Subgroups — C₆₁×#18716 #29522149

Order 15616.

1060 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3548	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1622	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1621	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1543	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1484	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
11	C₆₁×#562	—	2	2	no	—
12	C₆₁×#554	—	2	1	no	—
13	C₆₁×#554	C₂²	1	2	yes	complemented
14	C₆₁×#550	—	2	1	no	—
15	C₆₁×#550	C₂²	1	2	yes	complemented
16	C₆₁×#547	—	2	1	no	—
17	C₆₁×#547	—	2	1	no	—
18	C₆₁×#540	—	2	2	no	—
19	C₆₁×#537	—	2	2	no	—
20	C₆₁×#537	—	2	2	no	—
21	C₆₁×#533	C₂²	1	1	yes	characteristic
22	C₆₁×#532	C₂²	1	1	yes	characteristic
23	C₆₁×#532	C₂²	1	1	yes	characteristic
24	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#496	—	2	2	no	—
26	C₆₁×#493	—	2	1	no	—
27	C₆₁×#493	C₂²	1	2	yes	—
28	C₆₁×#492	—	2	1	no	—
29	C₆₁×#492	C₂²	1	2	yes	—
30	C₆₁×#486	—	2	1	no	—
31	C₆₁×#486	—	2	1	no	—
32	C₆₁×#460	C₂²	1	4	yes	—
33	C₆₁×#459	C₂²	1	4	yes	—
34	C₆₁×#450	—	2	2	no	—
35	C₆₁×#447	—	2	2	no	—
36	C₆₁×#431	C₂²	1	1	yes	characteristic
37	C₆₁×#431	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
39	C₆₁×#333	C₂²	1	2	yes	complemented
40	C₆₁×#331	C₂²	1	2	yes	complemented
41	C₆₁×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₆₁×D₄⋊C₈	C₂²	1	4	yes	complemented
43	C₂×#41×C₆₁	—	2	2	no	—
44	C₂×#41×C₆₁	—	2	2	no	—
45	C₂×#41×C₆₁	—	2	1	no	—
46	C₂×#41×C₆₁	D₄	1	2	yes	—
47	C₂²×D₄×C₆₁	—	2	1	no	—
48	C₂²×D₄×C₆₁	—	2	1	no	—
49	C₂²×D₄×C₆₁	—	2	1	no	—
50	AGL(1,ℤ/8)×C₆₁	—	4	8	no	—
51	C₂×SD₁₆×C₆₁	—	2	4	no	—
52	C₂×SD₁₆×C₆₁	—	2	4	no	—
53	C₂×D₈×C₆₁	—	2	4	no	—
54	C₂×D₈×C₆₁	—	2	4	no	—
55	C₂×M₁₆×C₆₁	—	2	1	no	—
56	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
57	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
58	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
60	#127×C₆₁	—	2	2	no	—
61	#127×C₆₁	—	2	1	no	—
62	#127×C₆₁	—	2	1	no	—
63	#127×C₆₁	—	2	1	no	—
64	#127×C₆₁	D₄	1	1	yes	characteristic
65	#127×C₆₁	C₂³	1	2	yes	—
66	#127×C₆₁	C₂³	1	1	yes	characteristic
67	#125×C₆₁	—	2	2	no	—
68	#125×C₆₁	—	2	1	no	—
69	#124×C₆₁	—	2	1	no	—
70	#124×C₆₁	D₄	1	2	yes	—
71	#123×C₆₁	—	2	4	no	—
72	#123×C₆₁	—	2	4	no	—
73	C₂²⋊Q₈×C₆₁	—	2	2	no	—
74	C₂²⋊Q₈×C₆₁	—	2	4	no	—
75	C₂²⋊Q₈×C₆₁	—	2	2	no	—
76	C₂²⋊Q₈×C₆₁	—	2	4	no	—
77	C₂²⋊Q₈×C₆₁	—	2	1	no	—
78	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
79	#121×C₆₁	—	2	4	no	—
80	#121×C₆₁	—	2	4	no	—
81	#121×C₆₁	—	2	2	no	—
82	#121×C₆₁	—	2	4	no	—
83	#121×C₆₁	—	2	4	no	—
84	#121×C₆₁	—	2	1	no	—
85	#121×C₆₁	—	2	1	no	—
86	#121×C₆₁	—	2	1	no	—
87	#121×C₆₁	—	2	2	no	—
88	#121×C₆₁	D₄	1	2	yes	—
89	C₄×Q₈×C₆₁	—	2	1	no	—
90	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
91	C₄×D₄×C₆₁	—	2	2	no	—
92	C₄×D₄×C₆₁	—	2	2	no	—
93	C₄×D₄×C₆₁	—	2	1	no	—
94	C₄×D₄×C₆₁	—	2	1	no	—
95	C₄×D₄×C₆₁	—	2	1	no	—
96	C₄×D₄×C₆₁	C₂³	1	2	yes	—
97	#117×C₆₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
102	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
103	#102×C₆₁	—	2	4	no	—
104	#102×C₆₁	—	2	4	no	—
105	C₂²⋊C₈×C₆₁	—	2	1	no	—
106	C₂²⋊C₈×C₆₁	—	2	1	no	—
107	#97×C₆₁	—	2	1	no	—
108	C₄×C₈×C₆₁	—	2	1	no	—
109	C₄×C₈×C₆₁	—	2	1	no	—
110	C₄×C₈×C₆₁	C₂³	1	2	yes	—
111	C₂⁴×C₆₁	—	4	1	no	—
112	#41×C₆₁	—	4	4	no	—
113	#41×C₆₁	—	8	2	no	—
114	#41×C₆₁	—	8	4	no	—
115	#41×C₆₁	—	8	4	no	—
116	C₂×Q₈×C₆₁	—	2	1	no	—
117	C₂×Q₈×C₆₁	—	2	1	no	—
118	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
119	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₆₁	—	2	4	no	—
121	C₂×D₄×C₆₁	—	2	4	no	—
122	C₂×D₄×C₆₁	—	2	2	no	—
123	C₂×D₄×C₆₁	—	2	2	no	—
124	C₂×D₄×C₆₁	—	2	1	no	—
125	C₂×D₄×C₆₁	—	2	1	no	—
126	C₂×D₄×C₆₁	—	2	1	no	—
127	C₂×D₄×C₆₁	—	2	1	no	—
128	C₂×D₄×C₆₁	—	2	2	no	—
129	C₂×D₄×C₆₁	—	2	2	no	—
130	C₂×D₄×C₆₁	—	2	2	no	—
131	C₂×D₄×C₆₁	—	4	4	no	—
132	C₂×D₄×C₆₁	—	4	1	no	—
133	C₂×D₄×C₆₁	—	4	1	no	—
134	C₂×D₄×C₆₁	—	4	1	no	—
135	C₂×D₄×C₆₁	—	8	2	no	—
136	C₂×D₄×C₆₁	—	8	2	no	—
137	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₆₁	—	2	4	no	—
140	C₂²×C₄×C₆₁	—	2	4	no	—
141	C₂²×C₄×C₆₁	—	2	2	no	—
142	C₂²×C₄×C₆₁	—	2	2	no	—
143	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
146	SD₁₆×C₆₁	—	4	8	no	—
147	SD₁₆×C₆₁	—	8	4	no	—
148	D₈×C₆₁	—	4	8	no	—
149	D₈×C₆₁	—	8	4	no	—
150	M₁₆×C₆₁	—	2	4	no	—
151	C₂×C₈×C₆₁	—	2	2	no	—
152	C₂×C₈×C₆₁	—	2	2	no	—
153	C₂×C₈×C₆₁	—	2	2	no	—
154	C₄⋊C₄×C₆₁	—	2	4	no	—
155	C₄⋊C₄×C₆₁	—	2	4	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	1	no	—
160	C₄⋊C₄×C₆₁	—	2	2	no	—
161	C₄⋊C₄×C₆₁	—	2	1	no	—
162	C₄⋊C₄×C₆₁	—	2	1	no	—
163	C₄⋊C₄×C₆₁	—	2	2	no	—
164	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₆₁	—	2	4	no	—
167	C₂²⋊C₄×C₆₁	—	2	4	no	—
168	C₂²⋊C₄×C₆₁	—	2	2	no	—
169	C₂²⋊C₄×C₆₁	—	2	2	no	—
170	C₂²⋊C₄×C₆₁	—	2	2	no	—
171	C₂²⋊C₄×C₆₁	—	2	2	no	—
172	C₂²⋊C₄×C₆₁	—	2	2	no	—
173	C₄²×C₆₁	—	2	2	no	—
174	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
178	C₂³×C₆₁	—	2	2	no	—
179	C₂³×C₆₁	—	2	2	no	—
180	C₂³×C₆₁	—	4	2	no	—
181	C₂³×C₆₁	—	8	2	no	—
182	C₂³×C₆₁	—	8	2	no	—
183	C₂³×C₆₁	—	8	2	no	—
184	C₂³×C₆₁	#120	1	1	yes	characteristic
185	Q₈×C₆₁	—	2	4	no	—
186	Q₈×C₆₁	—	4	2	no	—
187	Q₈×C₆₁	—	4	2	no	—
188	Q₈×C₆₁	—	8	1	no	—
189	D₄×C₆₁	—	2	4	no	—
190	D₄×C₆₁	—	4	2	no	—
191	D₄×C₆₁	—	4	4	no	—
192	D₄×C₆₁	—	4	2	no	—
193	D₄×C₆₁	—	4	4	no	—
194	D₄×C₆₁	—	4	2	no	—
195	D₄×C₆₁	—	4	4	no	—
196	D₄×C₆₁	—	4	2	no	—
197	D₄×C₆₁	—	8	1	no	—
198	D₄×C₆₁	—	8	2	no	—
199	D₄×C₆₁	—	8	2	no	—
200	D₄×C₆₁	—	8	2	no	—
201	D₄×C₆₁	—	8	2	no	—
202	D₄×C₆₁	—	8	4	no	—
203	D₄×C₆₁	—	8	4	no	—
204	D₄×C₆₁	—	8	2	no	—
205	C₂×C₄×C₆₁	—	2	2	no	—
206	C₂×C₄×C₆₁	—	2	2	no	—
207	C₂×C₄×C₆₁	—	2	2	no	—
208	C₂×C₄×C₆₁	—	2	2	no	—
209	C₂×C₄×C₆₁	—	2	2	no	—
210	C₂×C₄×C₆₁	—	2	2	no	—
211	C₂×C₄×C₆₁	—	2	2	no	—
212	C₂×C₄×C₆₁	—	4	4	no	—
213	C₂×C₄×C₆₁	—	4	4	no	—
214	C₂×C₄×C₆₁	—	8	4	no	—
215	C₂×C₄×C₆₁	—	8	4	no	—
216	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
217	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
218	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
219	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
222	C₈×C₆₁	—	2	4	no	—
223	C₈×C₆₁	—	4	2	no	—
224	C₈×C₆₁	—	4	2	no	—
225	#18716	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂²×C₆₁	—	2	1	no	—
227	C₂²×C₆₁	—	2	1	no	—
228	C₂²×C₆₁	—	2	1	no	—
229	C₂²×C₆₁	—	4	2	no	—
230	C₂²×C₆₁	—	4	2	no	—
231	C₂²×C₆₁	—	4	2	no	—
232	C₂²×C₆₁	—	4	2	no	—
233	C₂²×C₆₁	—	4	2	no	—
234	C₂²×C₆₁	—	4	2	no	—
235	C₂²×C₆₁	—	8	2	no	—
236	C₂²×C₆₁	—	8	2	no	—
237	C₂²×C₆₁	—	8	2	no	—
238	C₂²×C₆₁	—	16	4	no	—
239	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
240	C₄×C₆₁	—	2	1	no	—
241	C₄×C₆₁	—	2	1	no	—
242	C₄×C₆₁	—	2	1	no	—
243	C₄×C₆₁	—	4	2	no	—
244	C₄×C₆₁	—	4	2	no	—
245	C₄×C₆₁	—	4	2	no	—
246	C₄×C₆₁	—	4	2	no	—
247	C₄×C₆₁	#453	1	2	yes	—
248	C₄×C₆₁	#576	1	2	yes	—
249	C₄×C₆₁	#577	1	2	yes	—
250	#3548	C₂×C₆₁	1	1	yes	characteristic, complemented
251	#3208	C₂×C₆₁	1	1	yes	characteristic, complemented
252	#3205	C₂×C₆₁	1	1	yes	characteristic, complemented
253	#3151	C₂×C₆₁	1	1	yes	characteristic, complemented
254	#1622	C₂×C₆₁	1	4	yes	complemented
255	#1621	C₂×C₆₁	1	4	yes	complemented
256	#1543	C₂×C₆₁	1	1	yes	characteristic, complemented
257	#1485	C₂×C₆₁	1	1	yes	characteristic, complemented
258	#1484	C₂×C₆₁	1	1	yes	characteristic, complemented
259	C₂×C₆₁	—	4	1	no	—
260	C₂×C₆₁	—	8	2	no	—
261	C₂×C₆₁	—	8	2	no	—
262	C₂×C₆₁	—	16	2	no	—
263	C₂×C₆₁	#3008	1	1	yes	characteristic
264	C₂×C₆₁	#3010	1	1	yes	characteristic
265	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
266	C₂×AGL(1,ℤ/8)	—	2	2	no	—
267	#562	—	2	2	no	—
268	#554	—	2	1	no	—
269	#554	C₂²×C₆₁	1	2	yes	complemented
270	#550	—	2	1	no	—
271	#550	C₂²×C₆₁	1	2	yes	complemented
272	#547	—	2	1	no	—
273	#547	—	2	1	no	—
274	#540	—	2	2	no	—
275	#537	—	2	2	no	—
276	#537	—	2	2	no	—
277	#533	C₂²×C₆₁	1	1	yes	characteristic
278	#532	C₂²×C₆₁	1	1	yes	characteristic
279	#532	C₂²×C₆₁	1	1	yes	characteristic
280	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
281	#496	—	2	2	no	—
282	#493	—	2	1	no	—
283	#493	C₂²×C₆₁	1	2	yes	—
284	#492	—	2	1	no	—
285	#492	C₂²×C₆₁	1	2	yes	—
286	#486	—	2	1	no	—
287	#486	—	2	1	no	—
288	#460	C₂²×C₆₁	1	4	yes	—
289	#459	C₂²×C₆₁	1	4	yes	—
290	#450	—	2	2	no	—
291	#447	—	2	2	no	—
292	#431	C₂²×C₆₁	1	1	yes	characteristic
293	#431	C₂²×C₆₁	1	1	yes	characteristic
294	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
295	#333	C₂²×C₆₁	1	2	yes	complemented
296	#331	C₂²×C₆₁	1	2	yes	complemented
297	Q₈⋊C₈	C₂²×C₆₁	1	4	yes	complemented
298	D₄⋊C₈	C₂²×C₆₁	1	4	yes	complemented
299	C₆₁	#18716	1	1	yes	characteristic, direct product factor, Sylow
300	C₂×#41	—	2	2	no	—
301	C₂×#41	—	2	2	no	—
302	C₂×#41	—	2	1	no	—
303	C₂×#41	D₄×C₆₁	1	2	yes	—
304	C₂²×D₄	—	2	1	no	—
305	C₂²×D₄	—	2	1	no	—
306	C₂²×D₄	—	2	1	no	—
307	AGL(1,ℤ/8)	—	4	8	no	—
308	C₂×SD₁₆	—	2	4	no	—
309	C₂×SD₁₆	—	2	4	no	—
310	C₂×D₈	—	2	4	no	—
311	C₂×D₈	—	2	4	no	—
312	C₂×M₁₆	—	2	1	no	—
313	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
314	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
315	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
316	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
317	#127	—	2	2	no	—
318	#127	—	2	1	no	—
319	#127	—	2	1	no	—
320	#127	—	2	1	no	—
321	#127	D₄×C₆₁	1	1	yes	characteristic
322	#127	C₂³×C₆₁	1	2	yes	—
323	#127	C₂³×C₆₁	1	1	yes	characteristic
324	#125	—	2	2	no	—
325	#125	—	2	1	no	—
326	#124	—	2	1	no	—
327	#124	D₄×C₆₁	1	2	yes	—
328	#123	—	2	4	no	—
329	#123	—	2	4	no	—
330	C₂²⋊Q₈	—	2	2	no	—
331	C₂²⋊Q₈	—	2	4	no	—
332	C₂²⋊Q₈	—	2	2	no	—
333	C₂²⋊Q₈	—	2	4	no	—
334	C₂²⋊Q₈	—	2	1	no	—
335	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
336	#121	—	2	4	no	—
337	#121	—	2	4	no	—
338	#121	—	2	2	no	—
339	#121	—	2	4	no	—
340	#121	—	2	4	no	—
341	#121	—	2	1	no	—
342	#121	—	2	1	no	—
343	#121	—	2	1	no	—
344	#121	—	2	2	no	—
345	#121	D₄×C₆₁	1	2	yes	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
348	C₄×D₄	—	2	2	no	—
349	C₄×D₄	—	2	2	no	—
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	C₂³×C₆₁	1	2	yes	—
354	#117	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
358	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
359	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
360	#102	—	2	4	no	—
361	#102	—	2	4	no	—
362	C₂²⋊C₈	—	2	1	no	—
363	C₂²⋊C₈	—	2	1	no	—
364	#97	—	2	1	no	—
365	C₄×C₈	—	2	1	no	—
366	C₄×C₈	—	2	1	no	—
367	C₄×C₈	C₂³×C₆₁	1	2	yes	—
368	C₂⁴	—	4	1	no	—
369	#41	—	4	4	no	—
370	#41	—	8	2	no	—
371	#41	—	8	4	no	—
372	#41	—	8	4	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
376	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
377	C₂×D₄	—	2	4	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	4	4	no	—
389	C₂×D₄	—	4	1	no	—
390	C₂×D₄	—	4	1	no	—
391	C₂×D₄	—	4	1	no	—
392	C₂×D₄	—	8	2	no	—
393	C₂×D₄	—	8	2	no	—
394	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
395	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
403	SD₁₆	—	4	8	no	—
404	SD₁₆	—	8	4	no	—
405	D₈	—	4	8	no	—
406	D₈	—	8	4	no	—
407	M₁₆	—	2	4	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
422	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	2	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
433	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
434	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
435	C₂³	—	2	2	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	8	2	no	—
439	C₂³	—	8	2	no	—
440	C₂³	—	8	2	no	—
441	C₂³	#120×C₆₁	1	1	yes	characteristic
442	Q₈	—	2	4	no	—
443	Q₈	—	4	2	no	—
444	Q₈	—	4	2	no	—
445	Q₈	—	8	1	no	—
446	D₄	—	2	4	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	4	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	4	no	—
451	D₄	—	4	2	no	—
452	D₄	—	4	4	no	—
453	D₄	—	4	2	no	—
454	D₄	—	8	1	no	—
455	D₄	—	8	2	no	—
456	D₄	—	8	2	no	—
457	D₄	—	8	2	no	—
458	D₄	—	8	2	no	—
459	D₄	—	8	4	no	—
460	D₄	—	8	4	no	—
461	D₄	—	8	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	4	no	—
471	C₂×C₄	—	8	4	no	—
472	C₂×C₄	—	8	4	no	—
473	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
474	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
475	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
479	C₈	—	2	4	no	—
480	C₈	—	4	2	no	—
481	C₈	—	4	2	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	8	2	no	—
492	C₂²	—	8	2	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	16	4	no	—
495	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
496	C₄	—	2	1	no	—
497	C₄	—	2	1	no	—
498	C₄	—	2	1	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	C₆₁×#453	1	2	yes	—
504	C₄	C₆₁×#576	1	2	yes	—
505	C₄	C₆₁×#577	1	2	yes	—
506	C₂	—	4	1	no	—
507	C₂	—	8	2	no	—
508	C₂	—	8	2	no	—
509	C₂	—	16	2	no	—
510	C₂	C₆₁×#3008	1	1	yes	characteristic
511	C₂	C₆₁×#3010	1	1	yes	characteristic
512	C₂	C₂×C₆₁×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
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×#3548	C₂	semidirect
2. C₆₁×#3208	C₂	semidirect
3. C₆₁×#3205	C₂	semidirect
4. C₆₁×#3151	C₂	semidirect
5. C₆₁×#1622	C₂	semidirect
6. C₆₁×#1621	C₂	semidirect
7. C₆₁×#1543	C₂	semidirect
8. C₆₁×#1485	C₂	semidirect
9. C₆₁×#1484	C₂	semidirect
13. C₆₁×#554	C₂²	semidirect
15. C₆₁×#550	C₂²	semidirect
39. C₆₁×#333	C₂²	semidirect
40. C₆₁×#331	C₂²	semidirect
41. C₆₁×Q₈⋊C₈	C₂²	semidirect
42. C₆₁×D₄⋊C₈	C₂²	semidirect
225. #18716	C₆₁	direct (N × H)
250. #3548	C₂×C₆₁	semidirect
251. #3208	C₂×C₆₁	semidirect
252. #3205	C₂×C₆₁	semidirect
253. #3151	C₂×C₆₁	semidirect
254. #1622	C₂×C₆₁	semidirect
255. #1621	C₂×C₆₁	semidirect
256. #1543	C₂×C₆₁	semidirect
257. #1485	C₂×C₆₁	semidirect
258. #1484	C₂×C₆₁	semidirect
269. #554	C₂²×C₆₁	semidirect
271. #550	C₂²×C₆₁	semidirect
295. #333	C₂²×C₆₁	semidirect
296. #331	C₂²×C₆₁	semidirect
297. Q₈⋊C₈	C₂²×C₆₁	semidirect
298. D₄⋊C₈	C₂²×C₆₁	semidirect
299. C₆₁	#18716	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×#3548	2
C₆₁×#3208	2
C₆₁×#3205	2
C₆₁×#3151	2
C₆₁×#1622	2
C₆₁×#1621	2
C₆₁×#1543	2
C₆₁×#1485	2
C₆₁×#1484	2
#18716	61

Subgroups — C61×#18716 #29522149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18716 #29522149