Subgroups — C₆₁×#19356 #29522789

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3251	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2966	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1686	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1685	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1670	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1557	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₆₁	—	2	1	no	—
13	C₂×#135×C₆₁	—	2	1	no	—
14	C₆₁×#532	C₂²	1	2	yes	—
15	C₆₁×#529	C₂²	1	1	yes	characteristic
16	C₆₁×#495	—	2	1	no	—
17	C₆₁×#495	—	2	1	no	—
18	C₆₁×#494	C₂²	1	1	yes	characteristic
19	C₆₁×#493	C₂²	1	1	yes	characteristic
20	C₆₁×#492	C₂²	1	2	yes	—
21	C₆₁×#478	—	2	2	no	—
22	C₆₁×#475	C₂²	1	2	yes	—
23	C₆₁×#474	C₂²	1	2	yes	—
24	C₆₁×#469	—	2	2	no	—
25	C₆₁×#468	—	2	2	no	—
26	C₆₁×#463	—	2	2	no	—
27	C₆₁×#460	C₂²	1	2	yes	—
28	C₆₁×#459	C₂²	1	2	yes	—
29	C₆₁×#446	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₆₁×#436	C₂²	1	1	yes	characteristic
32	C₆₁×#435	—	2	1	no	—
33	C₈×D₄×C₆₁	—	2	1	no	—
34	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#433	—	2	1	no	—
36	C₆₁×#432	C₂²	1	1	yes	characteristic
37	C₆₁×#423	C₂²	1	2	yes	—
38	C₆₁×#405	—	2	1	no	—
39	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
40	C₆₁×#335	C₂²	1	1	yes	characteristic
41	C₆₁×#335	C₂²	1	1	yes	characteristic
42	C₆₁×#335	C₂²	1	2	yes	—
43	C₆₁×#332	C₂²	1	2	yes	—
44	C₆₁×#332	C₂²	1	2	yes	—
45	C₆₁×#331	C₂²	1	2	yes	—
46	C₆₁×#331	C₂²	1	2	yes	—
47	C₂×#41×C₆₁	—	2	2	no	—
48	#135×C₆₁	—	4	4	no	—
49	#135×C₆₁	—	4	4	no	—
50	C₂×Q₁₆×C₆₁	—	2	1	no	—
51	C₂×Q₁₆×C₆₁	—	2	1	no	—
52	C₂×SD₁₆×C₆₁	—	2	2	no	—
53	C₂×SD₁₆×C₆₁	—	2	2	no	—
54	C₂×SD₁₆×C₆₁	—	2	2	no	—
55	C₂×D₈×C₆₁	—	2	2	no	—
56	C₂×D₈×C₆₁	—	2	1	no	—
57	C₂×D₈×C₆₁	—	2	1	no	—
58	C₂×M₁₆×C₆₁	—	2	2	no	—
59	C₂×M₁₆×C₆₁	—	2	1	no	—
60	C₂²×C₈×C₆₁	—	2	1	no	—
61	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
64	#127×C₆₁	C₂³	1	2	yes	—
65	#126×C₆₁	—	2	1	no	—
66	#125×C₆₁	D₄	1	1	yes	characteristic
67	#124×C₆₁	—	2	2	no	—
68	#124×C₆₁	D₄	1	1	yes	characteristic
69	#123×C₆₁	—	2	1	no	—
70	C₂²⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
71	#121×C₆₁	—	2	2	no	—
72	#121×C₆₁	—	2	2	no	—
73	#121×C₆₁	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₄×D₄×C₆₁	—	2	1	no	—
76	C₄×D₄×C₆₁	—	2	2	no	—
77	C₄×D₄×C₆₁	—	2	2	no	—
78	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
79	#117×C₆₁	—	2	1	no	—
80	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
81	#107×C₆₁	—	2	2	no	—
82	#106×C₆₁	—	2	2	no	—
83	C₄⋊C₈×C₆₁	—	2	1	no	—
84	C₄⋊C₈×C₆₁	—	2	1	no	—
85	C₄⋊C₈×C₆₁	—	2	2	no	—
86	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
88	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
89	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
90	#103×C₆₁	—	2	2	no	—
91	#102×C₆₁	—	2	2	no	—
92	#102×C₆₁	—	2	2	no	—
93	#102×C₆₁	—	2	2	no	—
94	C₂²⋊C₈×C₆₁	—	2	1	no	—
95	C₂²⋊C₈×C₆₁	—	2	1	no	—
96	C₂²⋊C₈×C₆₁	—	2	1	no	—
97	#97×C₆₁	—	2	1	no	—
98	#97×C₆₁	—	2	1	no	—
99	C₄×C₈×C₆₁	—	2	1	no	—
100	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
101	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
102	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
103	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
104	#41×C₆₁	—	4	4	no	—
105	#41×C₆₁	—	4	4	no	—
106	C₂×Q₈×C₆₁	—	2	2	no	—
107	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄×C₆₁	—	2	1	no	—
109	C₂×D₄×C₆₁	—	2	2	no	—
110	C₂×D₄×C₆₁	—	2	2	no	—
111	C₂×D₄×C₆₁	—	2	2	no	—
112	C₂×D₄×C₆₁	—	4	2	no	—
113	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₁	—	2	1	no	—
115	C₂²×C₄×C₆₁	—	4	2	no	—
116	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
119	Q₁₆×C₆₁	—	4	2	no	—
120	Q₁₆×C₆₁	—	4	2	no	—
121	SD₁₆×C₆₁	—	4	4	no	—
122	SD₁₆×C₆₁	—	4	4	no	—
123	SD₁₆×C₆₁	—	8	2	no	—
124	D₈×C₆₁	—	4	2	no	—
125	D₈×C₆₁	—	4	2	no	—
126	D₈×C₆₁	—	8	2	no	—
127	M₁₆×C₆₁	—	4	2	no	—
128	M₁₆×C₆₁	—	8	2	no	—
129	C₂×C₈×C₆₁	—	2	2	no	—
130	C₂×C₈×C₆₁	—	2	2	no	—
131	C₂×C₈×C₆₁	—	2	1	no	—
132	C₂×C₈×C₆₁	—	2	1	no	—
133	C₂×C₈×C₆₁	—	2	2	no	—
134	C₂×C₈×C₆₁	—	2	2	no	—
135	C₂×C₈×C₆₁	—	4	2	no	—
136	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₆₁	—	2	1	no	—
141	C₄⋊C₄×C₆₁	—	2	1	no	—
142	C₄⋊C₄×C₆₁	—	2	1	no	—
143	C₄⋊C₄×C₆₁	—	2	1	no	—
144	C₄⋊C₄×C₆₁	—	2	2	no	—
145	C₄⋊C₄×C₆₁	—	2	2	no	—
146	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
147	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
148	C₂²⋊C₄×C₆₁	—	2	1	no	—
149	C₂²⋊C₄×C₆₁	—	2	1	no	—
150	C₂²⋊C₄×C₆₁	—	2	1	no	—
151	C₂²⋊C₄×C₆₁	—	2	1	no	—
152	C₂²⋊C₄×C₆₁	—	2	1	no	—
153	C₂²⋊C₄×C₆₁	—	4	2	no	—
154	C₂²⋊C₄×C₆₁	—	4	2	no	—
155	C₄²×C₆₁	—	2	1	no	—
156	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₆₁	#41	1	1	yes	characteristic
159	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
160	C₂³×C₆₁	—	2	1	no	—
161	C₂³×C₆₁	—	4	2	no	—
162	C₂³×C₆₁	#121	1	1	yes	characteristic
163	Q₈×C₆₁	—	2	1	no	—
164	Q₈×C₆₁	—	2	1	no	—
165	Q₈×C₆₁	—	4	2	no	—
166	Q₈×C₆₁	—	4	2	no	—
167	D₄×C₆₁	—	2	1	no	—
168	D₄×C₆₁	—	2	1	no	—
169	D₄×C₆₁	—	4	2	no	—
170	D₄×C₆₁	—	4	2	no	—
171	D₄×C₆₁	—	4	2	no	—
172	D₄×C₆₁	—	4	2	no	—
173	D₄×C₆₁	—	4	2	no	—
174	D₄×C₆₁	—	8	2	no	—
175	D₄×C₆₁	—	8	4	no	—
176	C₂×C₄×C₆₁	—	2	1	no	—
177	C₂×C₄×C₆₁	—	2	1	no	—
178	C₂×C₄×C₆₁	—	2	1	no	—
179	C₂×C₄×C₆₁	—	2	1	no	—
180	C₂×C₄×C₆₁	—	2	1	no	—
181	C₂×C₄×C₆₁	—	2	1	no	—
182	C₂×C₄×C₆₁	—	2	1	no	—
183	C₂×C₄×C₆₁	—	4	1	no	—
184	C₂×C₄×C₆₁	—	4	1	no	—
185	C₂×C₄×C₆₁	—	4	1	no	—
186	C₂×C₄×C₆₁	—	4	2	no	—
187	C₂×C₄×C₆₁	—	8	4	no	—
188	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
189	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
190	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
191	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
194	C₈×C₆₁	—	2	1	no	—
195	C₈×C₆₁	—	2	1	no	—
196	C₈×C₆₁	—	4	1	no	—
197	C₈×C₆₁	—	4	1	no	—
198	C₈×C₆₁	—	4	2	no	—
199	C₈×C₆₁	—	4	2	no	—
200	C₈×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
201	C₈×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
202	#19356	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂²×C₆₁	—	2	1	no	—
204	C₂²×C₆₁	—	2	1	no	—
205	C₂²×C₆₁	—	2	1	no	—
206	C₂²×C₆₁	—	4	1	no	—
207	C₂²×C₆₁	—	4	1	no	—
208	C₂²×C₆₁	—	4	1	no	—
209	C₂²×C₆₁	—	8	2	no	—
210	C₂²×C₆₁	—	8	2	no	—
211	C₂²×C₆₁	—	8	2	no	—
212	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
213	C₄×C₆₁	—	2	1	no	—
214	C₄×C₆₁	—	2	1	no	—
215	C₄×C₆₁	—	2	1	no	—
216	C₄×C₆₁	—	2	1	no	—
217	C₄×C₆₁	—	4	1	no	—
218	C₄×C₆₁	—	4	1	no	—
219	C₄×C₆₁	—	4	1	no	—
220	C₄×C₆₁	—	8	2	no	—
221	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₄×C₆₁	#576	1	1	yes	characteristic
224	C₄×C₆₁	#578	1	1	yes	characteristic
225	#3251	C₂×C₆₁	1	2	yes	complemented
226	#3147	C₂×C₆₁	1	1	yes	characteristic, complemented
227	#2966	C₂×C₆₁	1	1	yes	characteristic, complemented
228	#1686	C₂×C₆₁	1	2	yes	complemented
229	#1685	C₂×C₆₁	1	2	yes	complemented
230	#1670	C₂×C₆₁	1	1	yes	characteristic, complemented
231	#1669	C₂×C₆₁	1	1	yes	characteristic, complemented
232	#1668	C₂×C₆₁	1	1	yes	characteristic, complemented
233	#1667	C₂×C₆₁	1	1	yes	characteristic, complemented
234	#1567	C₂×C₆₁	1	1	yes	characteristic, complemented
235	#1557	C₂×C₆₁	1	2	yes	complemented
236	C₂×C₆₁	—	4	1	no	—
237	C₂×C₆₁	—	8	1	no	—
238	C₂×C₆₁	—	16	2	no	—
239	C₂×C₆₁	#3038	1	1	yes	characteristic
240	C₂×C₆₁	#3061	1	1	yes	characteristic
241	C₂×C₆₁	#3067	1	1	yes	characteristic
242	C₂×#135	—	2	1	no	—
243	C₂×#135	—	2	1	no	—
244	#532	C₂²×C₆₁	1	2	yes	—
245	#529	C₂²×C₆₁	1	1	yes	characteristic
246	#495	—	2	1	no	—
247	#495	—	2	1	no	—
248	#494	C₂²×C₆₁	1	1	yes	characteristic
249	#493	C₂²×C₆₁	1	1	yes	characteristic
250	#492	C₂²×C₆₁	1	2	yes	—
251	#478	—	2	2	no	—
252	#475	C₂²×C₆₁	1	2	yes	—
253	#474	C₂²×C₆₁	1	2	yes	—
254	#469	—	2	2	no	—
255	#468	—	2	2	no	—
256	#463	—	2	2	no	—
257	#460	C₂²×C₆₁	1	2	yes	—
258	#459	C₂²×C₆₁	1	2	yes	—
259	#446	C₂²×C₆₁	1	1	yes	characteristic
260	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
261	#436	C₂²×C₆₁	1	1	yes	characteristic
262	#435	—	2	1	no	—
263	C₈×D₄	—	2	1	no	—
264	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic
265	#433	—	2	1	no	—
266	#432	C₂²×C₆₁	1	1	yes	characteristic
267	#423	C₂²×C₆₁	1	2	yes	—
268	#405	—	2	1	no	—
269	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
270	#335	C₂²×C₆₁	1	1	yes	characteristic
271	#335	C₂²×C₆₁	1	1	yes	characteristic
272	#335	C₂²×C₆₁	1	2	yes	—
273	#332	C₂²×C₆₁	1	2	yes	—
274	#332	C₂²×C₆₁	1	2	yes	—
275	#331	C₂²×C₆₁	1	2	yes	—
276	#331	C₂²×C₆₁	1	2	yes	—
277	C₆₁	#19356	1	1	yes	characteristic, direct product factor, Sylow
278	C₂×#41	—	2	2	no	—
279	#135	—	4	4	no	—
280	#135	—	4	4	no	—
281	C₂×Q₁₆	—	2	1	no	—
282	C₂×Q₁₆	—	2	1	no	—
283	C₂×SD₁₆	—	2	2	no	—
284	C₂×SD₁₆	—	2	2	no	—
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×D₈	—	2	2	no	—
287	C₂×D₈	—	2	1	no	—
288	C₂×D₈	—	2	1	no	—
289	C₂×M₁₆	—	2	2	no	—
290	C₂×M₁₆	—	2	1	no	—
291	C₂²×C₈	—	2	1	no	—
292	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
293	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
294	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
295	#127	C₂³×C₆₁	1	2	yes	—
296	#126	—	2	1	no	—
297	#125	D₄×C₆₁	1	1	yes	characteristic
298	#124	—	2	2	no	—
299	#124	D₄×C₆₁	1	1	yes	characteristic
300	#123	—	2	1	no	—
301	C₂²⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
302	#121	—	2	2	no	—
303	#121	—	2	2	no	—
304	#121	D₄×C₆₁	1	1	yes	characteristic
305	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	—	2	2	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
310	#117	—	2	1	no	—
311	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
312	#107	—	2	2	no	—
313	#106	—	2	2	no	—
314	C₄⋊C₈	—	2	1	no	—
315	C₄⋊C₈	—	2	1	no	—
316	C₄⋊C₈	—	2	2	no	—
317	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
318	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
319	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
320	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
321	#103	—	2	2	no	—
322	#102	—	2	2	no	—
323	#102	—	2	2	no	—
324	#102	—	2	2	no	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₂²⋊C₈	—	2	1	no	—
328	#97	—	2	1	no	—
329	#97	—	2	1	no	—
330	C₄×C₈	—	2	1	no	—
331	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
332	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
333	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
334	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
335	#41	—	4	4	no	—
336	#41	—	4	4	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	4	2	no	—
344	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	4	2	no	—
347	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
349	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
350	Q₁₆	—	4	2	no	—
351	Q₁₆	—	4	2	no	—
352	SD₁₆	—	4	4	no	—
353	SD₁₆	—	4	4	no	—
354	SD₁₆	—	8	2	no	—
355	D₈	—	4	2	no	—
356	D₈	—	4	2	no	—
357	D₈	—	8	2	no	—
358	M₁₆	—	4	2	no	—
359	M₁₆	—	8	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	4	2	no	—
367	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
368	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
369	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
370	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
378	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
379	C₂²⋊C₄	—	2	1	no	—
380	C₂²⋊C₄	—	2	1	no	—
381	C₂²⋊C₄	—	2	1	no	—
382	C₂²⋊C₄	—	2	1	no	—
383	C₂²⋊C₄	—	2	1	no	—
384	C₂²⋊C₄	—	4	2	no	—
385	C₂²⋊C₄	—	4	2	no	—
386	C₄²	—	2	1	no	—
387	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
388	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
389	C₄²	#41×C₆₁	1	1	yes	characteristic
390	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
391	C₂³	—	2	1	no	—
392	C₂³	—	4	2	no	—
393	C₂³	#121×C₆₁	1	1	yes	characteristic
394	Q₈	—	2	1	no	—
395	Q₈	—	2	1	no	—
396	Q₈	—	4	2	no	—
397	Q₈	—	4	2	no	—
398	D₄	—	2	1	no	—
399	D₄	—	2	1	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	2	no	—
405	D₄	—	8	2	no	—
406	D₄	—	8	4	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	8	4	no	—
419	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
420	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
421	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
425	C₈	—	2	1	no	—
426	C₈	—	2	1	no	—
427	C₈	—	4	1	no	—
428	C₈	—	4	1	no	—
429	C₈	—	4	2	no	—
430	C₈	—	4	2	no	—
431	C₈	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
432	C₈	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
433	C₂²	—	2	1	no	—
434	C₂²	—	2	1	no	—
435	C₂²	—	2	1	no	—
436	C₂²	—	4	1	no	—
437	C₂²	—	4	1	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	8	2	no	—
442	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	4	1	no	—
448	C₄	—	4	1	no	—
449	C₄	—	4	1	no	—
450	C₄	—	8	2	no	—
451	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
452	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
453	C₄	C₆₁×#576	1	1	yes	characteristic
454	C₄	C₆₁×#578	1	1	yes	characteristic
455	C₂	—	4	1	no	—
456	C₂	—	8	1	no	—
457	C₂	—	16	2	no	—
458	C₂	C₆₁×#3038	1	1	yes	characteristic
459	C₂	C₆₁×#3061	1	1	yes	characteristic
460	C₂	C₆₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
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×#3251	C₂	semidirect
2. C₆₁×#3147	C₂	semidirect
3. C₆₁×#2966	C₂	semidirect
4. C₆₁×#1686	C₂	semidirect
5. C₆₁×#1685	C₂	semidirect
6. C₆₁×#1670	C₂	semidirect
7. C₆₁×#1669	C₂	semidirect
8. C₆₁×#1668	C₂	semidirect
9. C₆₁×#1667	C₂	semidirect
10. C₆₁×#1567	C₂	semidirect
11. C₆₁×#1557	C₂	semidirect
202. #19356	C₆₁	direct (N × H)
225. #3251	C₂×C₆₁	semidirect
226. #3147	C₂×C₆₁	semidirect
227. #2966	C₂×C₆₁	semidirect
228. #1686	C₂×C₆₁	semidirect
229. #1685	C₂×C₆₁	semidirect
230. #1670	C₂×C₆₁	semidirect
231. #1669	C₂×C₆₁	semidirect
232. #1668	C₂×C₆₁	semidirect
233. #1667	C₂×C₆₁	semidirect
234. #1567	C₂×C₆₁	semidirect
235. #1557	C₂×C₆₁	semidirect
277. C₆₁	#19356	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×#3251	2
C₆₁×#3147	2
C₆₁×#2966	2
C₆₁×#1686	2
C₆₁×#1685	2
C₆₁×#1670	2
C₆₁×#1669	2
C₆₁×#1668	2
C₆₁×#1667	2
C₆₁×#1567	2
C₆₁×#1557	2
#19356	61

Subgroups — C61×#19356 #29522789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19356 #29522789