C₂×D₄	—	4	8	no	—
439	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
440	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
441	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
442	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
443	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
444	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
445	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
446	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	4	2	no	—
449	C₂²×C₄	—	4	8	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
454	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
458	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
459	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
460	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
461	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
462	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
463	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
464	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
465	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
466	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
467	C₂²×C₄	C₂²×C₄×C₆₁	1	8	yes	—
468	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
471	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
472	M₁₆	—	2	8	no	—
473	M₁₆	—	4	16	no	—
474	C₂×C₈	—	2	8	no	—
475	C₂×C₈	—	4	16	no	—
476	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
477	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
478	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
479	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
480	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
481	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
482	C₂²⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
483	C₂²⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
484	C₂²⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
485	C₂²⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
486	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
487	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
488	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
489	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
490	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
491	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
492	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
493	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
494	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
495	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
496	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
497	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
498	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
499	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
500	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
501	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
502	C₂³	—	2	2	no	—
503	C₂³	—	2	2	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	4	1	no	—
506	C₂³	—	4	4	no	—
507	C₂³	—	4	4	no	—
508	C₂³	C₂³×C₄×C₆₁	1	8	yes	—
509	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
510	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
511	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
512	Q₈	—	2	8	no	—
513	Q₈	—	4	4	no	—
514	D₄	—	2	16	no	—
515	D₄	—	2	8	no	—
516	D₄	—	2	8	no	—
517	D₄	—	4	4	no	—
518	D₄	—	4	4	no	—
519	D₄	—	4	4	no	—
520	D₄	—	4	4	no	—
521	D₄	—	4	8	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	16	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	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	4	no	—
541	C₂×C₄	—	4	8	no	—
542	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
543	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
544	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
545	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
546	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
547	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
548	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
549	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
550	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
551	C₂×C₄	C₂³×C₄×C₆₁	1	8	yes	—
552	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
553	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
554	C₂×C₄	H(5,𝔽₂)×C₆₁	1	4	yes	—
555	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
556	C₈	—	2	8	no	—
557	C₈	—	2	8	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	1	no	—
562	C₂²	—	2	1	no	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	8	no	—
565	C₂²	—	2	8	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	8	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	8	8	no	—
570	C₂²	C₆₁×#518	1	2	yes	—
571	C₂²	C₆₁×#568	1	2	yes	—
572	C₂²	C₂⁴×C₄×C₆₁	1	1	yes	characteristic
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	1	no	—
576	C₄	—	2	1	no	—
577	C₄	—	2	1	no	—
578	C₄	—	2	1	no	—
579	C₄	—	2	1	no	—
580	C₄	—	2	1	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	8	no	—
583	C₄	C₆₁×#518	1	2	yes	—
584	C₄	C₆₁×#568	1	2	yes	—
585	C₄	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
586	C₂	—	2	2	no	—
587	C₂	—	2	2	no	—
588	C₂	—	4	1	no	—
589	C₂	—	4	8	no	—
590	C₂	C₂×C₆₁×#518	1	1	yes	characteristic
591	C₂	C₂×C₆₁×#568	1	1	yes	characteristic
592	C₂	C₂×C₆₁×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
socle	C₂²×C₆₁	C₂⁴×C₄
prime radical	C₂×C₆₁×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2981	C₂	semidirect
3. C₆₁×#2980	C₂	semidirect
4. C₆₁×#2974	C₂	semidirect
5. C₆₁×#2973	C₂	semidirect
6. C₆₁×#2935	C₂	semidirect
7. C₆₁×#2932	C₂	semidirect
8. C₆₁×#2920	C₂	semidirect
9. C₂×C₆₁×#432	C₂	semidirect
10. C₆₁×#2880	C₂	semidirect
12. C₂×#131×C₆₁	C₂²	semidirect
29. C₆₁×#436	C₂²	semidirect
30. C₆₁×#435	C₂²	semidirect
31. C₆₁×#435	C₂²	semidirect
32. C₆₁×#435	C₂²	semidirect
33. C₈×D₄×C₆₁	C₂²	semidirect
37. C₆₁×#432	C₂²	semidirect
38. C₆₁×#432	C₂²	semidirect
39. C₆₁×#432	C₂²	semidirect
41. C₆₁×#431	C₂²	semidirect
42. C₆₁×#423	C₂²	semidirect
43. C₂×C₄⋊C₈×C₆₁	C₂²	semidirect
44. C₆₁×#408	C₂²	semidirect
45. C₆₁×#408	C₂²	semidirect
50. C₄×M₁₆×C₆₁	C₂²	semidirect
51. C₂×#97×C₆₁	C₂²	semidirect
261. #60237	C₆₁	direct (N × H)
291. #2981	C₂×C₆₁	semidirect
292. #2980	C₂×C₆₁	semidirect
293. #2974	C₂×C₆₁	semidirect
294. #2973	C₂×C₆₁	semidirect
295. #2935	C₂×C₆₁	semidirect
296. #2932	C₂×C₆₁	semidirect
297. #2920	C₂×C₆₁	semidirect
298. C₂×#432	C₂×C₆₁	semidirect
299. #2880	C₂×C₆₁	semidirect
308. C₂×#131	C₂²×C₆₁	semidirect
325. #436	C₂²×C₆₁	semidirect
326. #435	C₂²×C₆₁	semidirect
327. #435	C₂²×C₆₁	semidirect
328. #435	C₂²×C₆₁	semidirect
329. C₈×D₄	C₂²×C₆₁	semidirect
333. #432	C₂²×C₆₁	semidirect
334. #432	C₂²×C₆₁	semidirect
335. #432	C₂²×C₆₁	semidirect
337. #431	C₂²×C₆₁	semidirect
338. #423	C₂²×C₆₁	semidirect
339. C₂×C₄⋊C₈	C₂²×C₆₁	semidirect
340. #408	C₂²×C₆₁	semidirect
341. #408	C₂²×C₆₁	semidirect
346. C₄×M₁₆	C₂²×C₆₁	semidirect
347. C₂×#97	C₂²×C₆₁	semidirect
348. C₆₁	#60237	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×#532	2
C₆₁×#2981	2
C₆₁×#2980	2
C₆₁×#2974	2
C₆₁×#2973	2
C₆₁×#2935	2
C₆₁×#2932	2
C₆₁×#2920	2
C₂×C₆₁×#432	2
C₆₁×#2880	2
#60237	61

Subgroups — C61×#60237 #29563670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60237 #29563670