Subgroups — C₆₁×#8727 #29512160

Order 15616.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2285	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1797	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1797	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1296	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1296	C₂	1	2	yes	maximal, complemented
6	C₂×#117×C₆₁	—	2	1	no	—
7	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
11	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
12	C₆₁×#384	C₂²	1	1	yes	characteristic
13	C₆₁×#384	C₂²	1	1	yes	characteristic
14	C₆₁×#382	—	2	1	no	—
15	C₆₁×#380	—	2	1	no	—
16	C₆₁×#380	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
18	C₄×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
19	C₆₁×#336	—	2	1	no	—
20	C₆₁×#336	—	2	1	no	—
21	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	#117×C₆₁	—	4	4	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
34	C₂×C₄²×C₆₁	—	2	4	no	—
35	C₂×C₄²×C₆₁	—	2	2	no	—
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	—	2	4	no	—
41	C₄⋊C₈×C₆₁	—	2	4	no	—
42	C₂²⋊C₈×C₆₁	—	4	2	no	—
43	C₂²⋊C₈×C₆₁	—	4	2	no	—
44	#95×C₆₁	—	2	2	no	—
45	#95×C₆₁	—	2	2	no	—
46	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	—	2	2	no	—
48	C₂²×C₄×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	—	2	2	no	—
50	C₂²×C₄×C₆₁	—	2	2	no	—
51	C₂²×C₄×C₆₁	—	2	2	no	—
52	C₂²×C₄×C₆₁	—	2	1	no	—
53	C₂²×C₄×C₆₁	—	2	1	no	—
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	—	2	2	no	—
56	C₂²×C₄×C₆₁	—	4	1	no	—
57	C₂²×C₄×C₆₁	—	4	1	no	—
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	—	4	2	no	—
67	C₂×C₈×C₆₁	—	4	2	no	—
68	C₂×C₈×C₆₁	—	4	2	no	—
69	C₂×C₈×C₆₁	—	4	2	no	—
70	C₂×C₈×C₆₁	—	4	2	no	—
71	C₂×C₈×C₆₁	—	4	2	no	—
72	C₄⋊C₄×C₆₁	—	4	2	no	—
73	C₄⋊C₄×C₆₁	—	4	2	no	—
74	C₄⋊C₄×C₆₁	—	4	2	no	—
75	C₄⋊C₄×C₆₁	—	4	1	no	—
76	C₄⋊C₄×C₆₁	—	4	2	no	—
77	C₄⋊C₄×C₆₁	—	4	2	no	—
78	C₄⋊C₄×C₆₁	—	4	1	no	—
79	C₂²⋊C₄×C₆₁	—	4	4	no	—
80	C₄²×C₆₁	—	2	2	no	—
81	C₄²×C₆₁	—	2	2	no	—
82	C₄²×C₆₁	—	2	2	no	—
83	C₄²×C₆₁	—	4	4	no	—
84	C₄²×C₆₁	#41	1	4	yes	—
85	C₂³×C₆₁	—	2	1	no	—
86	C₂³×C₆₁	—	2	1	no	—
87	C₂³×C₆₁	—	2	1	no	—
88	C₂³×C₆₁	—	2	1	no	—
89	C₂³×C₆₁	—	2	1	no	—
90	C₂³×C₆₁	—	2	1	no	—
91	C₂³×C₆₁	#103	1	2	yes	—
92	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	—	2	1	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	4	no	—
98	C₂×C₄×C₆₁	—	2	4	no	—
99	C₂×C₄×C₆₁	—	2	1	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	1	no	—
105	C₂×C₄×C₆₁	—	4	4	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	4	2	no	—
110	C₂×C₄×C₆₁	—	4	2	no	—
111	C₂×C₄×C₆₁	—	4	2	no	—
112	C₂×C₄×C₆₁	—	4	2	no	—
113	C₂×C₄×C₆₁	—	4	2	no	—
114	C₂×C₄×C₆₁	#103	1	2	yes	—
115	C₂×C₄×C₆₁	#117	1	2	yes	—
116	C₂×C₄×C₆₁	#123	1	2	yes	—
117	C₂×C₄×C₆₁	#123	1	2	yes	—
118	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#135	1	2	yes	—
121	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
124	C₈×C₆₁	—	8	2	no	—
125	C₈×C₆₁	—	8	2	no	—
126	#8727	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	—	2	2	no	—
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	—	4	1	no	—
131	C₂²×C₆₁	—	4	1	no	—
132	C₂²×C₆₁	—	4	1	no	—
133	C₂²×C₆₁	—	4	1	no	—
134	C₂²×C₆₁	#381	1	1	yes	characteristic
135	C₂²×C₆₁	#413	1	1	yes	characteristic
136	C₂²×C₆₁	#413	1	1	yes	characteristic
137	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
138	C₂²×C₆₁	#416	1	1	yes	characteristic
139	C₂²×C₆₁	#417	1	1	yes	characteristic
140	C₂²×C₆₁	#418	1	1	yes	characteristic
141	C₄×C₆₁	—	2	2	no	—
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	—	4	1	no	—
144	C₄×C₆₁	—	4	1	no	—
145	C₄×C₆₁	—	8	2	no	—
146	C₄×C₆₁	—	8	2	no	—
147	C₄×C₆₁	#481	1	2	yes	—
148	C₄×C₆₁	#482	1	2	yes	—
149	C₄×C₆₁	#484	1	2	yes	—
150	C₄×C₆₁	#485	1	2	yes	—
151	#2285	C₂×C₆₁	1	1	yes	characteristic
152	#1797	C₂×C₆₁	1	1	yes	characteristic
153	#1797	C₂×C₆₁	1	1	yes	characteristic
154	#1296	C₂×C₆₁	1	2	yes	complemented
155	#1296	C₂×C₆₁	1	2	yes	complemented
156	C₂×C₆₁	—	4	2	no	—
157	C₂×C₆₁	#1542	1	1	yes	characteristic
158	C₂×C₆₁	#1550	1	1	yes	characteristic
159	C₂×C₆₁	#1553	1	1	yes	characteristic
160	C₂×C₆₁	#1557	1	1	yes	characteristic
161	C₂×C₆₁	#1822	1	1	yes	characteristic
162	C₂×C₆₁	#1824	1	1	yes	characteristic
163	C₂×C₆₁	#1825	1	1	yes	characteristic
164	C₂×#117	—	2	1	no	—
165	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
166	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
168	C₂×C₂²⋊C₈	—	2	1	no	—
169	C₂×C₂²⋊C₈	—	2	1	no	—
170	#384	C₂²×C₆₁	1	1	yes	characteristic
171	#384	C₂²×C₆₁	1	1	yes	characteristic
172	#382	—	2	1	no	—
173	#380	—	2	1	no	—
174	#380	C₄×C₆₁	1	2	yes	—
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄×C₄⋊C₄	C₄×C₆₁	1	2	yes	—
177	#336	—	2	1	no	—
178	#336	—	2	1	no	—
179	C₆₁	#8727	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
181	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
182	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	#117	—	4	4	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	2	4	no	—
201	C₂²⋊C₈	—	4	2	no	—
202	C₂²⋊C₈	—	4	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
220	C₂²×C₄	SD₁₆×C₆₁	1	2	yes	—
221	C₂²×C₄	Q₁₆×C₆₁	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	4	no	—
243	C₄²	#41×C₆₁	1	4	yes	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	#103×C₆₁	1	2	yes	—
251	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	#103×C₆₁	1	2	yes	—
274	C₂×C₄	#117×C₆₁	1	2	yes	—
275	C₂×C₄	#123×C₆₁	1	2	yes	—
276	C₂×C₄	#123×C₆₁	1	2	yes	—
277	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
279	C₂×C₄	#135×C₆₁	1	2	yes	—
280	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
281	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
282	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
283	C₈	—	8	2	no	—
284	C₈	—	8	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	C₆₁×#381	1	1	yes	characteristic
293	C₂²	C₆₁×#413	1	1	yes	characteristic
294	C₂²	C₆₁×#413	1	1	yes	characteristic
295	C₂²	C₂×#103×C₆₁	1	1	yes	characteristic
296	C₂²	C₆₁×#416	1	1	yes	characteristic
297	C₂²	C₆₁×#417	1	1	yes	characteristic
298	C₂²	C₆₁×#418	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₆₁×#481	1	2	yes	—
306	C₄	C₆₁×#482	1	2	yes	—
307	C₄	C₆₁×#484	1	2	yes	—
308	C₄	C₆₁×#485	1	2	yes	—
309	C₂	—	4	2	no	—
310	C₂	C₆₁×#1542	1	1	yes	characteristic
311	C₂	C₆₁×#1550	1	1	yes	characteristic
312	C₂	C₆₁×#1553	1	1	yes	characteristic
313	C₂	C₆₁×#1557	1	1	yes	characteristic
314	C₂	C₆₁×#1822	1	1	yes	characteristic
315	C₂	C₆₁×#1824	1	1	yes	characteristic
316	C₂	C₆₁×#1825	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1296	C₂	semidirect
5. C₆₁×#1296	C₂	semidirect
126. #8727	C₆₁	direct (N × H)
154. #1296	C₂×C₆₁	semidirect
155. #1296	C₂×C₆₁	semidirect
179. C₆₁	#8727	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×#2285	2
C₆₁×#1797	2
C₆₁×#1296	2
#8727	61

Subgroups — C61×#8727 #29512160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8727 #29512160