Subgroups — C₆₁×#60236 #29563669

Order 15616.

2212 conjugacy classes of subgroups.

The subgroup lattice has 612 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×#2979	C₂	1	4	yes	maximal, complemented
4	C₆₁×#2975	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2974	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2972	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2935	C₂	1	4	yes	maximal, complemented
8	C₆₁×#2931	C₂	1	4	yes	maximal, complemented
9	C₆₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2880	C₂	1	4	yes	maximal, complemented
12	C₂²×#41×C₆₁	C₂²	1	2	yes	—
13	C₂×#131×C₆₁	C₂²	1	8	yes	complemented
14	C₆₁×#532	C₄	1	2	yes	—
15	C₆₁×#532	C₄	1	2	yes	—
16	C₆₁×#532	C₄	1	2	yes	—
17	C₆₁×#532	C₄	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	C₆₁×#532	C₂²	1	2	yes	—
22	C₂×C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
23	C₂×#127×C₆₁	C₄	1	1	yes	characteristic
24	C₂×#124×C₆₁	C₄	1	2	yes	—
25	C₂×#121×C₆₁	C₄	1	2	yes	—
26	C₂×#121×C₆₁	C₄	1	2	yes	—
27	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
29	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#436	C₂²	1	8	yes	complemented
31	C₆₁×#436	C₂²	1	8	yes	complemented
32	C₆₁×#436	C₂²	1	8	yes	complemented
33	C₆₁×#435	C₂²	1	16	yes	complemented
34	C₈×D₄×C₆₁	C₂²	1	8	yes	complemented
35	C₈×D₄×C₆₁	C₂²	1	8	yes	complemented
36	C₈×D₄×C₆₁	C₂²	1	8	yes	complemented
37	C₆₁×#432	C₂²	1	2	yes	—
38	C₆₁×#432	C₂²	1	2	yes	complemented
39	C₆₁×#431	C₂²	1	1	yes	characteristic
40	C₆₁×#431	C₂²	1	2	yes	—
41	C₆₁×#431	C₂²	1	1	yes	characteristic
42	C₆₁×#431	C₂²	1	2	yes	—
43	C₆₁×#431	C₂²	1	2	yes	complemented
44	C₆₁×#431	C₂²	1	2	yes	complemented
45	C₆₁×#431	C₂²	1	2	yes	complemented
46	C₆₁×#423	C₂²	1	2	yes	complemented
47	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	complemented
48	C₆₁×#408	C₂²	1	8	yes	complemented
49	C₆₁×#408	C₂²	1	16	yes	complemented
50	C₆₁×#407	C₂²	1	2	yes	—
51	C₆₁×#407	C₂²	1	8	yes	—
52	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
53	C₂×C₂²⋊C₈×C₆₁	C₂²	1	8	yes	—
54	C₄×M₁₆×C₆₁	C₂²	1	2	yes	complemented
55	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	complemented
56	C₂×#41×C₆₁	—	4	8	no	—
57	C₂×#41×C₆₁	C₂×C₄	1	4	yes	—
58	C₂×#41×C₆₁	C₂×C₄	1	4	yes	—
59	C₂×#41×C₆₁	C₂³	1	4	yes	—
60	C₂×#41×C₆₁	C₂³	1	4	yes	—
61	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
62	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
63	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
64	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
65	C₂²×D₄×C₆₁	C₂×C₄	1	4	yes	—
66	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
70	#131×C₆₁	—	4	16	no	—
71	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
72	C₂×M₁₆×C₆₁	C₂³	1	16	yes	—
73	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
74	C₂²×C₈×C₆₁	C₂³	1	16	yes	—
75	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
76	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
77	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
78	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
79	#127×C₆₁	C₂×C₄	1	2	yes	—
80	#127×C₆₁	C₂×C₄	1	2	yes	—
81	#127×C₆₁	C₂×C₄	1	2	yes	—
82	#127×C₆₁	C₂×C₄	1	2	yes	—
83	#124×C₆₁	C₂×C₄	1	4	yes	—
84	#124×C₆₁	C₂×C₄	1	4	yes	—
85	#124×C₆₁	C₂×C₄	1	4	yes	—
86	#124×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	#121×C₆₁	C₂×C₄	1	4	yes	—
92	#121×C₆₁	C₂×C₄	1	4	yes	—
93	#121×C₆₁	C₂×C₄	1	4	yes	—
94	#121×C₆₁	C₂×C₄	1	4	yes	—
95	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
96	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
97	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
98	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
99	C₄×D₄×C₆₁	C₂³	1	4	yes	—
100	C₄×D₄×C₆₁	C₂³	1	4	yes	—
101	C₄×D₄×C₆₁	C₂³	1	4	yes	—
102	C₄×D₄×C₆₁	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
105	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
107	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
119	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
120	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
121	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
122	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
123	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
124	C₂²⋊C₈×C₆₁	C₂³	1	2	yes	—
125	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
126	C₂²⋊C₈×C₆₁	C₂³	1	2	yes	—
127	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
128	C₂²⋊C₈×C₆₁	C₂³	1	16	yes	—
129	C₂²⋊C₈×C₆₁	C₂³	1	4	yes	—
130	C₂²⋊C₈×C₆₁	C₂³	1	16	yes	—
131	#97×C₆₁	C₂³	1	4	yes	—
132	C₄×C₈×C₆₁	C₂³	1	2	yes	—
133	C₄×C₈×C₆₁	C₂³	1	4	yes	—
134	C₄×C₈×C₆₁	C₂³	1	2	yes	—
135	C₄×C₈×C₆₁	C₂³	1	4	yes	—
136	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
137	C₂⁴×C₆₁	C₂²×C₄	1	4	yes	—
138	#41×C₆₁	—	2	16	no	—
139	#41×C₆₁	—	4	8	no	—
140	#41×C₆₁	—	4	8	no	—
141	#41×C₆₁	—	4	8	no	—
142	C₂×Q₈×C₆₁	—	4	4	no	—
143	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
144	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
145	C₂×D₄×C₆₁	—	2	8	no	—
146	C₂×D₄×C₆₁	—	4	4	no	—
147	C₂×D₄×C₆₁	—	4	4	no	—
148	C₂×D₄×C₆₁	—	4	8	no	—
149	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
150	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
151	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
152	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
153	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
154	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
155	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
156	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₆₁	—	2	4	no	—
158	C₂²×C₄×C₆₁	—	2	4	no	—
159	C₂²×C₄×C₆₁	—	4	2	no	—
160	C₂²×C₄×C₆₁	—	4	8	no	—
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	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
176	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
177	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
181	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
182	M₁₆×C₆₁	—	2	8	no	—
183	M₁₆×C₆₁	—	4	16	no	—
184	C₂×C₈×C₆₁	—	2	8	no	—
185	C₂×C₈×C₆₁	—	4	16	no	—
186	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
187	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
188	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
189	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
190	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
191	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
192	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
193	C₂²⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
194	C₂²⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
195	C₂²⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
196	C₂²⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
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	C₂²×C₄	1	1	yes	characteristic
202	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
203	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
204	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
206	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
207	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
208	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
210	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
211	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
212	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
213	C₂³×C₆₁	—	2	2	no	—
214	C₂³×C₆₁	—	2	2	no	—
215	C₂³×C₆₁	—	2	8	no	—
216	C₂³×C₆₁	—	4	1	no	—
217	C₂³×C₆₁	—	4	4	no	—
218	C₂³×C₆₁	—	4	4	no	—
219	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
220	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
221	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
222	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
223	Q₈×C₆₁	—	2	8	no	—
224	Q₈×C₆₁	—	4	4	no	—
225	D₄×C₆₁	—	2	16	no	—
226	D₄×C₆₁	—	2	8	no	—
227	D₄×C₆₁	—	2	8	no	—
228	D₄×C₆₁	—	2	8	no	—
229	D₄×C₆₁	—	2	8	no	—
230	D₄×C₆₁	—	4	4	no	—
231	D₄×C₆₁	—	4	4	no	—
232	D₄×C₆₁	—	4	8	no	—
233	C₂×C₄×C₆₁	—	2	2	no	—
234	C₂×C₄×C₆₁	—	2	2	no	—
235	C₂×C₄×C₆₁	—	2	2	no	—
236	C₂×C₄×C₆₁	—	2	2	no	—
237	C₂×C₄×C₆₁	—	2	2	no	—
238	C₂×C₄×C₆₁	—	2	2	no	—
239	C₂×C₄×C₆₁	—	2	2	no	—
240	C₂×C₄×C₆₁	—	2	2	no	—
241	C₂×C₄×C₆₁	—	2	2	no	—
242	C₂×C₄×C₆₁	—	2	2	no	—
243	C₂×C₄×C₆₁	—	2	2	no	—
244	C₂×C₄×C₆₁	—	2	16	no	—
245	C₂×C₄×C₆₁	—	2	2	no	—
246	C₂×C₄×C₆₁	—	2	2	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	4	no	—
251	C₂×C₄×C₆₁	—	4	8	no	—
252	C₂×C₄×C₆₁	#131	1	4	yes	complemented
253	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
255	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
258	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
259	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
260	C₂×C₄×C₆₁	C₂³×C₄	1	8	yes	—
261	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
262	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
263	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
264	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
265	C₂×C₄×C₆₁	H(5,𝔽₂)	1	4	yes	—
266	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
267	C₈×C₆₁	—	2	4	no	—
268	C₈×C₆₁	—	2	4	no	—
269	C₈×C₆₁	—	2	8	no	—
270	#60236	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂²×C₆₁	—	2	1	no	—
272	C₂²×C₆₁	—	2	1	no	—
273	C₂²×C₆₁	—	2	1	no	—
274	C₂²×C₆₁	—	2	1	no	—
275	C₂²×C₆₁	—	2	1	no	—
276	C₂²×C₆₁	—	2	8	no	—
277	C₂²×C₆₁	—	2	1	no	—
278	C₂²×C₆₁	—	2	8	no	—
279	C₂²×C₆₁	—	2	1	no	—
280	C₂²×C₆₁	—	2	8	no	—
281	C₂²×C₆₁	—	4	4	no	—
282	C₂²×C₆₁	—	8	8	no	—
283	C₂²×C₆₁	#518	1	2	yes	—
284	C₂²×C₆₁	C₂×#131	1	2	yes	—
285	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
286	C₄×C₆₁	—	2	1	no	—
287	C₄×C₆₁	—	2	1	no	—
288	C₄×C₆₁	—	2	8	no	—
289	C₄×C₆₁	—	2	1	no	—
290	C₄×C₆₁	—	2	1	no	—
291	C₄×C₆₁	—	2	1	no	—
292	C₄×C₆₁	—	2	1	no	—
293	C₄×C₆₁	—	2	1	no	—
294	C₄×C₆₁	#518	1	2	yes	—
295	C₄×C₆₁	C₂×#131	1	2	yes	—
296	C₄×C₆₁	C₂×#131	1	2	yes	—
297	C₄×C₆₁	#568	1	2	yes	—
298	C₄×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
299	C₂×#532	C₂×C₆₁	1	1	yes	characteristic
300	#2981	C₂×C₆₁	1	4	yes	complemented
301	#2979	C₂×C₆₁	1	4	yes	complemented
302	#2975	C₂×C₆₁	1	2	yes	complemented
303	#2974	C₂×C₆₁	1	4	yes	complemented
304	#2972	C₂×C₆₁	1	2	yes	complemented
305	#2935	C₂×C₆₁	1	4	yes	complemented
306	#2931	C₂×C₆₁	1	4	yes	complemented
307	#2920	C₂×C₆₁	1	1	yes	characteristic, complemented
308	C₂×#431	C₂×C₆₁	1	1	yes	characteristic, complemented
309	#2880	C₂×C₆₁	1	4	yes	complemented
310	C₂×C₆₁	—	2	2	no	—
311	C₂×C₆₁	—	2	2	no	—
312	C₂×C₆₁	—	4	1	no	—
313	C₂×C₆₁	—	4	8	no	—
314	C₂×C₆₁	C₂×#518	1	1	yes	characteristic
315	C₂×C₆₁	C₂²×#131	1	1	yes	characteristic
316	C₂×C₆₁	C₂×#568	1	1	yes	characteristic
317	C₂²×#41	C₂²×C₆₁	1	2	yes	—
318	C₂×#131	C₂²×C₆₁	1	8	yes	complemented
319	#532	C₄×C₆₁	1	2	yes	—
320	#532	C₄×C₆₁	1	2	yes	—
321	#532	C₄×C₆₁	1	2	yes	—
322	#532	C₄×C₆₁	1	2	yes	—
323	#532	C₂²×C₆₁	1	2	yes	—
324	#532	C₂²×C₆₁	1	2	yes	—
325	#532	C₂²×C₆₁	1	2	yes	—
326	#532	C₂²×C₆₁	1	2	yes	—
327	C₂×C₄⋊Q₈	C₄×C₆₁	1	1	yes	characteristic
328	C₂×#127	C₄×C₆₁	1	1	yes	characteristic
329	C₂×#124	C₄×C₆₁	1	2	yes	—
330	C₂×#121	C₄×C₆₁	1	2	yes	—
331	C₂×#121	C₄×C₆₁	1	2	yes	—
332	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
333	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
334	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
335	#436	C₂²×C₆₁	1	8	yes	complemented
336	#436	C₂²×C₆₁	1	8	yes	complemented
337	#436	C₂²×C₆₁	1	8	yes	complemented
338	#435	C₂²×C₆₁	1	16	yes	complemented
339	C₈×D₄	C₂²×C₆₁	1	8	yes	complemented
340	C₈×D₄	C₂²×C₆₁	1	8	yes	complemented
341	C₈×D₄	C₂²×C₆₁	1	8	yes	complemented
342	#432	C₂²×C₆₁	1	2	yes	—
343	#432	C₂²×C₆₁	1	2	yes	complemented
344	#431	C₂²×C₆₁	1	1	yes	characteristic
345	#431	C₂²×C₆₁	1	2	yes	—
346	#431	C₂²×C₆₁	1	1	yes	characteristic
347	#431	C₂²×C₆₁	1	2	yes	—
348	#431	C₂²×C₆₁	1	2	yes	complemented
349	#431	C₂²×C₆₁	1	2	yes	complemented
350	#431	C₂²×C₆₁	1	2	yes	complemented
351	#423	C₂²×C₆₁	1	2	yes	complemented
352	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	complemented
353	#408	C₂²×C₆₁	1	8	yes	complemented
354	#408	C₂²×C₆₁	1	16	yes	complemented
355	#407	C₂²×C₆₁	1	2	yes	—
356	#407	C₂²×C₆₁	1	8	yes	—
357	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
358	C₂×C₂²⋊C₈	C₂²×C₆₁	1	8	yes	—
359	C₄×M₁₆	C₂²×C₆₁	1	2	yes	complemented
360	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	complemented
361	C₆₁	#60236	1	1	yes	characteristic, direct product factor, Sylow
362	C₂×#41	—	4	8	no	—
363	C₂×#41	C₂×C₄×C₆₁	1	4	yes	—
364	C₂×#41	C₂×C₄×C₆₁	1	4	yes	—
365	C₂×#41	C₂³×C₆₁	1	4	yes	—
366	C₂×#41	C₂³×C₆₁	1	4	yes	—
367	C₂²×Q₈	C₂×C₄×C₆₁	1	2	yes	—
368	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
369	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
370	C₂²×D₄	C₂×C₄×C₆₁	1	4	yes	—
371	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
372	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
373	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
374	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
375	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
376	#131	—	4	16	no	—
377	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
378	C₂×M₁₆	C₂³×C₆₁	1	16	yes	—
379	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
380	C₂²×C₈	C₂³×C₆₁	1	16	yes	—
381	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
382	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
383	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
384	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
385	#127	C₂×C₄×C₆₁	1	2	yes	—
386	#127	C₂×C₄×C₆₁	1	2	yes	—
387	#127	C₂×C₄×C₆₁	1	2	yes	—
388	#127	C₂×C₄×C₆₁	1	2	yes	—
389	#124	C₂×C₄×C₆₁	1	4	yes	—
390	#124	C₂×C₄×C₆₁	1	4	yes	—
391	#124	C₂×C₄×C₆₁	1	4	yes	—
392	#124	C₂×C₄×C₆₁	1	4	yes	—
393	#121	C₂×C₄×C₆₁	1	4	yes	—
394	#121	C₂×C₄×C₆₁	1	4	yes	—
395	#121	C₂×C₄×C₆₁	1	4	yes	—
396	#121	C₂×C₄×C₆₁	1	4	yes	—
397	#121	C₂×C₄×C₆₁	1	4	yes	—
398	#121	C₂×C₄×C₆₁	1	4	yes	—
399	#121	C₂×C₄×C₆₁	1	4	yes	—
400	#121	C₂×C₄×C₆₁	1	4	yes	—
401	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
402	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
403	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
404	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
405	C₄×D₄	C₂³×C₆₁	1	4	yes	—
406	C₄×D₄	C₂³×C₆₁	1	4	yes	—
407	C₄×D₄	C₂³×C₆₁	1	4	yes	—
408	C₄×D₄	C₂³×C₆₁	1	4	yes	—
409	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
410	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
412	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
413	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
414	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
415	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
416	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
417	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
418	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
425	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
426	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
427	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
428	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
429	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
430	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
431	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
432	C₂²⋊C₈	C₂³×C₆₁	1	2	yes	—
433	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
434	C₂²⋊C₈	C₂³×C₆₁	1	16	yes	—
435	C₂²⋊C₈	C₂³×C₆₁	1	4	yes	—
436	C₂²⋊C₈	C₂³×C₆₁	1	16	yes	—
437	#97	C₂³×C₆₁	1	4	yes	—
438	C₄×C₈	C₂³×C₆₁	1	2	yes	—
439	C₄×C₈	C₂³×C₆₁	1	4	yes	—
440	C₄×C₈	C₂³×C₆₁	1	2	yes	—
441	C₄×C₈	C₂³×C₆₁	1	4	yes	—
442	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
443	C₂⁴	C₂²×C₄×C₆₁	1	4	yes	—
444	#41	—	2	16	no	—
445	#41	—	4	8	no	—
446	#41	—	4	8	no	—
447	#41	—	4	8	no	—
448	C₂×Q₈	—	4	4	no	—
449	C₂×Q₈	C₂²×C₄×C₆₁	1	4	yes	—
450	C₂×Q₈	C₂²×C₄×C₆₁	1	4	yes	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	4	4	no	—
453	C₂×D₄	—	4	4	no	—
454	C₂×D₄	—	4	8	no	—
455	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
456	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
457	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
458	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
459	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
460	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
461	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
462	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	4	8	no	—
467	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
468	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
469	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
472	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
473	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
474	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
475	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
476	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
477	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
480	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
481	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
482	C₂²×C₄	C₂²×C₄×C₆₁	1	8	yes	—
483	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
487	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
488	M₁₆	—	2	8	no	—
489	M₁₆	—	4	16	no	—
490	C₂×C₈	—	2	8	no	—
491	C₂×C₈	—	4	16	no	—
492	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
493	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
494	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
495	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
496	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
497	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
498	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
499	C₂²⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
500	C₂²⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
501	C₂²⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
502	C₂²⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
503	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
504	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
505	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
506	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
507	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
508	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
509	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
510	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
511	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
512	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
513	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
514	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
515	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
516	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
517	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
518	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
519	C₂³	—	2	2	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	8	no	—
522	C₂³	—	4	1	no	—
523	C₂³	—	4	4	no	—
524	C₂³	—	4	4	no	—
525	C₂³	C₂³×C₄×C₆₁	1	8	yes	—
526	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
527	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
528	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
529	Q₈	—	2	8	no	—
530	Q₈	—	4	4	no	—
531	D₄	—	2	16	no	—
532	D₄	—	2	8	no	—
533	D₄	—	2	8	no	—
534	D₄	—	2	8	no	—
535	D₄	—	2	8	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	4	no	—
538	D₄	—	4	8	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	16	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	#131×C₆₁	1	4	yes	complemented
559	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
560	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
561	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
562	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
563	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
564	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
565	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
566	C₂×C₄	C₂³×C₄×C₆₁	1	8	yes	—
567	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
568	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
569	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
570	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
571	C₂×C₄	H(5,𝔽₂)×C₆₁	1	4	yes	—
572	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
573	C₈	—	2	4	no	—
574	C₈	—	2	4	no	—
575	C₈	—	2	8	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	2	8	no	—
582	C₂²	—	2	1	no	—
583	C₂²	—	2	8	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	8	no	—
586	C₂²	—	4	4	no	—
587	C₂²	—	8	8	no	—
588	C₂²	C₆₁×#518	1	2	yes	—
589	C₂²	C₂×#131×C₆₁	1	2	yes	—
590	C₂²	C₂⁴×C₄×C₆₁	1	1	yes	characteristic
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	2	8	no	—
599	C₄	C₆₁×#518	1	2	yes	—
600	C₄	C₂×#131×C₆₁	1	2	yes	—
601	C₄	C₂×#131×C₆₁	1	2	yes	—
602	C₄	C₆₁×#568	1	2	yes	—
603	C₄	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
604	C₂	—	2	2	no	—
605	C₂	—	2	2	no	—
606	C₂	—	4	1	no	—
607	C₂	—	4	8	no	—
608	C₂	C₂×C₆₁×#518	1	1	yes	characteristic
609	C₂	C₂²×#131×C₆₁	1	1	yes	characteristic
610	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×#2979	C₂	semidirect
4. C₆₁×#2975	C₂	semidirect
5. C₆₁×#2974	C₂	semidirect
6. C₆₁×#2972	C₂	semidirect
7. C₆₁×#2935	C₂	semidirect
8. C₆₁×#2931	C₂	semidirect
9. C₆₁×#2920	C₂	semidirect
10. C₂×C₆₁×#431	C₂	semidirect
11. C₆₁×#2880	C₂	semidirect
13. C₂×#131×C₆₁	C₂²	semidirect
30. C₆₁×#436	C₂²	semidirect
31. C₆₁×#436	C₂²	semidirect
32. C₆₁×#436	C₂²	semidirect
33. C₆₁×#435	C₂²	semidirect
34. C₈×D₄×C₆₁	C₂²	semidirect
35. C₈×D₄×C₆₁	C₂²	semidirect
36. C₈×D₄×C₆₁	C₂²	semidirect
38. C₆₁×#432	C₂²	semidirect
43. C₆₁×#431	C₂²	semidirect
44. C₆₁×#431	C₂²	semidirect
45. C₆₁×#431	C₂²	semidirect
46. C₆₁×#423	C₂²	semidirect
47. C₂×C₄⋊C₈×C₆₁	C₂²	semidirect
48. C₆₁×#408	C₂²	semidirect
49. C₆₁×#408	C₂²	semidirect
54. C₄×M₁₆×C₆₁	C₂²	semidirect
55. C₂×C₄×C₈×C₆₁	C₂²	semidirect
252. C₂×C₄×C₆₁	#131	semidirect
270. #60236	C₆₁	direct (N × H)
300. #2981	C₂×C₆₁	semidirect
301. #2979	C₂×C₆₁	semidirect
302. #2975	C₂×C₆₁	semidirect
303. #2974	C₂×C₆₁	semidirect
304. #2972	C₂×C₆₁	semidirect
305. #2935	C₂×C₆₁	semidirect
306. #2931	C₂×C₆₁	semidirect
307. #2920	C₂×C₆₁	semidirect
308. C₂×#431	C₂×C₆₁	semidirect
309. #2880	C₂×C₆₁	semidirect
318. C₂×#131	C₂²×C₆₁	semidirect
335. #436	C₂²×C₆₁	semidirect
336. #436	C₂²×C₆₁	semidirect
337. #436	C₂²×C₆₁	semidirect
338. #435	C₂²×C₆₁	semidirect
339. C₈×D₄	C₂²×C₆₁	semidirect
340. C₈×D₄	C₂²×C₆₁	semidirect
341. C₈×D₄	C₂²×C₆₁	semidirect
343. #432	C₂²×C₆₁	semidirect
348. #431	C₂²×C₆₁	semidirect
349. #431	C₂²×C₆₁	semidirect
350. #431	C₂²×C₆₁	semidirect
351. #423	C₂²×C₆₁	semidirect
352. C₂×C₄⋊C₈	C₂²×C₆₁	semidirect
353. #408	C₂²×C₆₁	semidirect
354. #408	C₂²×C₆₁	semidirect
359. C₄×M₁₆	C₂²×C₆₁	semidirect
360. C₂×C₄×C₈	C₂²×C₆₁	semidirect
361. C₆₁	#60236	direct (N × H)
558. C₂×C₄	#131×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×#532	2
C₆₁×#2981	2
C₆₁×#2979	2
C₆₁×#2975	2
C₆₁×#2974	2
C₆₁×#2972	2
C₆₁×#2935	2
C₆₁×#2931	2
C₆₁×#2920	2
C₂×C₆₁×#431	2
C₆₁×#2880	2
#60236	61

Subgroups — C61×#60236 #29563669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60236 #29563669