Subgroups — C₆₁×#19352 #29522785

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3239	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3217	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2966	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1686	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1685	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1666	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1665	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1556	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1555	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₁	—	2	1	no	—
13	C₂²×SD₁₆×C₆₁	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#529	C₂²	1	1	yes	characteristic
17	C₆₁×#501	—	2	1	no	—
18	C₆₁×#496	—	2	1	no	—
19	C₆₁×#492	—	2	1	no	—
20	C₆₁×#492	—	2	1	no	—
21	C₆₁×#492	C₂²	1	2	yes	—
22	C₆₁×#492	C₂²	1	2	yes	—
23	C₆₁×#490	—	2	1	no	—
24	C₆₁×#489	—	2	1	no	—
25	C₆₁×#484	—	2	1	no	—
26	C₆₁×#481	—	2	1	no	—
27	C₆₁×#475	C₂²	1	2	yes	complemented
28	C₆₁×#474	C₂²	1	2	yes	complemented
29	C₆₁×#460	C₂²	1	2	yes	—
30	C₆₁×#459	C₂²	1	2	yes	—
31	C₆₁×#450	—	2	1	no	—
32	C₆₁×#447	—	2	1	no	—
33	C₆₁×#446	C₂²	1	1	yes	characteristic, complemented
34	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
35	C₆₁×#436	C₂²	1	1	yes	characteristic, complemented
36	C₆₁×#435	—	2	1	no	—
37	C₈×D₄×C₆₁	—	2	1	no	—
38	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
39	C₆₁×#433	—	2	1	no	—
40	C₆₁×#432	C₂²	1	1	yes	characteristic
41	C₆₁×#432	C₂²	1	1	yes	characteristic
42	C₆₁×#432	C₂²	1	1	yes	characteristic
43	C₆₁×#405	—	2	1	no	—
44	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
45	C₆₁×#335	C₂²	1	2	yes	complemented
46	C₆₁×#334	C₂²	1	2	yes	complemented
47	C₆₁×#332	C₂²	1	2	yes	complemented
48	C₆₁×#332	C₂²	1	2	yes	complemented
49	C₆₁×#331	C₂²	1	2	yes	—
50	C₆₁×#331	C₂²	1	2	yes	—
51	C₂²×Q₈×C₆₁	—	2	1	no	—
52	C₂²×D₄×C₆₁	—	2	1	no	—
53	C₂²×D₄×C₆₁	—	2	1	no	—
54	C₂²×D₄×C₆₁	—	2	1	no	—
55	C₂×SD₁₆×C₆₁	—	2	2	no	—
56	C₂×SD₁₆×C₆₁	—	2	2	no	—
57	C₂×SD₁₆×C₆₁	—	2	2	no	—
58	C₂×SD₁₆×C₆₁	—	2	2	no	—
59	C₂×SD₁₆×C₆₁	—	2	2	no	—
60	C₂×SD₁₆×C₆₁	—	4	4	no	—
61	C₂×SD₁₆×C₆₁	—	4	4	no	—
62	C₂×D₈×C₆₁	—	2	2	no	—
63	C₂×M₁₆×C₆₁	—	2	1	no	—
64	C₂²×C₈×C₆₁	—	2	1	no	—
65	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₁	—	2	1	no	—
68	C₄⋊Q₈×C₆₁	—	2	1	no	—
69	C₄⋊Q₈×C₆₁	—	2	1	no	—
70	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
71	#127×C₆₁	—	2	1	no	—
72	#127×C₆₁	—	2	1	no	—
73	#127×C₆₁	—	2	1	no	—
74	#127×C₆₁	C₂³	1	2	yes	—
75	#126×C₆₁	—	2	1	no	—
76	#125×C₆₁	D₄	1	1	yes	characteristic
77	#124×C₆₁	D₄	1	1	yes	characteristic
78	#123×C₆₁	—	2	1	no	—
79	C₂²⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
80	#121×C₆₁	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₆₁	—	2	1	no	—
83	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
84	#117×C₆₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
88	#107×C₆₁	—	2	2	no	—
89	#106×C₆₁	—	2	2	no	—
90	C₄⋊C₈×C₆₁	—	2	1	no	—
91	C₄⋊C₈×C₆₁	—	2	1	no	—
92	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
95	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
96	#103×C₆₁	—	2	2	no	—
97	#102×C₆₁	—	2	2	no	—
98	#102×C₆₁	—	2	2	no	—
99	#102×C₆₁	—	2	2	no	—
100	C₂²⋊C₈×C₆₁	—	2	1	no	—
101	C₂²⋊C₈×C₆₁	—	2	1	no	—
102	C₂²⋊C₈×C₆₁	—	2	1	no	—
103	C₂²⋊C₈×C₆₁	—	2	1	no	—
104	C₂²⋊C₈×C₆₁	—	2	1	no	—
105	#97×C₆₁	—	2	1	no	—
106	#97×C₆₁	—	2	1	no	—
107	#97×C₆₁	—	2	1	no	—
108	#97×C₆₁	—	2	1	no	—
109	C₄×C₈×C₆₁	—	2	1	no	—
110	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
111	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
112	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
114	C₂⁴×C₆₁	—	4	1	no	—
115	C₂×Q₈×C₆₁	—	2	2	no	—
116	C₂×Q₈×C₆₁	—	4	2	no	—
117	C₂×Q₈×C₆₁	—	4	2	no	—
118	C₂×Q₈×C₆₁	—	4	1	no	—
119	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
120	C₂×D₄×C₆₁	—	2	1	no	—
121	C₂×D₄×C₆₁	—	2	2	no	—
122	C₂×D₄×C₆₁	—	2	2	no	—
123	C₂×D₄×C₆₁	—	2	2	no	—
124	C₂×D₄×C₆₁	—	4	2	no	—
125	C₂×D₄×C₆₁	—	4	2	no	—
126	C₂×D₄×C₆₁	—	4	1	no	—
127	C₂×D₄×C₆₁	—	4	1	no	—
128	C₂×D₄×C₆₁	—	4	1	no	—
129	C₂×D₄×C₆₁	—	8	2	no	—
130	C₂×D₄×C₆₁	—	8	2	no	—
131	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₁	—	2	1	no	—
133	C₂²×C₄×C₆₁	—	4	1	no	—
134	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
137	SD₁₆×C₆₁	—	4	4	no	—
138	SD₁₆×C₆₁	—	4	4	no	—
139	SD₁₆×C₆₁	—	4	4	no	—
140	SD₁₆×C₆₁	—	4	4	no	—
141	SD₁₆×C₆₁	—	8	2	no	—
142	D₈×C₆₁	—	8	2	no	—
143	M₁₆×C₆₁	—	4	2	no	—
144	C₂×C₈×C₆₁	—	2	2	no	—
145	C₂×C₈×C₆₁	—	2	2	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	2	no	—
150	C₂×C₈×C₆₁	—	4	2	no	—
151	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
155	C₄⋊C₄×C₆₁	—	2	1	no	—
156	C₄⋊C₄×C₆₁	—	2	1	no	—
157	C₄⋊C₄×C₆₁	—	2	1	no	—
158	C₄⋊C₄×C₆₁	—	2	1	no	—
159	C₄⋊C₄×C₆₁	—	2	2	no	—
160	C₄⋊C₄×C₆₁	—	2	2	no	—
161	C₄⋊C₄×C₆₁	—	4	1	no	—
162	C₄⋊C₄×C₆₁	—	4	1	no	—
163	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
165	C₂²⋊C₄×C₆₁	—	2	1	no	—
166	C₂²⋊C₄×C₆₁	—	2	1	no	—
167	C₂²⋊C₄×C₆₁	—	2	1	no	—
168	C₂²⋊C₄×C₆₁	—	2	1	no	—
169	C₂²⋊C₄×C₆₁	—	2	1	no	—
170	C₄²×C₆₁	—	2	1	no	—
171	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₆₁	#41	1	1	yes	characteristic
174	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
175	C₂³×C₆₁	—	2	1	no	—
176	C₂³×C₆₁	—	4	2	no	—
177	C₂³×C₆₁	—	8	2	no	—
178	C₂³×C₆₁	—	8	2	no	—
179	C₂³×C₆₁	—	8	2	no	—
180	C₂³×C₆₁	#121	1	1	yes	characteristic
181	Q₈×C₆₁	—	2	2	no	—
182	Q₈×C₆₁	—	4	2	no	—
183	Q₈×C₆₁	—	4	2	no	—
184	Q₈×C₆₁	—	8	2	no	—
185	D₄×C₆₁	—	2	2	no	—
186	D₄×C₆₁	—	4	2	no	—
187	D₄×C₆₁	—	4	2	no	—
188	D₄×C₆₁	—	4	2	no	—
189	D₄×C₆₁	—	4	2	no	—
190	D₄×C₆₁	—	4	2	no	—
191	D₄×C₆₁	—	8	2	no	—
192	D₄×C₆₁	—	8	2	no	—
193	D₄×C₆₁	—	8	2	no	—
194	D₄×C₆₁	—	8	2	no	—
195	C₂×C₄×C₆₁	—	2	2	no	—
196	C₂×C₄×C₆₁	—	2	1	no	—
197	C₂×C₄×C₆₁	—	2	1	no	—
198	C₂×C₄×C₆₁	—	2	2	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	—	4	1	no	—
203	C₂×C₄×C₆₁	—	4	1	no	—
204	C₂×C₄×C₆₁	—	4	2	no	—
205	C₂×C₄×C₆₁	—	8	2	no	—
206	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
207	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
208	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
209	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
212	C₈×C₆₁	—	2	1	no	—
213	C₈×C₆₁	—	2	1	no	—
214	C₈×C₆₁	—	4	1	no	—
215	C₈×C₆₁	—	4	1	no	—
216	C₈×C₆₁	—	4	2	no	—
217	C₈×C₆₁	—	4	2	no	—
218	C₈×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
219	C₈×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
220	#19352	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂²×C₆₁	—	2	1	no	—
222	C₂²×C₆₁	—	2	1	no	—
223	C₂²×C₆₁	—	4	1	no	—
224	C₂²×C₆₁	—	4	1	no	—
225	C₂²×C₆₁	—	4	1	no	—
226	C₂²×C₆₁	—	8	2	no	—
227	C₂²×C₆₁	—	8	2	no	—
228	C₂²×C₆₁	—	8	2	no	—
229	C₂²×C₆₁	—	16	4	no	—
230	C₂²×C₆₁	#471	1	2	yes	—
231	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
232	C₄×C₆₁	—	2	1	no	—
233	C₄×C₆₁	—	2	1	no	—
234	C₄×C₆₁	—	2	1	no	—
235	C₄×C₆₁	—	2	1	no	—
236	C₄×C₆₁	—	4	1	no	—
237	C₄×C₆₁	—	4	1	no	—
238	C₄×C₆₁	—	4	1	no	—
239	C₄×C₆₁	—	8	2	no	—
240	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₄×C₆₁	#577	1	2	yes	—
243	#3239	C₂×C₆₁	1	1	yes	characteristic, complemented
244	#3217	C₂×C₆₁	1	1	yes	characteristic
245	C₂×#492	C₂×C₆₁	1	1	yes	characteristic, complemented
246	#2966	C₂×C₆₁	1	1	yes	characteristic, complemented
247	#1686	C₂×C₆₁	1	2	yes	complemented
248	#1685	C₂×C₆₁	1	2	yes	complemented
249	#1666	C₂×C₆₁	1	2	yes	complemented
250	#1665	C₂×C₆₁	1	2	yes	complemented
251	#1568	C₂×C₆₁	1	1	yes	characteristic, complemented
252	#1556	C₂×C₆₁	1	1	yes	characteristic, complemented
253	#1555	C₂×C₆₁	1	1	yes	characteristic, complemented
254	C₂×C₆₁	—	2	2	no	—
255	C₂×C₆₁	—	8	1	no	—
256	C₂×C₆₁	—	16	2	no	—
257	C₂×C₆₁	#3038	1	1	yes	characteristic
258	C₂×C₆₁	#3057	1	1	yes	characteristic
259	C₂×C₆₁	C₂×#471	1	1	yes	characteristic
260	C₂²×SD₁₆	—	2	1	no	—
261	C₂²×SD₁₆	—	2	1	no	—
262	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
263	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
264	#529	C₂²×C₆₁	1	1	yes	characteristic
265	#501	—	2	1	no	—
266	#496	—	2	1	no	—
267	#492	—	2	1	no	—
268	#492	—	2	1	no	—
269	#492	C₂²×C₆₁	1	2	yes	—
270	#492	C₂²×C₆₁	1	2	yes	—
271	#490	—	2	1	no	—
272	#489	—	2	1	no	—
273	#484	—	2	1	no	—
274	#481	—	2	1	no	—
275	#475	C₂²×C₆₁	1	2	yes	complemented
276	#474	C₂²×C₆₁	1	2	yes	complemented
277	#460	C₂²×C₆₁	1	2	yes	—
278	#459	C₂²×C₆₁	1	2	yes	—
279	#450	—	2	1	no	—
280	#447	—	2	1	no	—
281	#446	C₂²×C₆₁	1	1	yes	characteristic, complemented
282	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
283	#436	C₂²×C₆₁	1	1	yes	characteristic, complemented
284	#435	—	2	1	no	—
285	C₈×D₄	—	2	1	no	—
286	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic, complemented
287	#433	—	2	1	no	—
288	#432	C₂²×C₆₁	1	1	yes	characteristic
289	#432	C₂²×C₆₁	1	1	yes	characteristic
290	#432	C₂²×C₆₁	1	1	yes	characteristic
291	#405	—	2	1	no	—
292	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
293	#335	C₂²×C₆₁	1	2	yes	complemented
294	#334	C₂²×C₆₁	1	2	yes	complemented
295	#332	C₂²×C₆₁	1	2	yes	complemented
296	#332	C₂²×C₆₁	1	2	yes	complemented
297	#331	C₂²×C₆₁	1	2	yes	—
298	#331	C₂²×C₆₁	1	2	yes	—
299	C₆₁	#19352	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²×Q₈	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂×SD₁₆	—	2	2	no	—
305	C₂×SD₁₆	—	2	2	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	2	no	—
308	C₂×SD₁₆	—	2	2	no	—
309	C₂×SD₁₆	—	4	4	no	—
310	C₂×SD₁₆	—	4	4	no	—
311	C₂×D₈	—	2	2	no	—
312	C₂×M₁₆	—	2	1	no	—
313	C₂²×C₈	—	2	1	no	—
314	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
315	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
316	C₄⋊Q₈	—	2	1	no	—
317	C₄⋊Q₈	—	2	1	no	—
318	C₄⋊Q₈	—	2	1	no	—
319	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
320	#127	—	2	1	no	—
321	#127	—	2	1	no	—
322	#127	—	2	1	no	—
323	#127	C₂³×C₆₁	1	2	yes	—
324	#126	—	2	1	no	—
325	#125	D₄×C₆₁	1	1	yes	characteristic
326	#124	D₄×C₆₁	1	1	yes	characteristic
327	#123	—	2	1	no	—
328	C₂²⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
329	#121	D₄×C₆₁	1	1	yes	characteristic
330	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
331	C₄×D₄	—	2	1	no	—
332	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
333	#117	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
337	#107	—	2	2	no	—
338	#106	—	2	2	no	—
339	C₄⋊C₈	—	2	1	no	—
340	C₄⋊C₈	—	2	1	no	—
341	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
342	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
343	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
344	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
345	#103	—	2	2	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#102	—	2	2	no	—
349	C₂²⋊C₈	—	2	1	no	—
350	C₂²⋊C₈	—	2	1	no	—
351	C₂²⋊C₈	—	2	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	C₂²⋊C₈	—	2	1	no	—
354	#97	—	2	1	no	—
355	#97	—	2	1	no	—
356	#97	—	2	1	no	—
357	#97	—	2	1	no	—
358	C₄×C₈	—	2	1	no	—
359	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
360	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
361	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
362	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
363	C₂⁴	—	4	1	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	4	2	no	—
366	C₂×Q₈	—	4	2	no	—
367	C₂×Q₈	—	4	1	no	—
368	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	4	2	no	—
375	C₂×D₄	—	4	1	no	—
376	C₂×D₄	—	4	1	no	—
377	C₂×D₄	—	4	1	no	—
378	C₂×D₄	—	8	2	no	—
379	C₂×D₄	—	8	2	no	—
380	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	4	1	no	—
383	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
385	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
386	SD₁₆	—	4	4	no	—
387	SD₁₆	—	4	4	no	—
388	SD₁₆	—	4	4	no	—
389	SD₁₆	—	4	4	no	—
390	SD₁₆	—	8	2	no	—
391	D₈	—	8	2	no	—
392	M₁₆	—	4	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	4	2	no	—
400	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
401	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
402	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
403	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	4	1	no	—
411	C₄⋊C₄	—	4	1	no	—
412	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
413	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
414	C₂²⋊C₄	—	2	1	no	—
415	C₂²⋊C₄	—	2	1	no	—
416	C₂²⋊C₄	—	2	1	no	—
417	C₂²⋊C₄	—	2	1	no	—
418	C₂²⋊C₄	—	2	1	no	—
419	C₄²	—	2	1	no	—
420	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
421	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
422	C₄²	#41×C₆₁	1	1	yes	characteristic
423	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
424	C₂³	—	2	1	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	8	2	no	—
427	C₂³	—	8	2	no	—
428	C₂³	—	8	2	no	—
429	C₂³	#121×C₆₁	1	1	yes	characteristic
430	Q₈	—	2	2	no	—
431	Q₈	—	4	2	no	—
432	Q₈	—	4	2	no	—
433	Q₈	—	8	2	no	—
434	D₄	—	2	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	8	2	no	—
441	D₄	—	8	2	no	—
442	D₄	—	8	2	no	—
443	D₄	—	8	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	2	no	—
455	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
456	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
457	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	4	1	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	2	no	—
466	C₈	—	4	2	no	—
467	C₈	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
468	C₈	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	4	1	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	16	4	no	—
478	C₂²	C₆₁×#471	1	2	yes	—
479	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	1	no	—
483	C₄	—	2	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	1	no	—
487	C₄	—	8	2	no	—
488	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
489	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
490	C₄	C₆₁×#577	1	2	yes	—
491	C₂	—	2	2	no	—
492	C₂	—	8	1	no	—
493	C₂	—	16	2	no	—
494	C₂	C₆₁×#3038	1	1	yes	characteristic
495	C₂	C₆₁×#3057	1	1	yes	characteristic
496	C₂	C₂×C₆₁×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3217	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3239	C₂	semidirect
3. C₂×C₆₁×#492	C₂	semidirect
4. C₆₁×#2966	C₂	semidirect
5. C₆₁×#1686	C₂	semidirect
6. C₆₁×#1685	C₂	semidirect
7. C₆₁×#1666	C₂	semidirect
8. C₆₁×#1665	C₂	semidirect
9. C₆₁×#1568	C₂	semidirect
10. C₆₁×#1556	C₂	semidirect
11. C₆₁×#1555	C₂	semidirect
27. C₆₁×#475	C₂²	semidirect
28. C₆₁×#474	C₂²	semidirect
33. C₆₁×#446	C₂²	semidirect
34. C₈×Q₈×C₆₁	C₂²	semidirect
35. C₆₁×#436	C₂²	semidirect
38. C₈×D₄×C₆₁	C₂²	semidirect
45. C₆₁×#335	C₂²	semidirect
46. C₆₁×#334	C₂²	semidirect
47. C₆₁×#332	C₂²	semidirect
48. C₆₁×#332	C₂²	semidirect
220. #19352	C₆₁	direct (N × H)
243. #3239	C₂×C₆₁	semidirect
245. C₂×#492	C₂×C₆₁	semidirect
246. #2966	C₂×C₆₁	semidirect
247. #1686	C₂×C₆₁	semidirect
248. #1685	C₂×C₆₁	semidirect
249. #1666	C₂×C₆₁	semidirect
250. #1665	C₂×C₆₁	semidirect
251. #1568	C₂×C₆₁	semidirect
252. #1556	C₂×C₆₁	semidirect
253. #1555	C₂×C₆₁	semidirect
275. #475	C₂²×C₆₁	semidirect
276. #474	C₂²×C₆₁	semidirect
281. #446	C₂²×C₆₁	semidirect
282. C₈×Q₈	C₂²×C₆₁	semidirect
283. #436	C₂²×C₆₁	semidirect
286. C₈×D₄	C₂²×C₆₁	semidirect
293. #335	C₂²×C₆₁	semidirect
294. #334	C₂²×C₆₁	semidirect
295. #332	C₂²×C₆₁	semidirect
296. #332	C₂²×C₆₁	semidirect
299. C₆₁	#19352	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×#3239	2
C₆₁×#3217	2
C₂×C₆₁×#492	2
C₆₁×#2966	2
C₆₁×#1686	2
C₆₁×#1685	2
C₆₁×#1666	2
C₆₁×#1665	2
C₆₁×#1568	2
C₆₁×#1556	2
C₆₁×#1555	2
#19352	61

Subgroups — C61×#19352 #29522785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19352 #29522785