Subgroups — C₆₁×#24808 #29528241

Order 15616.

820 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3069	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3068	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#471	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₆₁×#471	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2955	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2070	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2069	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#2049	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#2048	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1977	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₁×#1903	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₁×#1902	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₁×#1813	C₂	1	1	yes	maximal, characteristic, complemented
16	C₆₁×#575	C₂²	1	1	yes	characteristic
17	C₂×#137×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#137×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#568	C₂²	1	1	yes	characteristic
23	C₆₁×#568	C₂²	1	1	yes	characteristic
24	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#473	—	2	2	no	—
27	C₆₁×#472	—	2	2	no	—
28	C₆₁×#471	—	2	2	no	—
29	C₆₁×#471	—	2	2	no	—
30	C₆₁×#471	—	2	2	no	—
31	C₆₁×#471	—	2	2	no	—
32	C₆₁×#471	—	2	2	no	—
33	C₆₁×#471	—	2	2	no	—
34	C₆₁×#444	—	2	2	no	—
35	C₆₁×#443	—	2	2	no	—
36	C₆₁×#430	—	2	2	no	—
37	C₆₁×#430	C₂²	1	1	yes	characteristic
38	C₆₁×#430	C₂²	1	1	yes	characteristic
39	C₆₁×#430	C₂²	1	1	yes	characteristic
40	C₆₁×#430	C₂²	1	1	yes	characteristic
41	C₆₁×#430	C₂²	1	1	yes	characteristic
42	C₂×#108×C₆₁	—	2	2	no	—
43	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
44	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
45	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
46	C₆₁×#421	C₂²	1	1	yes	characteristic
47	C₆₁×#421	C₂²	1	1	yes	characteristic
48	C₆₁×#419	C₂²	1	1	yes	characteristic
49	C₆₁×#418	C₂²	1	1	yes	characteristic
50	C₆₁×#413	C₂²	1	1	yes	characteristic
51	C₆₁×#413	C₂²	1	1	yes	characteristic
52	C₂×#101×C₆₁	C₂²	1	1	yes	characteristic
53	C₂×#101×C₆₁	C₂²	1	1	yes	characteristic
54	C₂×#101×C₆₁	C₂²	1	1	yes	characteristic
55	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
56	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
57	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
58	C₆₁×#405	C₂²	1	1	yes	characteristic
59	C₆₁×#344	C₂²	1	1	yes	characteristic
60	C₆₁×#344	C₂²	1	1	yes	characteristic
61	C₆₁×#338	C₂²	1	1	yes	characteristic
62	C₆₁×#338	C₂²	1	1	yes	characteristic
63	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
64	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
67	#137×C₆₁	—	2	1	no	—
68	#137×C₆₁	—	2	2	no	—
69	#137×C₆₁	—	2	2	no	—
70	#137×C₆₁	—	2	2	no	—
71	#137×C₆₁	—	4	1	no	—
72	#137×C₆₁	D₄	1	2	yes	—
73	AGL(1,ℤ/8)×C₆₁	—	2	1	no	—
74	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
75	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
76	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
77	AGL(1,ℤ/8)×C₆₁	—	4	1	no	—
78	AGL(1,ℤ/8)×C₆₁	D₄	1	2	yes	—
79	#135×C₆₁	—	2	2	no	—
80	C₂×Q₁₆×C₆₁	—	2	1	no	—
81	C₂×Q₁₆×C₆₁	—	2	1	no	—
82	C₂×SD₁₆×C₆₁	—	2	1	no	—
83	C₂×SD₁₆×C₆₁	—	2	1	no	—
84	C₂×SD₁₆×C₆₁	—	2	1	no	—
85	C₂×SD₁₆×C₆₁	—	2	1	no	—
86	C₂×SD₁₆×C₆₁	—	2	1	no	—
87	C₂×SD₁₆×C₆₁	—	2	1	no	—
88	C₂×SD₁₆×C₆₁	—	2	1	no	—
89	C₂×SD₁₆×C₆₁	—	2	1	no	—
90	C₂×SD₁₆×C₆₁	—	2	1	no	—
91	C₂×SD₁₆×C₆₁	—	2	1	no	—
92	C₂×D₈×C₆₁	—	2	1	no	—
93	C₂×D₈×C₆₁	—	2	1	no	—
94	#131×C₆₁	—	2	1	no	—
95	#131×C₆₁	—	2	1	no	—
96	#131×C₆₁	—	2	2	no	—
97	#131×C₆₁	—	2	2	no	—
98	#131×C₆₁	—	2	1	no	—
99	#131×C₆₁	—	2	1	no	—
100	#131×C₆₁	—	2	1	no	—
101	#131×C₆₁	—	2	1	no	—
102	#131×C₆₁	—	4	1	no	—
103	#131×C₆₁	—	4	1	no	—
104	#131×C₆₁	D₄	1	2	yes	—
105	#131×C₆₁	D₄	1	2	yes	—
106	C₂×M₁₆×C₆₁	—	2	1	no	—
107	C₂×M₁₆×C₆₁	—	2	1	no	—
108	C₂×M₁₆×C₆₁	—	2	2	no	—
109	C₂×M₁₆×C₆₁	—	2	2	no	—
110	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
112	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
113	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
114	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
115	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
116	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
117	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
118	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
119	C₂²×C₈×C₆₁	—	2	1	no	—
120	C₂²×C₈×C₆₁	—	2	1	no	—
121	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
122	#117×C₆₁	C₂³	1	1	yes	characteristic
123	#108×C₆₁	—	2	2	no	—
124	#108×C₆₁	—	2	2	no	—
125	#108×C₆₁	—	2	2	no	—
126	#108×C₆₁	—	2	2	no	—
127	#108×C₆₁	—	2	2	no	—
128	#108×C₆₁	—	2	1	no	—
129	#108×C₆₁	—	2	1	no	—
130	#108×C₆₁	—	2	1	no	—
131	#108×C₆₁	—	2	1	no	—
132	#108×C₆₁	D₄	1	2	yes	—
133	#108×C₆₁	D₄	1	2	yes	—
134	#104×C₆₁	—	2	2	no	—
135	#104×C₆₁	—	2	2	no	—
136	#103×C₆₁	—	4	1	no	—
137	#102×C₆₁	—	4	1	no	—
138	#101×C₆₁	—	2	1	no	—
139	#101×C₆₁	—	2	1	no	—
140	#101×C₆₁	—	2	2	no	—
141	#101×C₆₁	—	2	2	no	—
142	#101×C₆₁	—	2	2	no	—
143	#100×C₆₁	—	2	1	no	—
144	#100×C₆₁	—	2	1	no	—
145	#100×C₆₁	—	2	2	no	—
146	#100×C₆₁	—	2	2	no	—
147	#100×C₆₁	—	2	2	no	—
148	#97×C₆₁	—	2	1	no	—
149	C₄×C₈×C₆₁	—	2	1	no	—
150	C₂⁴×C₆₁	—	2	1	no	—
151	#41×C₆₁	—	2	1	no	—
152	#41×C₆₁	—	2	1	no	—
153	#41×C₆₁	—	2	1	no	—
154	#41×C₆₁	—	2	1	no	—
155	#41×C₆₁	—	2	1	no	—
156	#41×C₆₁	—	2	1	no	—
157	#41×C₆₁	C₂×D₄	1	2	yes	—
158	#41×C₆₁	C₂×D₄	1	2	yes	—
159	C₂×Q₈×C₆₁	—	2	1	no	—
160	C₂×Q₈×C₆₁	—	2	1	no	—
161	C₂×Q₈×C₆₁	—	2	1	no	—
162	C₂×Q₈×C₆₁	—	2	1	no	—
163	C₂×Q₈×C₆₁	—	2	1	no	—
164	C₂×Q₈×C₆₁	—	2	1	no	—
165	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₂×Q₈×C₆₁	#41	1	1	yes	characteristic
167	C₂×D₄×C₆₁	—	2	1	no	—
168	C₂×D₄×C₆₁	—	2	1	no	—
169	C₂×D₄×C₆₁	—	2	1	no	—
170	C₂×D₄×C₆₁	—	2	1	no	—
171	C₂×D₄×C₆₁	—	2	1	no	—
172	C₂×D₄×C₆₁	—	2	1	no	—
173	C₂×D₄×C₆₁	—	2	1	no	—
174	C₂×D₄×C₆₁	—	2	1	no	—
175	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
176	C₂×D₄×C₆₁	#41	1	1	yes	characteristic
177	C₂²×C₄×C₆₁	—	2	1	no	—
178	C₂²×C₄×C₆₁	—	2	1	no	—
179	C₂²×C₄×C₆₁	—	2	1	no	—
180	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
181	Q₁₆×C₆₁	—	2	2	no	—
182	Q₁₆×C₆₁	—	4	2	no	—
183	Q₁₆×C₆₁	—	4	2	no	—
184	SD₁₆×C₆₁	—	2	2	no	—
185	SD₁₆×C₆₁	—	2	2	no	—
186	SD₁₆×C₆₁	—	4	2	no	—
187	SD₁₆×C₆₁	—	4	2	no	—
188	SD₁₆×C₆₁	—	4	2	no	—
189	SD₁₆×C₆₁	—	4	2	no	—
190	SD₁₆×C₆₁	—	4	1	no	—
191	SD₁₆×C₆₁	—	4	1	no	—
192	SD₁₆×C₆₁	—	4	1	no	—
193	SD₁₆×C₆₁	—	4	1	no	—
194	D₈×C₆₁	—	2	2	no	—
195	D₈×C₆₁	—	4	2	no	—
196	D₈×C₆₁	—	4	2	no	—
197	M₁₆×C₆₁	—	2	1	no	—
198	M₁₆×C₆₁	—	2	1	no	—
199	M₁₆×C₆₁	—	2	1	no	—
200	M₁₆×C₆₁	—	2	1	no	—
201	M₁₆×C₆₁	—	2	2	no	—
202	M₁₆×C₆₁	—	2	2	no	—
203	M₁₆×C₆₁	—	2	1	no	—
204	M₁₆×C₆₁	—	2	1	no	—
205	M₁₆×C₆₁	—	2	1	no	—
206	M₁₆×C₆₁	—	2	1	no	—
207	M₁₆×C₆₁	—	2	1	no	—
208	M₁₆×C₆₁	—	2	1	no	—
209	M₁₆×C₆₁	—	2	1	no	—
210	M₁₆×C₆₁	—	2	1	no	—
211	M₁₆×C₆₁	—	2	1	no	—
212	M₁₆×C₆₁	—	2	1	no	—
213	M₁₆×C₆₁	—	4	2	no	—
214	M₁₆×C₆₁	—	4	2	no	—
215	M₁₆×C₆₁	—	4	1	no	—
216	M₁₆×C₆₁	—	4	1	no	—
217	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
218	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
219	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
220	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
221	C₂×C₈×C₆₁	—	2	1	no	—
222	C₂×C₈×C₆₁	—	2	1	no	—
223	C₂×C₈×C₆₁	—	2	1	no	—
224	C₂×C₈×C₆₁	—	2	1	no	—
225	C₂×C₈×C₆₁	—	2	2	no	—
226	C₂×C₈×C₆₁	—	2	2	no	—
227	C₂×C₈×C₆₁	—	2	1	no	—
228	C₂×C₈×C₆₁	—	2	1	no	—
229	C₂×C₈×C₆₁	—	2	1	no	—
230	C₂×C₈×C₆₁	—	2	1	no	—
231	C₂×C₈×C₆₁	—	2	1	no	—
232	C₂×C₈×C₆₁	—	4	2	no	—
233	C₂×C₈×C₆₁	—	4	2	no	—
234	C₂×C₈×C₆₁	—	4	1	no	—
235	C₂×C₈×C₆₁	—	4	1	no	—
236	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
237	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
238	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
239	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
240	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
241	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
242	C₄⋊C₄×C₆₁	—	2	1	no	—
243	C₂²⋊C₄×C₆₁	—	2	1	no	—
244	C₄²×C₆₁	—	2	1	no	—
245	C₂³×C₆₁	—	2	1	no	—
246	C₂³×C₆₁	—	2	1	no	—
247	C₂³×C₆₁	—	4	1	no	—
248	C₂³×C₆₁	—	4	1	no	—
249	C₂³×C₆₁	—	4	1	no	—
250	C₂³×C₆₁	—	16	1	no	—
251	C₂³×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
252	Q₈×C₆₁	—	2	1	no	—
253	Q₈×C₆₁	—	2	1	no	—
254	Q₈×C₆₁	—	2	1	no	—
255	Q₈×C₆₁	—	4	1	no	—
256	Q₈×C₆₁	—	4	1	no	—
257	Q₈×C₆₁	—	4	1	no	—
258	Q₈×C₆₁	—	4	1	no	—
259	Q₈×C₆₁	#121	1	2	yes	—
260	D₄×C₆₁	—	2	2	no	—
261	D₄×C₆₁	—	2	2	no	—
262	D₄×C₆₁	—	2	1	no	—
263	D₄×C₆₁	—	2	1	no	—
264	D₄×C₆₁	—	2	1	no	—
265	D₄×C₆₁	—	4	1	no	—
266	D₄×C₆₁	—	4	1	no	—
267	D₄×C₆₁	—	4	1	no	—
268	D₄×C₆₁	—	4	1	no	—
269	D₄×C₆₁	—	4	1	no	—
270	D₄×C₆₁	—	4	1	no	—
271	D₄×C₆₁	#121	1	2	yes	—
272	C₂×C₄×C₆₁	—	2	1	no	—
273	C₂×C₄×C₆₁	—	2	2	no	—
274	C₂×C₄×C₆₁	—	2	2	no	—
275	C₂×C₄×C₆₁	—	2	1	no	—
276	C₂×C₄×C₆₁	—	4	1	no	—
277	C₂×C₄×C₆₁	—	4	1	no	—
278	C₂×C₄×C₆₁	—	4	1	no	—
279	C₂×C₄×C₆₁	—	4	1	no	—
280	C₂×C₄×C₆₁	—	4	1	no	—
281	C₂×C₄×C₆₁	—	4	1	no	—
282	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
287	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
288	C₈×C₆₁	—	2	2	no	—
289	C₈×C₆₁	—	2	2	no	—
290	C₈×C₆₁	—	2	2	no	—
291	C₈×C₆₁	—	2	2	no	—
292	C₈×C₆₁	—	4	1	no	—
293	C₈×C₆₁	—	4	1	no	—
294	C₈×C₆₁	—	4	1	no	—
295	C₈×C₆₁	—	4	1	no	—
296	C₈×C₆₁	—	4	1	no	—
297	C₈×C₆₁	#121	1	1	yes	characteristic
298	C₈×C₆₁	#121	1	1	yes	characteristic
299	C₈×C₆₁	#121	1	1	yes	characteristic
300	C₈×C₆₁	#121	1	1	yes	characteristic
301	#24808	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂²×C₆₁	—	2	2	no	—
303	C₂²×C₆₁	—	4	1	no	—
304	C₂²×C₆₁	—	4	1	no	—
305	C₂²×C₆₁	—	8	1	no	—
306	C₂²×C₆₁	—	8	1	no	—
307	C₂²×C₆₁	—	8	1	no	—
308	C₂²×C₆₁	—	16	1	no	—
309	C₂²×C₆₁	—	16	1	no	—
310	C₂²×C₆₁	—	16	1	no	—
311	C₂²×C₆₁	#537	1	1	yes	characteristic
312	C₂²×C₆₁	#547	1	1	yes	characteristic
313	C₂²×C₆₁	#548	1	1	yes	characteristic
314	C₄×C₆₁	—	2	2	no	—
315	C₄×C₆₁	—	4	1	no	—
316	C₄×C₆₁	—	8	1	no	—
317	C₄×C₆₁	—	8	1	no	—
318	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
319	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
320	C₄×C₆₁	D₄²	1	1	yes	characteristic
321	C₄×C₆₁	#553	1	1	yes	characteristic
322	#3069	C₂×C₆₁	1	1	yes	characteristic, complemented
323	#3068	C₂×C₆₁	1	1	yes	characteristic
324	C₂×#471	C₂×C₆₁	1	1	yes	characteristic, complemented
325	C₂×#471	C₂×C₆₁	1	1	yes	characteristic, complemented
326	#2955	C₂×C₆₁	1	1	yes	characteristic, complemented
327	#2916	C₂×C₆₁	1	1	yes	characteristic, complemented
328	#2070	C₂×C₆₁	1	1	yes	characteristic, complemented
329	#2069	C₂×C₆₁	1	1	yes	characteristic, complemented
330	#2049	C₂×C₆₁	1	1	yes	characteristic, complemented
331	#2048	C₂×C₆₁	1	1	yes	characteristic, complemented
332	#1977	C₂×C₆₁	1	1	yes	characteristic, complemented
333	#1953	C₂×C₆₁	1	1	yes	characteristic, complemented
334	#1903	C₂×C₆₁	1	1	yes	characteristic, complemented
335	#1902	C₂×C₆₁	1	1	yes	characteristic, complemented
336	#1813	C₂×C₆₁	1	1	yes	characteristic, complemented
337	C₂×C₆₁	—	2	1	no	—
338	C₂×C₆₁	—	2	1	no	—
339	C₂×C₆₁	—	2	1	no	—
340	C₂×C₆₁	—	4	2	no	—
341	C₂×C₆₁	—	8	1	no	—
342	C₂×C₆₁	—	16	1	no	—
343	C₂×C₆₁	#2416	1	1	yes	characteristic
344	#575	C₂²×C₆₁	1	1	yes	characteristic
345	C₂×#137	C₂²×C₆₁	1	1	yes	characteristic
346	C₂×#137	C₂²×C₆₁	1	1	yes	characteristic
347	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	1	yes	characteristic
348	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	1	yes	characteristic
349	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
350	#568	C₂²×C₆₁	1	1	yes	characteristic
351	#568	C₂²×C₆₁	1	1	yes	characteristic
352	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
353	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
354	#473	—	2	2	no	—
355	#472	—	2	2	no	—
356	#471	—	2	2	no	—
357	#471	—	2	2	no	—
358	#471	—	2	2	no	—
359	#471	—	2	2	no	—
360	#471	—	2	2	no	—
361	#471	—	2	2	no	—
362	#444	—	2	2	no	—
363	#443	—	2	2	no	—
364	#430	—	2	2	no	—
365	#430	C₂²×C₆₁	1	1	yes	characteristic
366	#430	C₂²×C₆₁	1	1	yes	characteristic
367	#430	C₂²×C₆₁	1	1	yes	characteristic
368	#430	C₂²×C₆₁	1	1	yes	characteristic
369	#430	C₂²×C₆₁	1	1	yes	characteristic
370	C₂×#108	—	2	2	no	—
371	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
372	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
373	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
374	#421	C₂²×C₆₁	1	1	yes	characteristic
375	#421	C₂²×C₆₁	1	1	yes	characteristic
376	#419	C₂²×C₆₁	1	1	yes	characteristic
377	#418	C₂²×C₆₁	1	1	yes	characteristic
378	#413	C₂²×C₆₁	1	1	yes	characteristic
379	#413	C₂²×C₆₁	1	1	yes	characteristic
380	C₂×#101	C₂²×C₆₁	1	1	yes	characteristic
381	C₂×#101	C₂²×C₆₁	1	1	yes	characteristic
382	C₂×#101	C₂²×C₆₁	1	1	yes	characteristic
383	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
384	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
385	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
386	#405	C₂²×C₆₁	1	1	yes	characteristic
387	#344	C₂²×C₆₁	1	1	yes	characteristic
388	#344	C₂²×C₆₁	1	1	yes	characteristic
389	#338	C₂²×C₆₁	1	1	yes	characteristic
390	#338	C₂²×C₆₁	1	1	yes	characteristic
391	C₆₁	#24808	1	1	yes	characteristic, direct product factor, Sylow
392	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
393	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
394	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
395	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
396	#137	—	2	1	no	—
397	#137	—	2	2	no	—
398	#137	—	2	2	no	—
399	#137	—	2	2	no	—
400	#137	—	4	1	no	—
401	#137	D₄×C₆₁	1	2	yes	—
402	AGL(1,ℤ/8)	—	2	1	no	—
403	AGL(1,ℤ/8)	—	2	2	no	—
404	AGL(1,ℤ/8)	—	2	2	no	—
405	AGL(1,ℤ/8)	—	2	2	no	—
406	AGL(1,ℤ/8)	—	4	1	no	—
407	AGL(1,ℤ/8)	D₄×C₆₁	1	2	yes	—
408	#135	—	2	2	no	—
409	C₂×Q₁₆	—	2	1	no	—
410	C₂×Q₁₆	—	2	1	no	—
411	C₂×SD₁₆	—	2	1	no	—
412	C₂×SD₁₆	—	2	1	no	—
413	C₂×SD₁₆	—	2	1	no	—
414	C₂×SD₁₆	—	2	1	no	—
415	C₂×SD₁₆	—	2	1	no	—
416	C₂×SD₁₆	—	2	1	no	—
417	C₂×SD₁₆	—	2	1	no	—
418	C₂×SD₁₆	—	2	1	no	—
419	C₂×SD₁₆	—	2	1	no	—
420	C₂×SD₁₆	—	2	1	no	—
421	C₂×D₈	—	2	1	no	—
422	C₂×D₈	—	2	1	no	—
423	#131	—	2	1	no	—
424	#131	—	2	1	no	—
425	#131	—	2	2	no	—
426	#131	—	2	2	no	—
427	#131	—	2	1	no	—
428	#131	—	2	1	no	—
429	#131	—	2	1	no	—
430	#131	—	2	1	no	—
431	#131	—	4	1	no	—
432	#131	—	4	1	no	—
433	#131	D₄×C₆₁	1	2	yes	—
434	#131	D₄×C₆₁	1	2	yes	—
435	C₂×M₁₆	—	2	1	no	—
436	C₂×M₁₆	—	2	1	no	—
437	C₂×M₁₆	—	2	2	no	—
438	C₂×M₁₆	—	2	2	no	—
439	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
440	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
441	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
442	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
443	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
444	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
445	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
446	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
447	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
448	C₂²×C₈	—	2	1	no	—
449	C₂²×C₈	—	2	1	no	—
450	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
451	#117	C₂³×C₆₁	1	1	yes	characteristic
452	#108	—	2	2	no	—
453	#108	—	2	2	no	—
454	#108	—	2	2	no	—
455	#108	—	2	2	no	—
456	#108	—	2	2	no	—
457	#108	—	2	1	no	—
458	#108	—	2	1	no	—
459	#108	—	2	1	no	—
460	#108	—	2	1	no	—
461	#108	D₄×C₆₁	1	2	yes	—
462	#108	D₄×C₆₁	1	2	yes	—
463	#104	—	2	2	no	—
464	#104	—	2	2	no	—
465	#103	—	4	1	no	—
466	#102	—	4	1	no	—
467	#101	—	2	1	no	—
468	#101	—	2	1	no	—
469	#101	—	2	2	no	—
470	#101	—	2	2	no	—
471	#101	—	2	2	no	—
472	#100	—	2	1	no	—
473	#100	—	2	1	no	—
474	#100	—	2	2	no	—
475	#100	—	2	2	no	—
476	#100	—	2	2	no	—
477	#97	—	2	1	no	—
478	C₄×C₈	—	2	1	no	—
479	C₂⁴	—	2	1	no	—
480	#41	—	2	1	no	—
481	#41	—	2	1	no	—
482	#41	—	2	1	no	—
483	#41	—	2	1	no	—
484	#41	—	2	1	no	—
485	#41	—	2	1	no	—
486	#41	C₂×D₄×C₆₁	1	2	yes	—
487	#41	C₂×D₄×C₆₁	1	2	yes	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	1	no	—
493	C₂×Q₈	—	2	1	no	—
494	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
495	C₂×Q₈	#41×C₆₁	1	1	yes	characteristic
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
505	C₂×D₄	#41×C₆₁	1	1	yes	characteristic
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
510	Q₁₆	—	2	2	no	—
511	Q₁₆	—	4	2	no	—
512	Q₁₆	—	4	2	no	—
513	SD₁₆	—	2	2	no	—
514	SD₁₆	—	2	2	no	—
515	SD₁₆	—	4	2	no	—
516	SD₁₆	—	4	2	no	—
517	SD₁₆	—	4	2	no	—
518	SD₁₆	—	4	2	no	—
519	SD₁₆	—	4	1	no	—
520	SD₁₆	—	4	1	no	—
521	SD₁₆	—	4	1	no	—
522	SD₁₆	—	4	1	no	—
523	D₈	—	2	2	no	—
524	D₈	—	4	2	no	—
525	D₈	—	4	2	no	—
526	M₁₆	—	2	1	no	—
527	M₁₆	—	2	1	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	2	no	—
531	M₁₆	—	2	2	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	2	1	no	—
535	M₁₆	—	2	1	no	—
536	M₁₆	—	2	1	no	—
537	M₁₆	—	2	1	no	—
538	M₁₆	—	2	1	no	—
539	M₁₆	—	2	1	no	—
540	M₁₆	—	2	1	no	—
541	M₁₆	—	2	1	no	—
542	M₁₆	—	4	2	no	—
543	M₁₆	—	4	2	no	—
544	M₁₆	—	4	1	no	—
545	M₁₆	—	4	1	no	—
546	M₁₆	C₂×D₄×C₆₁	1	2	yes	—
547	M₁₆	C₂×D₄×C₆₁	1	2	yes	—
548	M₁₆	C₂×D₄×C₆₁	1	2	yes	—
549	M₁₆	C₂×D₄×C₆₁	1	2	yes	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	2	no	—
555	C₂×C₈	—	2	2	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	4	2	no	—
562	C₂×C₈	—	4	2	no	—
563	C₂×C₈	—	4	1	no	—
564	C₂×C₈	—	4	1	no	—
565	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
566	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
567	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
568	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
569	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
570	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
571	C₄⋊C₄	—	2	1	no	—
572	C₂²⋊C₄	—	2	1	no	—
573	C₄²	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	—	4	1	no	—
577	C₂³	—	4	1	no	—
578	C₂³	—	4	1	no	—
579	C₂³	—	16	1	no	—
580	C₂³	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
581	Q₈	—	2	1	no	—
582	Q₈	—	2	1	no	—
583	Q₈	—	2	1	no	—
584	Q₈	—	4	1	no	—
585	Q₈	—	4	1	no	—
586	Q₈	—	4	1	no	—
587	Q₈	—	4	1	no	—
588	Q₈	#121×C₆₁	1	2	yes	—
589	D₄	—	2	2	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	1	no	—
592	D₄	—	2	1	no	—
593	D₄	—	2	1	no	—
594	D₄	—	4	1	no	—
595	D₄	—	4	1	no	—
596	D₄	—	4	1	no	—
597	D₄	—	4	1	no	—
598	D₄	—	4	1	no	—
599	D₄	—	4	1	no	—
600	D₄	#121×C₆₁	1	2	yes	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
612	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
613	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
614	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
615	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
616	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
617	C₈	—	2	2	no	—
618	C₈	—	2	2	no	—
619	C₈	—	2	2	no	—
620	C₈	—	2	2	no	—
621	C₈	—	4	1	no	—
622	C₈	—	4	1	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	1	no	—
625	C₈	—	4	1	no	—
626	C₈	#121×C₆₁	1	1	yes	characteristic
627	C₈	#121×C₆₁	1	1	yes	characteristic
628	C₈	#121×C₆₁	1	1	yes	characteristic
629	C₈	#121×C₆₁	1	1	yes	characteristic
630	C₂²	—	2	2	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	8	1	no	—
634	C₂²	—	8	1	no	—
635	C₂²	—	8	1	no	—
636	C₂²	—	16	1	no	—
637	C₂²	—	16	1	no	—
638	C₂²	—	16	1	no	—
639	C₂²	C₆₁×#537	1	1	yes	characteristic
640	C₂²	C₆₁×#547	1	1	yes	characteristic
641	C₂²	C₆₁×#548	1	1	yes	characteristic
642	C₄	—	2	2	no	—
643	C₄	—	4	1	no	—
644	C₄	—	8	1	no	—
645	C₄	—	8	1	no	—
646	C₄	C₂×#121×C₆₁	1	1	yes	characteristic
647	C₄	C₂×#121×C₆₁	1	1	yes	characteristic
648	C₄	D₄²×C₆₁	1	1	yes	characteristic
649	C₄	C₆₁×#553	1	1	yes	characteristic
650	C₂	—	2	1	no	—
651	C₂	—	2	1	no	—
652	C₂	—	2	1	no	—
653	C₂	—	4	2	no	—
654	C₂	—	8	1	no	—
655	C₂	—	16	1	no	—
656	C₂	C₆₁×#2416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	#2416
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#3068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3069	C₂	semidirect
3. C₂×C₆₁×#471	C₂	semidirect
4. C₂×C₆₁×#471	C₂	semidirect
5. C₆₁×#2955	C₂	semidirect
6. C₆₁×#2916	C₂	semidirect
7. C₆₁×#2070	C₂	semidirect
8. C₆₁×#2069	C₂	semidirect
9. C₆₁×#2049	C₂	semidirect
10. C₆₁×#2048	C₂	semidirect
11. C₆₁×#1977	C₂	semidirect
12. C₆₁×#1953	C₂	semidirect
13. C₆₁×#1903	C₂	semidirect
14. C₆₁×#1902	C₂	semidirect
15. C₆₁×#1813	C₂	semidirect
301. #24808	C₆₁	direct (N × H)
322. #3069	C₂×C₆₁	semidirect
324. C₂×#471	C₂×C₆₁	semidirect
325. C₂×#471	C₂×C₆₁	semidirect
326. #2955	C₂×C₆₁	semidirect
327. #2916	C₂×C₆₁	semidirect
328. #2070	C₂×C₆₁	semidirect
329. #2069	C₂×C₆₁	semidirect
330. #2049	C₂×C₆₁	semidirect
331. #2048	C₂×C₆₁	semidirect
332. #1977	C₂×C₆₁	semidirect
333. #1953	C₂×C₆₁	semidirect
334. #1903	C₂×C₆₁	semidirect
335. #1902	C₂×C₆₁	semidirect
336. #1813	C₂×C₆₁	semidirect
391. C₆₁	#24808	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×#3069	2
C₆₁×#3068	2
C₂×C₆₁×#471	2
C₆₁×#2955	2
C₆₁×#2916	2
C₆₁×#2070	2
C₆₁×#2069	2
C₆₁×#2049	2
C₆₁×#2048	2
C₆₁×#1977	2
C₆₁×#1953	2
C₆₁×#1903	2
C₆₁×#1902	2
C₆₁×#1813	2
#24808	61

Subgroups — C61×#24808 #29528241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24808 #29528241