Subgroups — C₆₁×#22698 #29526131

Order 15616.

1032 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#435	C₂	1	2	yes	maximal
2	C₄²×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1940	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1851	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1850	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#435	—	2	8	no	—
20	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
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	2	yes	—
24	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
26	C₂×#97×C₆₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₆₁	C₄	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₆₁	C₄	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₆₁	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₆₁	C₄	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
36	C₄³×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#336	C₂²	1	1	yes	characteristic
38	C₆₁×#336	C₂²	1	1	yes	characteristic
39	C₆₁×#336	C₂²	1	1	yes	characteristic
40	C₆₁×#336	C₂²	1	1	yes	characteristic
41	C₆₁×#336	C₂²	1	2	yes	—
42	C₆₁×#336	C₂²	1	2	yes	—
43	C₆₁×#336	C₂²	1	2	yes	—
44	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₆₁	—	2	4	no	—
52	C₂×M₁₆×C₆₁	—	2	4	no	—
53	C₂×M₁₆×C₆₁	—	4	2	no	—
54	C₂²×C₈×C₆₁	—	2	4	no	—
55	C₂²×C₈×C₆₁	—	2	4	no	—
56	C₂²×C₈×C₆₁	—	4	2	no	—
57	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
61	C₄×D₄×C₆₁	—	2	4	no	—
62	C₄×D₄×C₆₁	—	2	4	no	—
63	C₄×D₄×C₆₁	D₄	1	4	yes	—
64	C₄×D₄×C₆₁	Q₈	1	4	yes	—
65	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₁	—	4	2	no	—
75	C₂×C₄²×C₆₁	—	4	2	no	—
76	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₆₁	—	2	4	no	—
90	C₄⋊C₈×C₆₁	—	2	4	no	—
91	C₄⋊C₈×C₆₁	—	4	2	no	—
92	C₄⋊C₈×C₆₁	—	4	2	no	—
93	C₂²⋊C₈×C₆₁	—	2	4	no	—
94	C₂²⋊C₈×C₆₁	—	2	4	no	—
95	#97×C₆₁	—	2	4	no	—
96	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
97	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
98	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
99	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
100	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
101	#95×C₆₁	C₂×C₄	1	1	yes	characteristic
102	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×D₄×C₆₁	—	2	2	no	—
105	C₂×D₄×C₆₁	—	2	2	no	—
106	C₂×D₄×C₆₁	C₄⋊C₄	1	4	yes	—
107	C₂²×C₄×C₆₁	—	2	2	no	—
108	C₂²×C₄×C₆₁	—	2	2	no	—
109	C₂²×C₄×C₆₁	—	2	2	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	—	4	1	no	—
117	C₂²×C₄×C₆₁	—	4	1	no	—
118	C₂²×C₄×C₆₁	—	4	1	no	—
119	C₂²×C₄×C₆₁	—	4	1	no	—
120	C₂²×C₄×C₆₁	—	4	1	no	—
121	C₂²×C₄×C₆₁	—	4	1	no	—
122	C₂²×C₄×C₆₁	M₁₆	1	4	yes	complemented
123	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
128	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
138	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
139	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
140	M₁₆×C₆₁	—	4	8	no	—
141	C₂×C₈×C₆₁	—	2	2	no	—
142	C₂×C₈×C₆₁	—	2	2	no	—
143	C₂×C₈×C₆₁	—	2	2	no	—
144	C₂×C₈×C₆₁	—	2	2	no	—
145	C₂×C₈×C₆₁	—	2	2	no	—
146	C₂×C₈×C₆₁	—	2	2	no	—
147	C₂×C₈×C₆₁	—	2	2	no	—
148	C₂×C₈×C₆₁	—	2	2	no	—
149	C₂×C₈×C₆₁	—	2	2	no	—
150	C₂×C₈×C₆₁	—	2	2	no	—
151	C₂×C₈×C₆₁	—	2	2	no	—
152	C₂×C₈×C₆₁	—	4	8	no	—
153	C₂×C₈×C₆₁	#41	1	4	yes	—
154	C₄⋊C₄×C₆₁	—	2	2	no	—
155	C₄⋊C₄×C₆₁	—	2	2	no	—
156	C₄⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
157	C₂²⋊C₄×C₆₁	—	2	2	no	—
158	C₂²⋊C₄×C₆₁	—	2	2	no	—
159	C₂²⋊C₄×C₆₁	—	2	2	no	—
160	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
161	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
162	C₂²⋊C₄×C₆₁	M₁₆	1	4	yes	complemented
163	C₄²×C₆₁	—	2	2	no	—
164	C₄²×C₆₁	—	2	2	no	—
165	C₄²×C₆₁	—	2	2	no	—
166	C₄²×C₆₁	—	2	2	no	—
167	C₄²×C₆₁	—	2	2	no	—
168	C₄²×C₆₁	—	2	2	no	—
169	C₄²×C₆₁	—	2	2	no	—
170	C₄²×C₆₁	—	4	1	no	—
171	C₄²×C₆₁	—	4	1	no	—
172	C₄²×C₆₁	—	4	1	no	—
173	C₄²×C₆₁	—	4	1	no	—
174	C₄²×C₆₁	—	4	1	no	—
175	C₄²×C₆₁	C₂×D₄	1	2	yes	—
176	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
177	C₂³×C₆₁	—	2	1	no	—
178	C₂³×C₆₁	—	2	1	no	—
179	C₂³×C₆₁	—	2	1	no	—
180	C₂³×C₆₁	—	2	1	no	—
181	C₂³×C₆₁	—	2	1	no	—
182	C₂³×C₆₁	—	2	1	no	—
183	C₂³×C₆₁	—	2	1	no	—
184	C₂³×C₆₁	—	2	1	no	—
185	C₂³×C₆₁	—	2	1	no	—
186	C₂³×C₆₁	—	2	1	no	—
187	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
188	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
189	C₂³×C₆₁	C₂×M₁₆	1	2	yes	—
190	C₂³×C₆₁	C₂×M₁₆	1	2	yes	—
191	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
192	D₄×C₆₁	—	2	8	no	—
193	C₂×C₄×C₆₁	—	2	1	no	—
194	C₂×C₄×C₆₁	—	2	1	no	—
195	C₂×C₄×C₆₁	—	2	1	no	—
196	C₂×C₄×C₆₁	—	2	1	no	—
197	C₂×C₄×C₆₁	—	2	1	no	—
198	C₂×C₄×C₆₁	—	2	1	no	—
199	C₂×C₄×C₆₁	—	2	1	no	—
200	C₂×C₄×C₆₁	—	2	1	no	—
201	C₂×C₄×C₆₁	—	2	1	no	—
202	C₂×C₄×C₆₁	—	2	1	no	—
203	C₂×C₄×C₆₁	—	2	1	no	—
204	C₂×C₄×C₆₁	—	2	1	no	—
205	C₂×C₄×C₆₁	—	2	1	no	—
206	C₂×C₄×C₆₁	—	2	1	no	—
207	C₂×C₄×C₆₁	—	2	1	no	—
208	C₂×C₄×C₆₁	—	2	1	no	—
209	C₂×C₄×C₆₁	—	2	1	no	—
210	C₂×C₄×C₆₁	—	2	1	no	—
211	C₂×C₄×C₆₁	—	2	4	no	—
212	C₂×C₄×C₆₁	—	2	4	no	—
213	C₂×C₄×C₆₁	—	2	4	no	—
214	C₂×C₄×C₆₁	—	2	4	no	—
215	C₂×C₄×C₆₁	—	2	1	no	—
216	C₂×C₄×C₆₁	—	2	1	no	—
217	C₂×C₄×C₆₁	—	2	1	no	—
218	C₂×C₄×C₆₁	—	2	1	no	—
219	C₂×C₄×C₆₁	—	2	1	no	—
220	C₂×C₄×C₆₁	—	2	1	no	—
221	C₂×C₄×C₆₁	—	2	1	no	—
222	C₂×C₄×C₆₁	—	2	1	no	—
223	C₂×C₄×C₆₁	—	2	1	no	—
224	C₂×C₄×C₆₁	—	4	4	no	—
225	C₂×C₄×C₆₁	—	4	4	no	—
226	C₂×C₄×C₆₁	—	4	4	no	—
227	C₂×C₄×C₆₁	—	4	4	no	—
228	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
229	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
230	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
231	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
232	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
233	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
234	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
235	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
236	C₂×C₄×C₆₁	#131	1	2	yes	—
237	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
239	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
240	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
241	C₈×C₆₁	—	2	4	no	—
242	C₈×C₆₁	—	4	2	no	—
243	C₈×C₆₁	—	4	2	no	—
244	C₈×C₆₁	—	4	2	no	—
245	#22698	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂²×C₆₁	—	2	2	no	—
247	C₂²×C₆₁	—	2	2	no	—
248	C₂²×C₆₁	—	2	2	no	—
249	C₂²×C₆₁	—	2	2	no	—
250	C₂²×C₆₁	—	2	2	no	—
251	C₂²×C₆₁	—	2	2	no	—
252	C₂²×C₆₁	—	2	2	no	—
253	C₂²×C₆₁	—	2	2	no	—
254	C₂²×C₆₁	—	2	2	no	—
255	C₂²×C₆₁	—	4	1	no	—
256	C₂²×C₆₁	—	4	1	no	—
257	C₂²×C₆₁	—	4	1	no	—
258	C₂²×C₆₁	—	4	1	no	—
259	C₂²×C₆₁	#423	1	4	yes	—
260	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
261	C₂²×C₆₁	C₄×#41	1	1	yes	characteristic
262	C₂²×C₆₁	#518	1	1	yes	characteristic
263	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
264	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
265	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
266	C₂²×C₆₁	#568	1	1	yes	characteristic
267	C₄×C₆₁	—	2	2	no	—
268	C₄×C₆₁	—	2	2	no	—
269	C₄×C₆₁	—	2	2	no	—
270	C₄×C₆₁	—	2	2	no	—
271	C₄×C₆₁	—	2	2	no	—
272	C₄×C₆₁	—	2	2	no	—
273	C₄×C₆₁	—	2	2	no	—
274	C₄×C₆₁	—	2	2	no	—
275	C₄×C₆₁	—	2	2	no	—
276	C₄×C₆₁	—	4	1	no	—
277	C₄×C₆₁	—	4	1	no	—
278	C₄×C₆₁	—	4	1	no	—
279	C₄×C₆₁	#546	1	2	yes	—
280	C₄×C₆₁	#554	1	2	yes	—
281	C₂×#435	C₂×C₆₁	1	2	yes	—
282	C₄²×D₄	C₂×C₆₁	1	1	yes	characteristic
283	#1940	C₂×C₆₁	1	2	yes	complemented
284	#1851	C₂×C₆₁	1	2	yes	complemented
285	#1850	C₂×C₆₁	1	2	yes	complemented
286	#1831	C₂×C₆₁	1	1	yes	characteristic, complemented
287	#1797	C₂×C₆₁	1	1	yes	characteristic, complemented
288	#1797	C₂×C₆₁	1	1	yes	characteristic, complemented
289	#1767	C₂×C₆₁	1	1	yes	characteristic, complemented
290	#1728	C₂×C₆₁	1	1	yes	characteristic, complemented
291	C₂×#336	C₂×C₆₁	1	1	yes	characteristic, complemented
292	C₂×C₆₁	—	2	4	no	—
293	C₂×C₆₁	—	4	2	no	—
294	C₂×C₆₁	#2349	1	1	yes	characteristic
295	C₂×C₆₁	C₂×#423	1	1	yes	characteristic
296	C₂×C₆₁	#2906	1	1	yes	characteristic
297	C₂×C₆₁	#2965	1	1	yes	characteristic
298	C₂×C₆₁	#2970	1	1	yes	characteristic
299	C₂×C₆₁	#2973	1	1	yes	characteristic
300	C₂×C₆₁	#2974	1	1	yes	characteristic
301	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
302	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
303	C₂×C₄×D₄	C₄×C₆₁	1	1	yes	characteristic
304	C₂×C₄×D₄	C₄×C₆₁	1	1	yes	characteristic
305	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
306	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
307	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
308	#435	—	2	8	no	—
309	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
310	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
311	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
312	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
313	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
314	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
315	C₂×#97	C₂²×C₆₁	1	2	yes	—
316	C₄×C₄⋊C₄	C₄×C₆₁	1	1	yes	characteristic
317	C₄×C₄⋊C₄	C₄×C₆₁	1	1	yes	characteristic
318	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
319	C₄×C₂²⋊C₄	C₄×C₆₁	1	1	yes	characteristic
320	C₄×C₂²⋊C₄	C₄×C₆₁	1	1	yes	characteristic
321	C₄×C₂²⋊C₄	C₄×C₆₁	1	1	yes	characteristic
322	C₄×C₂²⋊C₄	C₄×C₆₁	1	1	yes	characteristic
323	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
324	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
325	C₄³	C₂²×C₆₁	1	1	yes	characteristic
326	#336	C₂²×C₆₁	1	1	yes	characteristic
327	#336	C₂²×C₆₁	1	1	yes	characteristic
328	#336	C₂²×C₆₁	1	1	yes	characteristic
329	#336	C₂²×C₆₁	1	2	yes	—
330	#336	C₂²×C₆₁	1	2	yes	—
331	#336	C₂²×C₆₁	1	2	yes	—
332	#336	C₂²×C₆₁	1	1	yes	characteristic
333	C₆₁	#22698	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
335	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
336	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
337	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
338	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
341	C₂×M₁₆	—	2	4	no	—
342	C₂×M₁₆	—	2	4	no	—
343	C₂×M₁₆	—	4	2	no	—
344	C₂²×C₈	—	2	4	no	—
345	C₂²×C₈	—	2	4	no	—
346	C₂²×C₈	—	4	2	no	—
347	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
348	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
349	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
350	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
351	C₄×D₄	—	2	4	no	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	D₄×C₆₁	1	4	yes	—
354	C₄×D₄	Q₈×C₆₁	1	4	yes	—
355	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
360	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
361	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
362	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
363	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
364	C₂×C₄²	—	4	2	no	—
365	C₂×C₄²	—	4	2	no	—
366	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
367	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
368	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
369	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
370	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
371	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
372	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
373	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
374	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
379	C₄⋊C₈	—	2	4	no	—
380	C₄⋊C₈	—	2	4	no	—
381	C₄⋊C₈	—	4	2	no	—
382	C₄⋊C₈	—	4	2	no	—
383	C₂²⋊C₈	—	2	4	no	—
384	C₂²⋊C₈	—	2	4	no	—
385	#97	—	2	4	no	—
386	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
387	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
388	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
389	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
390	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
391	#95	C₂×C₄×C₆₁	1	1	yes	characteristic
392	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
393	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	C₄⋊C₄×C₆₁	1	4	yes	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	4	1	no	—
407	C₂²×C₄	—	4	1	no	—
408	C₂²×C₄	—	4	1	no	—
409	C₂²×C₄	—	4	1	no	—
410	C₂²×C₄	—	4	1	no	—
411	C₂²×C₄	—	4	1	no	—
412	C₂²×C₄	M₁₆×C₆₁	1	4	yes	complemented
413	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
414	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
415	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
416	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
417	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
418	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
419	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
420	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
421	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
422	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
423	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
424	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
427	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
428	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
430	M₁₆	—	4	8	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	4	8	no	—
443	C₂×C₈	#41×C₆₁	1	4	yes	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	C₄⋊C₄×C₆₁	1	4	yes	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	C₄⋊C₄×C₆₁	1	4	yes	—
451	C₂²⋊C₄	C₄⋊C₄×C₆₁	1	4	yes	—
452	C₂²⋊C₄	M₁₆×C₆₁	1	4	yes	complemented
453	C₄²	—	2	2	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	2	2	no	—
456	C₄²	—	2	2	no	—
457	C₄²	—	2	2	no	—
458	C₄²	—	2	2	no	—
459	C₄²	—	2	2	no	—
460	C₄²	—	4	1	no	—
461	C₄²	—	4	1	no	—
462	C₄²	—	4	1	no	—
463	C₄²	—	4	1	no	—
464	C₄²	—	4	1	no	—
465	C₄²	C₂×D₄×C₆₁	1	2	yes	—
466	C₄²	C₂×Q₈×C₆₁	1	2	yes	—
467	C₂³	—	2	1	no	—
468	C₂³	—	2	1	no	—
469	C₂³	—	2	1	no	—
470	C₂³	—	2	1	no	—
471	C₂³	—	2	1	no	—
472	C₂³	—	2	1	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	1	no	—
475	C₂³	—	2	1	no	—
476	C₂³	—	2	1	no	—
477	C₂³	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
478	C₂³	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
479	C₂³	C₂×M₁₆×C₆₁	1	2	yes	—
480	C₂³	C₂×M₁₆×C₆₁	1	2	yes	—
481	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
482	D₄	—	2	8	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
519	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
520	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
521	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
522	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
523	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
524	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
525	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
526	C₂×C₄	#131×C₆₁	1	2	yes	—
527	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
530	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
531	C₈	—	2	4	no	—
532	C₈	—	4	2	no	—
533	C₈	—	4	2	no	—
534	C₈	—	4	2	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	1	no	—
548	C₂²	C₆₁×#423	1	4	yes	—
549	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
550	C₂²	C₄×#41×C₆₁	1	1	yes	characteristic
551	C₂²	C₆₁×#518	1	1	yes	characteristic
552	C₂²	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
553	C₂²	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
554	C₂²	C₂×#131×C₆₁	1	1	yes	characteristic
555	C₂²	C₆₁×#568	1	1	yes	characteristic
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	2	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	2	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	C₆₁×#546	1	2	yes	—
569	C₄	C₆₁×#554	1	2	yes	—
570	C₂	—	2	4	no	—
571	C₂	—	4	2	no	—
572	C₂	C₆₁×#2349	1	1	yes	characteristic
573	C₂	C₂×C₆₁×#423	1	1	yes	characteristic
574	C₂	C₆₁×#2906	1	1	yes	characteristic
575	C₂	C₆₁×#2965	1	1	yes	characteristic
576	C₂	C₆₁×#2970	1	1	yes	characteristic
577	C₂	C₆₁×#2973	1	1	yes	characteristic
578	C₂	C₆₁×#2974	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	C₂²×D₄×C₆₁	C₂×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
3. C₆₁×#1940	C₂	semidirect
4. C₆₁×#1851	C₂	semidirect
5. C₆₁×#1850	C₂	semidirect
6. C₆₁×#1831	C₂	semidirect
7. C₆₁×#1797	C₂	semidirect
8. C₆₁×#1797	C₂	semidirect
9. C₆₁×#1767	C₂	semidirect
10. C₆₁×#1728	C₂	semidirect
11. C₂×C₆₁×#336	C₂	semidirect
122. C₂²×C₄×C₆₁	M₁₆	semidirect
162. C₂²⋊C₄×C₆₁	M₁₆	semidirect
245. #22698	C₆₁	direct (N × H)
283. #1940	C₂×C₆₁	semidirect
284. #1851	C₂×C₆₁	semidirect
285. #1850	C₂×C₆₁	semidirect
286. #1831	C₂×C₆₁	semidirect
287. #1797	C₂×C₆₁	semidirect
288. #1797	C₂×C₆₁	semidirect
289. #1767	C₂×C₆₁	semidirect
290. #1728	C₂×C₆₁	semidirect
291. C₂×#336	C₂×C₆₁	semidirect
333. C₆₁	#22698	direct (N × H)
412. C₂²×C₄	M₁₆×C₆₁	semidirect
452. C₂²⋊C₄	M₁₆×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×#435	2
C₄²×D₄×C₆₁	2
C₆₁×#1940	2
C₆₁×#1851	2
C₆₁×#1850	2
C₆₁×#1831	2
C₆₁×#1797	2
C₆₁×#1767	2
C₆₁×#1728	2
C₂×C₆₁×#336	2
#22698	61

Subgroups — C61×#22698 #29526131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#22698 #29526131