Subgroups — C₆₁×#60802 #29564235

Order 15616.

2186 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3216	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3214	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3212	C₂	1	8	yes	maximal, complemented
6	C₆₁×#3153	C₂	1	2	yes	maximal, complemented
7	C₆₁×#3140	C₂	1	2	yes	maximal, complemented
8	C₆₁×#3039	C₂	1	2	yes	maximal, complemented
9	C₆₁×#3038	C₂	1	2	yes	maximal, complemented
10	C₆₁×#3005	C₂	1	4	yes	maximal, complemented
11	C₆₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₁	C₂²	1	2	yes	—
13	C₂×#137×C₆₁	C₂²	1	4	yes	—
14	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	4	yes	—
15	C₆₁×#533	C₂²	1	1	yes	characteristic
16	C₆₁×#533	C₂²	1	2	yes	complemented
17	C₆₁×#532	—	2	2	no	—
18	C₆₁×#532	—	2	2	no	—
19	C₆₁×#532	C₂²	1	1	yes	characteristic
20	C₆₁×#532	C₂²	1	4	yes	—
21	C₆₁×#532	C₂²	1	4	yes	—
22	C₆₁×#532	C₂²	1	2	yes	complemented
23	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#529	—	2	2	no	—
26	C₆₁×#529	—	2	2	no	—
27	C₆₁×#529	C₂²	1	4	yes	complemented
28	C₂×#124×C₆₁	C₂²	1	2	yes	—
29	C₆₁×#525	—	2	2	no	—
30	C₆₁×#525	C₂²	1	2	yes	—
31	C₂×#121×C₆₁	—	2	2	no	—
32	C₄×#41×C₆₁	—	2	2	no	—
33	C₄×#41×C₆₁	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₆₁	—	2	2	no	—
35	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#497	C₂²	1	4	yes	complemented
37	C₆₁×#496	C₂²	1	4	yes	complemented
38	C₆₁×#495	C₂²	1	8	yes	complemented
39	C₆₁×#489	C₂²	1	8	yes	complemented
40	C₆₁×#488	C₂²	1	8	yes	complemented
41	C₆₁×#462	C₂²	1	8	yes	complemented
42	C₆₁×#461	C₂²	1	8	yes	complemented
43	C₆₁×#460	C₂²	1	8	yes	complemented
44	C₆₁×#459	C₂²	1	8	yes	complemented
45	C₆₁×#452	C₂²	1	16	yes	complemented
46	C₆₁×#449	C₂²	1	16	yes	complemented
47	C₆₁×#432	C₂²	1	4	yes	complemented
48	C₆₁×#431	C₂²	1	4	yes	complemented
49	C₆₁×#423	C₂²	1	2	yes	—
50	C₆₁×#419	C₂²	1	4	yes	—
51	C₆₁×#418	C₂²	1	4	yes	—
52	C₆₁×#407	C₂²	1	2	yes	—
53	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
54	C₂×#41×C₆₁	—	2	4	no	—
55	C₂×#41×C₆₁	—	2	2	no	—
56	C₂×#41×C₆₁	—	4	8	no	—
57	C₂×#41×C₆₁	C₂³	1	2	yes	—
58	C₂×#41×C₆₁	C₂³	1	8	yes	—
59	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
60	C₂²×D₄×C₆₁	—	2	2	no	—
61	C₂²×D₄×C₆₁	—	2	2	no	—
62	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₁	—	2	2	no	—
64	C₂³×C₄×C₆₁	D₄	1	2	yes	—
65	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
66	#137×C₆₁	—	4	8	no	—
67	AGL(1,ℤ/8)×C₆₁	—	4	8	no	—
68	C₂×Q₁₆×C₆₁	C₂³	1	8	yes	—
69	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
70	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
71	C₂×D₈×C₆₁	C₂³	1	8	yes	—
72	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
73	C₄⋊Q₈×C₆₁	—	2	1	no	—
74	C₄⋊Q₈×C₆₁	—	2	1	no	—
75	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
76	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
78	#127×C₆₁	—	2	1	no	—
79	#127×C₆₁	—	2	1	no	—
80	#127×C₆₁	C₂³	1	4	yes	—
81	#127×C₆₁	C₂³	1	1	yes	characteristic
82	#126×C₆₁	—	2	4	no	—
83	#125×C₆₁	—	2	2	no	—
84	#125×C₆₁	C₂³	1	2	yes	—
85	#124×C₆₁	—	2	4	no	—
86	#124×C₆₁	—	2	2	no	—
87	#124×C₆₁	C₂³	1	4	yes	—
88	#124×C₆₁	C₂³	1	4	yes	—
89	#124×C₆₁	C₂³	1	2	yes	—
90	#123×C₆₁	—	2	4	no	—
91	#123×C₆₁	—	2	2	no	—
92	C₂²⋊Q₈×C₆₁	—	2	2	no	—
93	C₂²⋊Q₈×C₆₁	—	2	2	no	—
94	C₂²⋊Q₈×C₆₁	C₂³	1	4	yes	—
95	#121×C₆₁	—	2	4	no	—
96	#121×C₆₁	—	2	4	no	—
97	#121×C₆₁	—	2	4	no	—
98	#121×C₆₁	—	2	4	no	—
99	#121×C₆₁	—	2	2	no	—
100	#121×C₆₁	—	2	2	no	—
101	#121×C₆₁	C₂³	1	4	yes	—
102	#120×C₆₁	—	2	2	no	—
103	C₄×Q₈×C₆₁	—	2	2	no	—
104	C₄×Q₈×C₆₁	—	2	2	no	—
105	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
106	C₄×D₄×C₆₁	—	2	4	no	—
107	C₄×D₄×C₆₁	—	2	4	no	—
108	C₄×D₄×C₆₁	—	2	4	no	—
109	C₄×D₄×C₆₁	—	2	4	no	—
110	C₄×D₄×C₆₁	—	2	4	no	—
111	C₄×D₄×C₆₁	—	2	4	no	—
112	C₄×D₄×C₆₁	—	2	4	no	—
113	C₄×D₄×C₆₁	—	2	2	no	—
114	C₄×D₄×C₆₁	—	2	2	no	—
115	C₄×D₄×C₆₁	C₂³	1	4	yes	—
116	#117×C₆₁	—	2	2	no	—
117	#117×C₆₁	—	2	4	no	—
118	#117×C₆₁	—	2	2	no	—
119	#117×C₆₁	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
122	C₂×C₄²×C₆₁	—	2	2	no	—
123	C₂×C₄²×C₆₁	D₄	1	2	yes	—
124	C₂×C₄²×C₆₁	D₄	1	2	yes	—
125	C₂×C₄²×C₆₁	D₄	1	2	yes	—
126	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
128	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
130	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
131	#103×C₆₁	C₂³	1	16	yes	—
132	#102×C₆₁	C₂³	1	16	yes	—
133	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
134	#97×C₆₁	C₂³	1	4	yes	—
135	C₄×C₈×C₆₁	C₂³	1	4	yes	—
136	C₂⁴×C₆₁	—	2	2	no	—
137	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
138	#41×C₆₁	—	2	4	no	—
139	#41×C₆₁	—	2	4	no	—
140	#41×C₆₁	—	4	16	no	—
141	#41×C₆₁	—	4	8	no	—
142	#41×C₆₁	—	4	16	no	—
143	C₂×Q₈×C₆₁	—	2	2	no	—
144	C₂×Q₈×C₆₁	—	2	2	no	—
145	C₂×Q₈×C₆₁	—	2	8	no	—
146	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
147	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
148	C₂×D₄×C₆₁	—	2	2	no	—
149	C₂×D₄×C₆₁	—	2	4	no	—
150	C₂×D₄×C₆₁	—	2	8	no	—
151	C₂×D₄×C₆₁	—	2	2	no	—
152	C₂×D₄×C₆₁	—	2	4	no	—
153	C₂×D₄×C₆₁	—	2	4	no	—
154	C₂×D₄×C₆₁	—	2	2	no	—
155	C₂×D₄×C₆₁	—	2	8	no	—
156	C₂×D₄×C₆₁	—	4	8	no	—
157	C₂×D₄×C₆₁	—	4	8	no	—
158	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
159	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₆₁	—	2	4	no	—
161	C₂²×C₄×C₆₁	—	2	2	no	—
162	C₂²×C₄×C₆₁	—	2	2	no	—
163	C₂²×C₄×C₆₁	—	2	4	no	—
164	C₂²×C₄×C₆₁	—	2	8	no	—
165	C₂²×C₄×C₆₁	—	4	4	no	—
166	C₂²×C₄×C₆₁	—	4	8	no	—
167	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
177	Q₁₆×C₆₁	—	4	8	no	—
178	SD₁₆×C₆₁	—	4	8	no	—
179	SD₁₆×C₆₁	—	4	8	no	—
180	D₈×C₆₁	—	4	8	no	—
181	M₁₆×C₆₁	—	2	8	no	—
182	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
183	C₄⋊C₄×C₆₁	—	2	4	no	—
184	C₄⋊C₄×C₆₁	—	2	4	no	—
185	C₄⋊C₄×C₆₁	—	2	2	no	—
186	C₄⋊C₄×C₆₁	—	2	2	no	—
187	C₄⋊C₄×C₆₁	—	2	4	no	—
188	C₄⋊C₄×C₆₁	—	2	4	no	—
189	C₄⋊C₄×C₆₁	—	2	4	no	—
190	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
191	C₂²⋊C₄×C₆₁	—	2	4	no	—
192	C₂²⋊C₄×C₆₁	—	2	2	no	—
193	C₂²⋊C₄×C₆₁	—	2	8	no	—
194	C₂²⋊C₄×C₆₁	—	2	8	no	—
195	C₂²⋊C₄×C₆₁	—	2	4	no	—
196	C₂²⋊C₄×C₆₁	—	2	4	no	—
197	C₂²⋊C₄×C₆₁	—	2	4	no	—
198	C₄²×C₆₁	—	2	4	no	—
199	C₄²×C₆₁	—	2	2	no	—
200	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₆₁	C₂×D₄	1	2	yes	—
202	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₆₁	C₂×D₄	1	2	yes	—
204	C₄²×C₆₁	C₂×D₄	1	2	yes	—
205	C₄²×C₆₁	C₂×D₄	1	2	yes	—
206	C₄²×C₆₁	C₂×D₄	1	2	yes	—
207	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
208	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
209	C₄²×C₆₁	C₂⁴	1	2	yes	—
210	C₂³×C₆₁	—	2	2	no	—
211	C₂³×C₆₁	—	2	2	no	—
212	C₂³×C₆₁	—	2	4	no	—
213	C₂³×C₆₁	—	4	2	no	—
214	C₂³×C₆₁	—	4	8	no	—
215	C₂³×C₆₁	—	4	4	no	—
216	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
217	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
218	Q₈×C₆₁	—	2	2	no	—
219	Q₈×C₆₁	—	2	8	no	—
220	Q₈×C₆₁	—	2	2	no	—
221	Q₈×C₆₁	—	4	4	no	—
222	Q₈×C₆₁	—	4	4	no	—
223	D₄×C₆₁	—	2	2	no	—
224	D₄×C₆₁	—	2	8	no	—
225	D₄×C₆₁	—	2	2	no	—
226	D₄×C₆₁	—	4	8	no	—
227	D₄×C₆₁	—	4	4	no	—
228	D₄×C₆₁	—	4	16	no	—
229	D₄×C₆₁	—	4	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	4	no	—
235	C₂×C₄×C₆₁	—	2	4	no	—
236	C₂×C₄×C₆₁	—	2	4	no	—
237	C₂×C₄×C₆₁	—	2	4	no	—
238	C₂×C₄×C₆₁	—	2	4	no	—
239	C₂×C₄×C₆₁	—	2	2	no	—
240	C₂×C₄×C₆₁	—	2	4	no	—
241	C₂×C₄×C₆₁	—	2	4	no	—
242	C₂×C₄×C₆₁	—	4	2	no	—
243	C₂×C₄×C₆₁	—	4	2	no	—
244	C₂×C₄×C₆₁	—	4	2	no	—
245	C₂×C₄×C₆₁	—	4	16	no	—
246	C₂×C₄×C₆₁	—	4	4	no	—
247	C₂×C₄×C₆₁	—	4	4	no	—
248	C₂×C₄×C₆₁	—	4	8	no	—
249	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₆₁	H(5,𝔽₂)	1	4	yes	—
256	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
257	C₈×C₆₁	—	2	8	no	—
258	#60802	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂²×C₆₁	—	2	2	no	—
260	C₂²×C₆₁	—	2	2	no	—
261	C₂²×C₆₁	—	2	2	no	—
262	C₂²×C₆₁	—	2	2	no	—
263	C₂²×C₆₁	—	4	4	no	—
264	C₂²×C₆₁	—	4	4	no	—
265	C₂²×C₆₁	—	4	2	no	—
266	C₂²×C₆₁	—	4	4	no	—
267	C₂²×C₆₁	—	4	8	no	—
268	C₂²×C₆₁	—	8	8	no	—
269	C₂²×C₆₁	#535	1	2	yes	—
270	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
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	—	4	4	no	—
276	C₄×C₆₁	—	4	2	no	—
277	C₄×C₆₁	—	4	4	no	—
278	C₄×C₆₁	#535	1	2	yes	—
279	C₄×C₆₁	#575	1	4	yes	—
280	C₄×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
281	#3444	C₂×C₆₁	1	1	yes	characteristic, complemented
282	C₂×#532	C₂×C₆₁	1	1	yes	characteristic, complemented
283	#3216	C₂×C₆₁	1	4	yes	complemented
284	#3214	C₂×C₆₁	1	4	yes	complemented
285	#3212	C₂×C₆₁	1	8	yes	complemented
286	#3153	C₂×C₆₁	1	2	yes	complemented
287	#3140	C₂×C₆₁	1	2	yes	complemented
288	#3039	C₂×C₆₁	1	2	yes	complemented
289	#3038	C₂×C₆₁	1	2	yes	complemented
290	#3005	C₂×C₆₁	1	4	yes	complemented
291	#2920	C₂×C₆₁	1	1	yes	characteristic, complemented
292	C₂×C₆₁	—	2	2	no	—
293	C₂×C₆₁	—	4	2	no	—
294	C₂×C₆₁	—	8	2	no	—
295	C₂×C₆₁	—	8	4	no	—
296	C₂×C₆₁	C₂×#535	1	1	yes	characteristic
297	C₂×C₆₁	C₂×#575	1	2	yes	—
298	C₂²×#41	C₂²×C₆₁	1	2	yes	—
299	C₂×#137	C₂²×C₆₁	1	4	yes	—
300	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	4	yes	—
301	#533	C₂²×C₆₁	1	1	yes	characteristic
302	#533	C₂²×C₆₁	1	2	yes	complemented
303	#532	—	2	2	no	—
304	#532	—	2	2	no	—
305	#532	C₂²×C₆₁	1	1	yes	characteristic
306	#532	C₂²×C₆₁	1	4	yes	—
307	#532	C₂²×C₆₁	1	4	yes	—
308	#532	C₂²×C₆₁	1	2	yes	complemented
309	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
310	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
311	#529	—	2	2	no	—
312	#529	—	2	2	no	—
313	#529	C₂²×C₆₁	1	4	yes	complemented
314	C₂×#124	C₂²×C₆₁	1	2	yes	—
315	#525	—	2	2	no	—
316	#525	C₂²×C₆₁	1	2	yes	—
317	C₂×#121	—	2	2	no	—
318	C₄×#41	—	2	2	no	—
319	C₄×#41	C₂²×C₆₁	1	2	yes	—
320	C₂×C₄×D₄	—	2	2	no	—
321	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
322	#497	C₂²×C₆₁	1	4	yes	complemented
323	#496	C₂²×C₆₁	1	4	yes	complemented
324	#495	C₂²×C₆₁	1	8	yes	complemented
325	#489	C₂²×C₆₁	1	8	yes	complemented
326	#488	C₂²×C₆₁	1	8	yes	complemented
327	#462	C₂²×C₆₁	1	8	yes	complemented
328	#461	C₂²×C₆₁	1	8	yes	complemented
329	#460	C₂²×C₆₁	1	8	yes	complemented
330	#459	C₂²×C₆₁	1	8	yes	complemented
331	#452	C₂²×C₆₁	1	16	yes	complemented
332	#449	C₂²×C₆₁	1	16	yes	complemented
333	#432	C₂²×C₆₁	1	4	yes	complemented
334	#431	C₂²×C₆₁	1	4	yes	complemented
335	#423	C₂²×C₆₁	1	2	yes	—
336	#419	C₂²×C₆₁	1	4	yes	—
337	#418	C₂²×C₆₁	1	4	yes	—
338	#407	C₂²×C₆₁	1	2	yes	—
339	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
340	C₆₁	#60802	1	1	yes	characteristic, direct product factor, Sylow
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	4	no	—
343	C₂×#41	—	4	8	no	—
344	C₂×#41	C₂³×C₆₁	1	2	yes	—
345	C₂×#41	C₂³×C₆₁	1	8	yes	—
346	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
347	C₂²×D₄	—	2	2	no	—
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
350	C₂³×C₄	—	2	2	no	—
351	C₂³×C₄	D₄×C₆₁	1	2	yes	—
352	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
353	#137	—	4	8	no	—
354	AGL(1,ℤ/8)	—	4	8	no	—
355	C₂×Q₁₆	C₂³×C₆₁	1	8	yes	—
356	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
357	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
358	C₂×D₈	C₂³×C₆₁	1	8	yes	—
359	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
360	C₄⋊Q₈	—	2	1	no	—
361	C₄⋊Q₈	—	2	1	no	—
362	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
363	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
364	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
365	#127	—	2	1	no	—
366	#127	—	2	1	no	—
367	#127	C₂³×C₆₁	1	4	yes	—
368	#127	C₂³×C₆₁	1	1	yes	characteristic
369	#126	—	2	4	no	—
370	#125	—	2	2	no	—
371	#125	C₂³×C₆₁	1	2	yes	—
372	#124	—	2	4	no	—
373	#124	—	2	2	no	—
374	#124	C₂³×C₆₁	1	4	yes	—
375	#124	C₂³×C₆₁	1	4	yes	—
376	#124	C₂³×C₆₁	1	2	yes	—
377	#123	—	2	4	no	—
378	#123	—	2	2	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	C₂³×C₆₁	1	4	yes	—
382	#121	—	2	4	no	—
383	#121	—	2	4	no	—
384	#121	—	2	4	no	—
385	#121	—	2	4	no	—
386	#121	—	2	2	no	—
387	#121	—	2	2	no	—
388	#121	C₂³×C₆₁	1	4	yes	—
389	#120	—	2	2	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	4	no	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	—	2	4	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	C₂³×C₆₁	1	4	yes	—
403	#117	—	2	2	no	—
404	#117	—	2	4	no	—
405	#117	—	2	2	no	—
406	#117	C₂³×C₆₁	1	2	yes	—
407	C₂×C₄⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	—	2	4	no	—
409	C₂×C₄²	—	2	2	no	—
410	C₂×C₄²	D₄×C₆₁	1	2	yes	—
411	C₂×C₄²	D₄×C₆₁	1	2	yes	—
412	C₂×C₄²	D₄×C₆₁	1	2	yes	—
413	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
416	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
417	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
418	#103	C₂³×C₆₁	1	16	yes	—
419	#102	C₂³×C₆₁	1	16	yes	—
420	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
421	#97	C₂³×C₆₁	1	4	yes	—
422	C₄×C₈	C₂³×C₆₁	1	4	yes	—
423	C₂⁴	—	2	2	no	—
424	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
425	#41	—	2	4	no	—
426	#41	—	2	4	no	—
427	#41	—	4	16	no	—
428	#41	—	4	16	no	—
429	#41	—	4	8	no	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	2	no	—
432	C₂×Q₈	—	2	8	no	—
433	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
434	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	8	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	4	8	no	—
444	C₂×D₄	—	4	8	no	—
445	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
446	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	8	no	—
452	C₂²×C₄	—	4	4	no	—
453	C₂²×C₄	—	4	8	no	—
454	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
457	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
458	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
460	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
461	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
464	Q₁₆	—	4	8	no	—
465	SD₁₆	—	4	8	no	—
466	SD₁₆	—	4	8	no	—
467	D₈	—	4	8	no	—
468	M₁₆	—	2	8	no	—
469	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	2	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₄²	—	2	4	no	—
486	C₄²	—	2	2	no	—
487	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
488	C₄²	C₂×D₄×C₆₁	1	2	yes	—
489	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
490	C₄²	C₂×D₄×C₆₁	1	2	yes	—
491	C₄²	C₂×D₄×C₆₁	1	2	yes	—
492	C₄²	C₂×D₄×C₆₁	1	2	yes	—
493	C₄²	C₂×D₄×C₆₁	1	2	yes	—
494	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
495	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
496	C₄²	C₂⁴×C₆₁	1	2	yes	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	8	no	—
502	C₂³	—	4	4	no	—
503	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
504	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
505	Q₈	—	2	2	no	—
506	Q₈	—	2	8	no	—
507	Q₈	—	2	2	no	—
508	Q₈	—	4	4	no	—
509	Q₈	—	4	4	no	—
510	D₄	—	2	2	no	—
511	D₄	—	2	8	no	—
512	D₄	—	2	2	no	—
513	D₄	—	4	8	no	—
514	D₄	—	4	4	no	—
515	D₄	—	4	16	no	—
516	D₄	—	4	8	no	—
517	D₄	—	4	4	no	—
518	D₄	—	4	4	no	—
519	D₄	—	4	8	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	16	no	—
533	C₂×C₄	—	4	4	no	—
534	C₂×C₄	—	4	4	no	—
535	C₂×C₄	—	4	8	no	—
536	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
537	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
538	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
540	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
542	C₂×C₄	H(5,𝔽₂)×C₆₁	1	4	yes	—
543	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
544	C₈	—	2	8	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	8	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	8	8	no	—
555	C₂²	C₆₁×#535	1	2	yes	—
556	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	4	4	no	—
562	C₄	—	4	4	no	—
563	C₄	—	4	2	no	—
564	C₄	C₆₁×#535	1	2	yes	—
565	C₄	C₆₁×#575	1	4	yes	—
566	C₄	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
567	C₂	—	2	2	no	—
568	C₂	—	4	2	no	—
569	C₂	—	8	4	no	—
570	C₂	—	8	2	no	—
571	C₂	C₂×C₆₁×#535	1	1	yes	characteristic
572	C₂	C₂×C₆₁×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3444	C₂	semidirect
2. C₂×C₆₁×#532	C₂	semidirect
3. C₆₁×#3216	C₂	semidirect
4. C₆₁×#3214	C₂	semidirect
5. C₆₁×#3212	C₂	semidirect
6. C₆₁×#3153	C₂	semidirect
7. C₆₁×#3140	C₂	semidirect
8. C₆₁×#3039	C₂	semidirect
9. C₆₁×#3038	C₂	semidirect
10. C₆₁×#3005	C₂	semidirect
11. C₆₁×#2920	C₂	semidirect
16. C₆₁×#533	C₂²	semidirect
22. C₆₁×#532	C₂²	semidirect
27. C₆₁×#529	C₂²	semidirect
36. C₆₁×#497	C₂²	semidirect
37. C₆₁×#496	C₂²	semidirect
38. C₆₁×#495	C₂²	semidirect
39. C₆₁×#489	C₂²	semidirect
40. C₆₁×#488	C₂²	semidirect
41. C₆₁×#462	C₂²	semidirect
42. C₆₁×#461	C₂²	semidirect
43. C₆₁×#460	C₂²	semidirect
44. C₆₁×#459	C₂²	semidirect
45. C₆₁×#452	C₂²	semidirect
46. C₆₁×#449	C₂²	semidirect
47. C₆₁×#432	C₂²	semidirect
48. C₆₁×#431	C₂²	semidirect
258. #60802	C₆₁	direct (N × H)
281. #3444	C₂×C₆₁	semidirect
282. C₂×#532	C₂×C₆₁	semidirect
283. #3216	C₂×C₆₁	semidirect
284. #3214	C₂×C₆₁	semidirect
285. #3212	C₂×C₆₁	semidirect
286. #3153	C₂×C₆₁	semidirect
287. #3140	C₂×C₆₁	semidirect
288. #3039	C₂×C₆₁	semidirect
289. #3038	C₂×C₆₁	semidirect
290. #3005	C₂×C₆₁	semidirect
291. #2920	C₂×C₆₁	semidirect
302. #533	C₂²×C₆₁	semidirect
308. #532	C₂²×C₆₁	semidirect
313. #529	C₂²×C₆₁	semidirect
322. #497	C₂²×C₆₁	semidirect
323. #496	C₂²×C₆₁	semidirect
324. #495	C₂²×C₆₁	semidirect
325. #489	C₂²×C₆₁	semidirect
326. #488	C₂²×C₆₁	semidirect
327. #462	C₂²×C₆₁	semidirect
328. #461	C₂²×C₆₁	semidirect
329. #460	C₂²×C₆₁	semidirect
330. #459	C₂²×C₆₁	semidirect
331. #452	C₂²×C₆₁	semidirect
332. #449	C₂²×C₆₁	semidirect
333. #432	C₂²×C₆₁	semidirect
334. #431	C₂²×C₆₁	semidirect
340. C₆₁	#60802	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3444	2
C₂×C₆₁×#532	2
C₆₁×#3216	2
C₆₁×#3214	2
C₆₁×#3212	2
C₆₁×#3153	2
C₆₁×#3140	2
C₆₁×#3039	2
C₆₁×#3038	2
C₆₁×#3005	2
C₆₁×#2920	2
#60802	61

Subgroups — C61×#60802 #29564235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#60802 #29564235