Subgroups — C₆₁×#8709 #29512142

Order 15616.

540 conjugacy classes of subgroups.

The subgroup lattice has 310 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	C₄×C₂²⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1279	C₂	1	4	yes	maximal, complemented
5	C₂×#117×C₆₁	—	2	1	no	—
6	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
8	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
9	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#384	C₄	1	1	yes	characteristic
12	C₆₁×#384	C₄	1	1	yes	characteristic
13	C₆₁×#382	—	2	1	no	—
14	C₆₁×#380	—	2	1	no	—
15	C₆₁×#380	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
17	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
18	C₆₁×#336	—	2	1	no	—
19	C₆₁×#336	—	2	1	no	—
20	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	—	2	2	no	—
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	#117×C₆₁	—	4	2	no	—
26	#117×C₆₁	—	4	2	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	C₂²⋊C₈×C₆₁	—	2	4	no	—
42	C₂²⋊C₈×C₆₁	—	4	2	no	—
43	C₄×C₈×C₆₁	—	2	4	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	2	no	—
53	C₂²×C₄×C₆₁	—	2	2	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	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	D₈	1	2	yes	—
61	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	—	2	4	no	—
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	C₄⋊C₄×C₆₁	—	2	4	no	—
72	C₄⋊C₄×C₆₁	—	2	4	no	—
73	C₄⋊C₄×C₆₁	—	4	2	no	—
74	C₄⋊C₄×C₆₁	—	4	2	no	—
75	C₄⋊C₄×C₆₁	—	4	2	no	—
76	C₄⋊C₄×C₆₁	—	4	2	no	—
77	C₂²⋊C₄×C₆₁	—	4	4	no	—
78	C₄²×C₆₁	—	2	2	no	—
79	C₄²×C₆₁	—	2	2	no	—
80	C₄²×C₆₁	—	2	2	no	—
81	C₄²×C₆₁	—	4	2	no	—
82	C₄²×C₆₁	—	4	2	no	—
83	C₄²×C₆₁	#41	1	4	yes	—
84	C₂³×C₆₁	—	2	1	no	—
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	#102	1	2	yes	—
91	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	—	2	1	no	—
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	1	no	—
98	C₂×C₄×C₆₁	—	2	4	no	—
99	C₂×C₄×C₆₁	—	2	4	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	—	4	4	no	—
104	C₂×C₄×C₆₁	—	4	4	no	—
105	C₂×C₄×C₆₁	—	4	2	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	#102	1	2	yes	—
112	C₂×C₄×C₆₁	#104	1	2	yes	—
113	C₂×C₄×C₆₁	#117	1	2	yes	—
114	C₂×C₄×C₆₁	#123	1	2	yes	—
115	C₂×C₄×C₆₁	#123	1	2	yes	—
116	C₂×C₄×C₆₁	#131	1	2	yes	—
117	C₂×C₄×C₆₁	#131	1	2	yes	—
118	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#137	1	2	yes	—
121	C₈×C₆₁	—	4	4	no	—
122	C₈×C₆₁	—	8	2	no	—
123	#8709	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₆₁	—	2	2	no	—
125	C₂²×C₆₁	—	2	2	no	—
126	C₂²×C₆₁	—	2	2	no	—
127	C₂²×C₆₁	—	4	1	no	—
128	C₂²×C₆₁	—	4	1	no	—
129	C₂²×C₆₁	—	4	1	no	—
130	C₂²×C₆₁	—	4	1	no	—
131	C₂²×C₆₁	#381	1	1	yes	characteristic
132	C₂²×C₆₁	#408	1	1	yes	characteristic
133	C₂²×C₆₁	#408	1	1	yes	characteristic
134	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
135	C₂²×C₆₁	#419	1	1	yes	characteristic
136	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
137	C₂²×C₆₁	#421	1	1	yes	characteristic
138	C₄×C₆₁	—	2	2	no	—
139	C₄×C₆₁	—	2	2	no	—
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	—	4	1	no	—
142	C₄×C₆₁	—	8	2	no	—
143	C₄×C₆₁	—	8	2	no	—
144	C₄×C₆₁	#433	1	4	yes	—
145	C₄×C₆₁	#480	1	2	yes	—
146	C₄×C₆₁	#483	1	2	yes	—
147	#2285	C₂×C₆₁	1	1	yes	characteristic
148	#1797	C₂×C₆₁	1	1	yes	characteristic
149	C₄×C₂²⋊C₈	C₂×C₆₁	1	1	yes	characteristic
150	#1279	C₂×C₆₁	1	4	yes	complemented
151	C₂×C₆₁	—	4	2	no	—
152	C₂×C₆₁	#1480	1	1	yes	characteristic
153	C₂×C₆₁	#1485	1	1	yes	characteristic
154	C₂×C₆₁	#1492	1	1	yes	characteristic
155	C₂×C₆₁	#1497	1	1	yes	characteristic
156	C₂×C₆₁	#1818	1	1	yes	characteristic
157	C₂×C₆₁	#1823	1	1	yes	characteristic
158	C₂×C₆₁	#1826	1	1	yes	characteristic
159	C₂×#117	—	2	1	no	—
160	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
162	C₂×C₂²⋊C₈	—	2	1	no	—
163	C₂×C₂²⋊C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
165	#384	C₄×C₆₁	1	1	yes	characteristic
166	#384	C₄×C₆₁	1	1	yes	characteristic
167	#382	—	2	1	no	—
168	#380	—	2	1	no	—
169	#380	C₄×C₆₁	1	2	yes	—
170	C₄×C₄⋊C₄	—	2	1	no	—
171	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
172	#336	—	2	1	no	—
173	#336	—	2	1	no	—
174	C₆₁	#8709	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	#117	—	4	2	no	—
181	#117	—	4	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
195	C₄⋊C₈	—	2	4	no	—
196	C₂²⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	—	4	2	no	—
198	C₄×C₈	—	2	4	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	C₂⁴	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
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	—	4	1	no	—
212	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
215	C₂²×C₄	D₈×C₆₁	1	2	yes	—
216	C₂²×C₄	SD₁₆×C₆₁	1	2	yes	—
217	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	#41×C₆₁	1	4	yes	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	#102×C₆₁	1	2	yes	—
246	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	#102×C₆₁	1	2	yes	—
267	C₂×C₄	#104×C₆₁	1	2	yes	—
268	C₂×C₄	#117×C₆₁	1	2	yes	—
269	C₂×C₄	#123×C₆₁	1	2	yes	—
270	C₂×C₄	#123×C₆₁	1	2	yes	—
271	C₂×C₄	#131×C₆₁	1	2	yes	—
272	C₂×C₄	#131×C₆₁	1	2	yes	—
273	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
274	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
275	C₂×C₄	#137×C₆₁	1	2	yes	—
276	C₈	—	4	4	no	—
277	C₈	—	8	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	C₆₁×#381	1	1	yes	characteristic
286	C₂²	C₆₁×#408	1	1	yes	characteristic
287	C₂²	C₆₁×#408	1	1	yes	characteristic
288	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
289	C₂²	C₆₁×#419	1	1	yes	characteristic
290	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
291	C₂²	C₆₁×#421	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₆₁×#433	1	4	yes	—
299	C₄	C₆₁×#480	1	2	yes	—
300	C₄	C₆₁×#483	1	2	yes	—
301	C₂	—	4	2	no	—
302	C₂	C₆₁×#1480	1	1	yes	characteristic
303	C₂	C₆₁×#1485	1	1	yes	characteristic
304	C₂	C₆₁×#1492	1	1	yes	characteristic
305	C₂	C₆₁×#1497	1	1	yes	characteristic
306	C₂	C₆₁×#1818	1	1	yes	characteristic
307	C₂	C₆₁×#1823	1	1	yes	characteristic
308	C₂	C₆₁×#1826	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂³×C₆₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₁	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1279	C₂	semidirect
123. #8709	C₆₁	direct (N × H)
150. #1279	C₂×C₆₁	semidirect
174. C₆₁	#8709	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
C₄×C₂²⋊C₈×C₆₁	2
C₆₁×#1279	2
#8709	61

Subgroups — C61×#8709 #29512142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8709 #29512142