Subgroups — C₆₁×#22990 #29526423

Order 15616.

1064 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#436	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#121×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1954	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1923	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1870	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1851	C₂	1	2	yes	maximal, complemented
10	C₈×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×#121×C₆₁	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#436	—	2	4	no	—
19	C₆₁×#436	—	2	8	no	—
20	C₆₁×#423	—	2	4	no	—
21	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₆₁×#390	C₄	1	1	yes	characteristic
34	C₆₁×#389	C₄	1	1	yes	characteristic
35	C₆₁×#388	C₄	1	2	yes	—
36	C₆₁×#386	C₄	1	2	yes	—
37	C₆₁×#380	C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
40	C₆₁×#336	C₂²	1	2	yes	—
41	C₆₁×#336	C₂²	1	1	yes	characteristic
42	C₆₁×#336	C₂²	1	2	yes	—
43	C₆₁×#336	C₂²	1	1	yes	characteristic
44	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₆₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₆₁	—	2	4	no	—
51	C₂×M₁₆×C₆₁	—	2	4	no	—
52	C₂×M₁₆×C₆₁	—	2	4	no	—
53	C₂×M₁₆×C₆₁	—	2	4	no	—
54	C₂×M₁₆×C₆₁	—	4	2	no	—
55	C₂×M₁₆×C₆₁	—	4	2	no	—
56	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
62	#121×C₆₁	—	2	4	no	—
63	C₄×D₄×C₆₁	—	2	4	no	—
64	C₄×D₄×C₆₁	—	2	4	no	—
65	C₄×D₄×C₆₁	—	2	2	no	—
66	C₄×D₄×C₆₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
73	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
74	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
75	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
76	C₂×C₄²×C₆₁	—	2	2	no	—
77	C₂×C₄²×C₆₁	—	2	2	no	—
78	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
82	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₆₁	—	2	4	no	—
84	C₄⋊C₈×C₆₁	—	2	4	no	—
85	C₄⋊C₈×C₆₁	—	2	2	no	—
86	C₄⋊C₈×C₆₁	D₄	1	4	yes	complemented
87	C₂²⋊C₈×C₆₁	—	2	2	no	—
88	C₂²⋊C₈×C₆₁	—	2	4	no	—
89	C₂²⋊C₈×C₆₁	—	2	4	no	—
90	C₂²⋊C₈×C₆₁	—	4	2	no	—
91	C₂²⋊C₈×C₆₁	D₄	1	4	yes	complemented
92	C₄×C₈×C₆₁	—	2	4	no	—
93	C₄×C₈×C₆₁	—	2	2	no	—
94	C₄×C₈×C₆₁	—	2	2	no	—
95	#95×C₆₁	C₂×C₄	1	2	yes	—
96	#95×C₆₁	C₂×C₄	1	2	yes	—
97	#95×C₆₁	C₂×C₄	1	2	yes	—
98	C₂⁴×C₆₁	C₂²×C₄	1	2	yes	—
99	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×D₄×C₆₁	—	2	4	no	—
101	C₂×D₄×C₆₁	—	2	2	no	—
102	C₂×D₄×C₆₁	—	2	4	no	—
103	C₂×D₄×C₆₁	—	2	2	no	—
104	C₂×D₄×C₆₁	—	2	2	no	—
105	C₂×D₄×C₆₁	—	4	2	no	—
106	C₂²×C₄×C₆₁	—	2	1	no	—
107	C₂²×C₄×C₆₁	—	2	1	no	—
108	C₂²×C₄×C₆₁	—	2	1	no	—
109	C₂²×C₄×C₆₁	—	2	1	no	—
110	C₂²×C₄×C₆₁	—	2	2	no	—
111	C₂²×C₄×C₆₁	—	2	2	no	—
112	C₂²×C₄×C₆₁	—	2	2	no	—
113	C₂²×C₄×C₆₁	—	2	2	no	—
114	C₂²×C₄×C₆₁	—	2	2	no	—
115	C₂²×C₄×C₆₁	—	2	2	no	—
116	C₂²×C₄×C₆₁	—	2	2	no	—
117	C₂²×C₄×C₆₁	—	2	2	no	—
118	C₂²×C₄×C₆₁	—	2	2	no	—
119	C₂²×C₄×C₆₁	—	2	2	no	—
120	C₂²×C₄×C₆₁	—	2	2	no	—
121	C₂²×C₄×C₆₁	—	4	1	no	—
122	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₆₁	#41	1	2	yes	—
132	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
133	M₁₆×C₆₁	—	4	8	no	—
134	M₁₆×C₆₁	—	4	8	no	—
135	C₂×C₈×C₆₁	—	2	2	no	—
136	C₂×C₈×C₆₁	—	2	1	no	—
137	C₂×C₈×C₆₁	—	2	1	no	—
138	C₂×C₈×C₆₁	—	2	2	no	—
139	C₂×C₈×C₆₁	—	2	1	no	—
140	C₂×C₈×C₆₁	—	2	1	no	—
141	C₂×C₈×C₆₁	—	2	1	no	—
142	C₂×C₈×C₆₁	—	2	2	no	—
143	C₂×C₈×C₆₁	—	2	2	no	—
144	C₂×C₈×C₆₁	—	2	1	no	—
145	C₂×C₈×C₆₁	—	2	1	no	—
146	C₂×C₈×C₆₁	—	2	1	no	—
147	C₂×C₈×C₆₁	—	2	1	no	—
148	C₂×C₈×C₆₁	—	2	2	no	—
149	C₂×C₈×C₆₁	—	2	1	no	—
150	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
152	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₆₁	—	2	4	no	—
154	C₄⋊C₄×C₆₁	—	2	2	no	—
155	C₄⋊C₄×C₆₁	—	2	2	no	—
156	C₄⋊C₄×C₆₁	M₁₆	1	4	yes	complemented
157	C₂²⋊C₄×C₆₁	—	2	4	no	—
158	C₂²⋊C₄×C₆₁	—	2	4	no	—
159	C₂²⋊C₄×C₆₁	—	2	4	no	—
160	C₂²⋊C₄×C₆₁	—	2	4	no	—
161	C₂²⋊C₄×C₆₁	—	2	4	no	—
162	C₄²×C₆₁	—	2	2	no	—
163	C₄²×C₆₁	—	2	2	no	—
164	C₄²×C₆₁	—	2	2	no	—
165	C₄²×C₆₁	—	2	1	no	—
166	C₄²×C₆₁	—	2	1	no	—
167	C₄²×C₆₁	—	2	2	no	—
168	C₄²×C₆₁	C₂×D₄	1	2	yes	—
169	C₄²×C₆₁	#41	1	2	yes	—
170	C₂³×C₆₁	—	2	1	no	—
171	C₂³×C₆₁	—	2	1	no	—
172	C₂³×C₆₁	—	2	1	no	—
173	C₂³×C₆₁	—	2	1	no	—
174	C₂³×C₆₁	—	2	1	no	—
175	C₂³×C₆₁	—	2	1	no	—
176	C₂³×C₆₁	—	2	2	no	—
177	C₂³×C₆₁	—	2	2	no	—
178	C₂³×C₆₁	—	2	2	no	—
179	C₂³×C₆₁	—	2	2	no	—
180	C₂³×C₆₁	—	2	2	no	—
181	C₂³×C₆₁	—	2	2	no	—
182	C₂³×C₆₁	—	2	2	no	—
183	C₂³×C₆₁	C₄×D₄	1	2	yes	—
184	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
185	D₄×C₆₁	—	4	8	no	—
186	D₄×C₆₁	—	4	4	no	—
187	C₂×C₄×C₆₁	—	2	1	no	—
188	C₂×C₄×C₆₁	—	2	1	no	—
189	C₂×C₄×C₆₁	—	2	2	no	—
190	C₂×C₄×C₆₁	—	2	1	no	—
191	C₂×C₄×C₆₁	—	2	1	no	—
192	C₂×C₄×C₆₁	—	2	1	no	—
193	C₂×C₄×C₆₁	—	2	1	no	—
194	C₂×C₄×C₆₁	—	2	2	no	—
195	C₂×C₄×C₆₁	—	2	2	no	—
196	C₂×C₄×C₆₁	—	2	2	no	—
197	C₂×C₄×C₆₁	—	2	2	no	—
198	C₂×C₄×C₆₁	—	2	2	no	—
199	C₂×C₄×C₆₁	—	2	1	no	—
200	C₂×C₄×C₆₁	—	2	2	no	—
201	C₂×C₄×C₆₁	—	2	2	no	—
202	C₂×C₄×C₆₁	—	2	2	no	—
203	C₂×C₄×C₆₁	—	2	2	no	—
204	C₂×C₄×C₆₁	—	2	1	no	—
205	C₂×C₄×C₆₁	—	2	2	no	—
206	C₂×C₄×C₆₁	—	2	2	no	—
207	C₂×C₄×C₆₁	—	2	1	no	—
208	C₂×C₄×C₆₁	—	4	4	no	—
209	C₂×C₄×C₆₁	—	4	2	no	—
210	C₂×C₄×C₆₁	—	4	2	no	—
211	C₂×C₄×C₆₁	—	4	2	no	—
212	C₂×C₄×C₆₁	—	4	2	no	—
213	C₂×C₄×C₆₁	—	4	4	no	—
214	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
215	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
216	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
217	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
218	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
219	C₂×C₄×C₆₁	#131	1	2	yes	—
220	C₂×C₄×C₆₁	#131	1	2	yes	—
221	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
226	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
227	C₈×C₆₁	—	2	4	no	—
228	C₈×C₆₁	—	2	2	no	—
229	C₈×C₆₁	—	2	2	no	—
230	C₈×C₆₁	—	4	2	no	—
231	C₈×C₆₁	—	4	1	no	—
232	C₈×C₆₁	—	4	1	no	—
233	#22990	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂²×C₆₁	—	2	2	no	—
235	C₂²×C₆₁	—	2	2	no	—
236	C₂²×C₆₁	—	2	2	no	—
237	C₂²×C₆₁	—	4	1	no	—
238	C₂²×C₆₁	—	4	1	no	—
239	C₂²×C₆₁	—	4	1	no	—
240	C₂²×C₆₁	—	4	1	no	—
241	C₂²×C₆₁	—	4	2	no	—
242	C₂²×C₆₁	—	4	2	no	—
243	C₂²×C₆₁	—	4	2	no	—
244	C₂²×C₆₁	—	4	2	no	—
245	C₂²×C₆₁	—	4	2	no	—
246	C₂²×C₆₁	—	4	2	no	—
247	C₂²×C₆₁	—	4	2	no	—
248	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
249	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
250	C₂²×C₆₁	#520	1	1	yes	characteristic
251	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
252	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
253	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
254	C₂²×C₆₁	#568	1	1	yes	characteristic
255	C₄×C₆₁	—	2	2	no	—
256	C₄×C₆₁	—	2	2	no	—
257	C₄×C₆₁	—	2	2	no	—
258	C₄×C₆₁	—	2	4	no	—
259	C₄×C₆₁	—	4	2	no	—
260	C₄×C₆₁	—	4	1	no	—
261	C₄×C₆₁	—	4	2	no	—
262	C₄×C₆₁	#436	1	4	yes	complemented
263	C₄×C₆₁	D₄²	1	1	yes	characteristic
264	C₄×C₆₁	#547	1	1	yes	characteristic
265	C₄×C₆₁	#548	1	1	yes	characteristic
266	C₄×C₆₁	#553	1	1	yes	characteristic
267	C₂×#436	C₂×C₆₁	1	2	yes	complemented
268	C₂×#436	C₂×C₆₁	1	1	yes	characteristic, complemented
269	C₂×#423	C₂×C₆₁	1	1	yes	characteristic, complemented
270	C₄×#121	C₂×C₆₁	1	1	yes	characteristic
271	#1954	C₂×C₆₁	1	1	yes	characteristic, complemented
272	#1939	C₂×C₆₁	1	1	yes	characteristic, complemented
273	#1923	C₂×C₆₁	1	2	yes	complemented
274	#1870	C₂×C₆₁	1	2	yes	complemented
275	#1851	C₂×C₆₁	1	2	yes	complemented
276	C₈×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
277	C₄×C₂²⋊C₈	C₂×C₆₁	1	1	yes	characteristic, complemented
278	C₂×C₆₁	—	4	2	no	—
279	C₂×C₆₁	—	8	2	no	—
280	C₂×C₆₁	#2361	1	1	yes	characteristic
281	C₂×C₆₁	C₂×#436	1	1	yes	characteristic
282	C₂×C₆₁	#2931	1	1	yes	characteristic
283	C₂×C₆₁	#2933	1	1	yes	characteristic
284	C₂×C₆₁	D₄×M₁₆	1	1	yes	characteristic
285	C₂×C₆₁	#2991	1	1	yes	characteristic
286	C₂×C₆₁	#2993	1	1	yes	characteristic
287	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
288	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
289	C₂×#121	C₄×C₆₁	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
291	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
292	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
293	#436	—	2	4	no	—
294	#436	—	2	8	no	—
295	#423	—	2	4	no	—
296	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
297	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
298	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
300	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
301	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
302	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
305	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
306	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
307	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
308	#390	C₄×C₆₁	1	1	yes	characteristic
309	#389	C₄×C₆₁	1	1	yes	characteristic
310	#388	C₄×C₆₁	1	2	yes	—
311	#386	C₄×C₆₁	1	2	yes	—
312	#380	C₄×C₆₁	1	1	yes	characteristic
313	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
314	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
315	#336	C₂²×C₆₁	1	1	yes	characteristic
316	#336	C₂²×C₆₁	1	2	yes	—
317	#336	C₂²×C₆₁	1	2	yes	—
318	#336	C₂²×C₆₁	1	1	yes	characteristic
319	C₆₁	#22990	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
321	C₂²×D₄	C₂×C₄×C₆₁	1	2	yes	—
322	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
323	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
324	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
325	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
326	C₂×M₁₆	—	2	4	no	—
327	C₂×M₁₆	—	2	4	no	—
328	C₂×M₁₆	—	2	4	no	—
329	C₂×M₁₆	—	2	4	no	—
330	C₂×M₁₆	—	4	2	no	—
331	C₂×M₁₆	—	4	2	no	—
332	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
333	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
336	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
338	#121	—	2	4	no	—
339	C₄×D₄	—	2	4	no	—
340	C₄×D₄	—	2	4	no	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	2	2	no	—
343	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
345	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
348	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
349	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
350	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
351	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
352	C₂×C₄²	—	2	2	no	—
353	C₂×C₄²	—	2	2	no	—
354	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
355	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
357	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
358	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
359	C₄⋊C₈	—	2	4	no	—
360	C₄⋊C₈	—	2	4	no	—
361	C₄⋊C₈	—	2	2	no	—
362	C₄⋊C₈	D₄×C₆₁	1	4	yes	complemented
363	C₂²⋊C₈	—	2	2	no	—
364	C₂²⋊C₈	—	2	4	no	—
365	C₂²⋊C₈	—	2	4	no	—
366	C₂²⋊C₈	—	4	2	no	—
367	C₂²⋊C₈	D₄×C₆₁	1	4	yes	complemented
368	C₄×C₈	—	2	4	no	—
369	C₄×C₈	—	2	2	no	—
370	C₄×C₈	—	2	2	no	—
371	#95	C₂×C₄×C₆₁	1	2	yes	—
372	#95	C₂×C₄×C₆₁	1	2	yes	—
373	#95	C₂×C₄×C₆₁	1	2	yes	—
374	C₂⁴	C₂²×C₄×C₆₁	1	2	yes	—
375	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	4	1	no	—
398	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
399	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
400	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
401	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
402	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
403	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
404	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
405	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
407	C₂²×C₄	#41×C₆₁	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
409	M₁₆	—	4	8	no	—
410	M₁₆	—	4	8	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
427	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
428	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	M₁₆×C₆₁	1	4	yes	complemented
433	C₂²⋊C₄	—	2	4	no	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	4	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	2	no	—
444	C₄²	C₂×D₄×C₆₁	1	2	yes	—
445	C₄²	#41×C₆₁	1	2	yes	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	1	no	—
448	C₂³	—	2	1	no	—
449	C₂³	—	2	1	no	—
450	C₂³	—	2	1	no	—
451	C₂³	—	2	1	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	2	2	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	C₄×D₄×C₆₁	1	2	yes	—
460	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
461	D₄	—	4	8	no	—
462	D₄	—	4	4	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
491	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
492	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
493	C₂×C₄	C₂×M₁₆×C₆₁	1	4	yes	—
494	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
495	C₂×C₄	#131×C₆₁	1	2	yes	—
496	C₂×C₄	#131×C₆₁	1	2	yes	—
497	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
499	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
500	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
501	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
502	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
503	C₈	—	2	4	no	—
504	C₈	—	2	2	no	—
505	C₈	—	2	2	no	—
506	C₈	—	4	2	no	—
507	C₈	—	4	1	no	—
508	C₈	—	4	1	no	—
509	C₂²	—	2	2	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	4	1	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	2	no	—
523	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
524	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
525	C₂²	C₆₁×#520	1	1	yes	characteristic
526	C₂²	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
527	C₂²	C₂×#131×C₆₁	1	1	yes	characteristic
528	C₂²	C₂×#131×C₆₁	1	1	yes	characteristic
529	C₂²	C₆₁×#568	1	1	yes	characteristic
530	C₄	—	2	2	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	4	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	1	no	—
536	C₄	—	4	2	no	—
537	C₄	C₆₁×#436	1	4	yes	complemented
538	C₄	D₄²×C₆₁	1	1	yes	characteristic
539	C₄	C₆₁×#547	1	1	yes	characteristic
540	C₄	C₆₁×#548	1	1	yes	characteristic
541	C₄	C₆₁×#553	1	1	yes	characteristic
542	C₂	—	4	2	no	—
543	C₂	—	8	2	no	—
544	C₂	C₆₁×#2361	1	1	yes	characteristic
545	C₂	C₂×C₆₁×#436	1	1	yes	characteristic
546	C₂	C₆₁×#2931	1	1	yes	characteristic
547	C₂	C₆₁×#2933	1	1	yes	characteristic
548	C₂	D₄×M₁₆×C₆₁	1	1	yes	characteristic
549	C₂	C₆₁×#2991	1	1	yes	characteristic
550	C₂	C₆₁×#2993	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂³×C₆₁	C₂³×C₄
prime radical	C₂×#121×C₆₁	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#436	C₂	semidirect
2. C₂×C₆₁×#436	C₂	semidirect
3. C₂×C₆₁×#423	C₂	semidirect
5. C₆₁×#1954	C₂	semidirect
6. C₆₁×#1939	C₂	semidirect
7. C₆₁×#1923	C₂	semidirect
8. C₆₁×#1870	C₂	semidirect
9. C₆₁×#1851	C₂	semidirect
10. C₈×C₄⋊C₄×C₆₁	C₂	semidirect
11. C₄×C₂²⋊C₈×C₆₁	C₂	semidirect
86. C₄⋊C₈×C₆₁	D₄	semidirect
91. C₂²⋊C₈×C₆₁	D₄	semidirect
156. C₄⋊C₄×C₆₁	M₁₆	semidirect
233. #22990	C₆₁	direct (N × H)
262. C₄×C₆₁	#436	semidirect
267. C₂×#436	C₂×C₆₁	semidirect
268. C₂×#436	C₂×C₆₁	semidirect
269. C₂×#423	C₂×C₆₁	semidirect
271. #1954	C₂×C₆₁	semidirect
272. #1939	C₂×C₆₁	semidirect
273. #1923	C₂×C₆₁	semidirect
274. #1870	C₂×C₆₁	semidirect
275. #1851	C₂×C₆₁	semidirect
276. C₈×C₄⋊C₄	C₂×C₆₁	semidirect
277. C₄×C₂²⋊C₈	C₂×C₆₁	semidirect
319. C₆₁	#22990	direct (N × H)
362. C₄⋊C₈	D₄×C₆₁	semidirect
367. C₂²⋊C₈	D₄×C₆₁	semidirect
432. C₄⋊C₄	M₁₆×C₆₁	semidirect
537. C₄	C₆₁×#436	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#436	2
C₂×C₆₁×#423	2
C₄×#121×C₆₁	2
C₆₁×#1954	2
C₆₁×#1939	2
C₆₁×#1923	2
C₆₁×#1870	2
C₆₁×#1851	2
C₈×C₄⋊C₄×C₆₁	2
C₄×C₂²⋊C₈×C₆₁	2
#22990	61

Subgroups — C61×#22990 #29526423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22990 #29526423