Subgroups — C₆₁×#20545 #29523978

Order 15616.

934 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2283	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1770	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₆₁	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1768	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1752	C₂	1	2	yes	maximal
8	C₈×C₂²⋊C₄×C₆₁	C₂	1	2	yes	maximal
9	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#431	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₈×C₆₁	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#97×C₆₁	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
22	C₆₁×#383	C₄	1	2	yes	—
23	C₆₁×#382	C₄	1	2	yes	—
24	C₆₁×#381	C₄	1	2	yes	—
25	C₆₁×#380	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
28	C₆₁×#376	C₂²	1	2	yes	—
29	C₆₁×#336	C₂²	1	4	yes	—
30	C₆₁×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	—	4	4	no	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
47	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₁	C₂×C₄	1	8	yes	—
50	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
51	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
52	C₄⋊C₈×C₆₁	C₂×C₄	1	8	yes	—
53	C₂²⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
54	C₂²⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
55	#97×C₆₁	—	4	4	no	—
56	C₄×C₈×C₆₁	—	4	4	no	—
57	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	—
58	#95×C₆₁	C₂×C₄	1	4	yes	—
59	#95×C₆₁	C₂×C₄	1	4	yes	—
60	#95×C₆₁	C₂×C₄	1	4	yes	—
61	#95×C₆₁	C₂×C₄	1	4	yes	—
62	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	—	2	4	no	—
64	C₂²×C₄×C₆₁	—	2	4	no	—
65	C₂²×C₄×C₆₁	—	2	2	no	—
66	C₂²×C₄×C₆₁	—	2	2	no	—
67	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
68	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
69	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₆₁	—	2	2	no	—
79	C₂×C₈×C₆₁	—	2	4	no	—
80	C₂×C₈×C₆₁	—	2	2	no	—
81	C₂×C₈×C₆₁	—	2	4	no	—
82	C₂×C₈×C₆₁	—	2	2	no	—
83	C₂×C₈×C₆₁	—	2	2	no	—
84	C₂×C₈×C₆₁	—	4	8	no	—
85	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
89	C₄⋊C₄×C₆₁	—	2	8	no	—
90	C₂²⋊C₄×C₆₁	—	2	8	no	—
91	C₄²×C₆₁	—	2	4	no	—
92	C₄²×C₆₁	—	2	4	no	—
93	C₄²×C₆₁	C₄²	1	4	yes	—
94	C₄²×C₆₁	C₄²	1	4	yes	—
95	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
96	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
97	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
98	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
99	C₂³×C₆₁	—	2	2	no	—
100	C₂³×C₆₁	—	2	2	no	—
101	C₂³×C₆₁	C₂×C₄²	1	2	yes	—
102	C₂³×C₆₁	#131	1	4	yes	—
103	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	2	no	—
107	C₂×C₄×C₆₁	—	2	2	no	—
108	C₂×C₄×C₆₁	—	2	8	no	—
109	C₂×C₄×C₆₁	—	2	2	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	2	4	no	—
112	C₂×C₄×C₆₁	—	2	8	no	—
113	C₂×C₄×C₆₁	—	2	8	no	—
114	C₂×C₄×C₆₁	—	2	8	no	—
115	C₂×C₄×C₆₁	—	2	8	no	—
116	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₆₁	#131	1	4	yes	—
122	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
125	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
126	C₈×C₆₁	—	2	4	no	—
127	C₈×C₆₁	—	2	4	no	—
128	C₈×C₆₁	—	2	4	no	—
129	C₈×C₆₁	—	2	4	no	—
130	#20545	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	—	2	8	no	—
133	C₂²×C₆₁	—	2	2	no	—
134	C₂²×C₆₁	#405	1	4	yes	—
135	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
136	C₂²×C₆₁	#520	1	2	yes	—
137	C₂²×C₆₁	C₂×#131	1	2	yes	—
138	C₂²×C₆₁	C₂×#131	1	2	yes	—
139	C₄×C₆₁	—	2	2	no	—
140	C₄×C₆₁	—	2	2	no	—
141	C₄×C₆₁	—	2	2	no	—
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	—	4	8	no	—
144	C₄×C₆₁	#405	1	4	yes	—
145	C₄×C₆₁	#518	1	1	yes	characteristic
146	C₄×C₆₁	#519	1	1	yes	characteristic
147	C₄×C₆₁	#520	1	2	yes	—
148	C₂×#431	C₂×C₆₁	1	1	yes	characteristic
149	C₂²×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
150	#2283	C₂×C₆₁	1	1	yes	characteristic
151	#1770	C₂×C₆₁	1	2	yes	complemented
152	C₈×C₄⋊C₄	C₂×C₆₁	1	2	yes	complemented
153	#1768	C₂×C₆₁	1	4	yes	complemented
154	#1752	C₂×C₆₁	1	2	yes	—
155	C₈×C₂²⋊C₄	C₂×C₆₁	1	2	yes	—
156	C₂×C₆₁	—	2	4	no	—
157	C₂×C₆₁	#2274	1	1	yes	characteristic
158	C₂×C₆₁	C₂×#405	1	1	yes	characteristic
159	C₂×C₆₁	C₂×#405	1	1	yes	characteristic
160	C₂×C₆₁	#2994	1	4	yes	—
161	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
162	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
163	#431	C₄×C₆₁	1	8	yes	complemented
164	C₂×C₄⋊C₈	C₄×C₆₁	1	8	yes	complemented
165	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
166	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
170	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
171	C₂×#97	C₂²×C₆₁	1	4	yes	—
172	C₂×C₄×C₈	C₂²×C₆₁	1	4	yes	—
173	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
174	#383	C₄×C₆₁	1	2	yes	—
175	#382	C₄×C₆₁	1	2	yes	—
176	#381	C₄×C₆₁	1	2	yes	—
177	#380	C₄×C₆₁	1	2	yes	—
178	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
179	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
180	#376	C₂²×C₆₁	1	2	yes	—
181	#336	C₂²×C₆₁	1	4	yes	—
182	#336	C₂²×C₆₁	1	4	yes	—
183	C₆₁	#20545	1	1	yes	characteristic, direct product factor, Sylow
184	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
185	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
187	C₂²×C₈	—	4	4	no	—
188	C₂²×C₈	C₂×C₄×C₆₁	1	8	yes	—
189	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
191	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
192	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
195	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
198	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
199	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
200	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
202	C₄⋊C₈	C₂×C₄×C₆₁	1	8	yes	—
203	C₄⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
204	C₄⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
205	C₄⋊C₈	C₂×C₄×C₆₁	1	8	yes	—
206	C₂²⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
207	C₂²⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
208	#97	—	4	4	no	—
209	C₄×C₈	—	4	4	no	—
210	C₄×C₈	C₂×C₄×C₆₁	1	8	yes	—
211	#95	C₂×C₄×C₆₁	1	4	yes	—
212	#95	C₂×C₄×C₆₁	1	4	yes	—
213	#95	C₂×C₄×C₆₁	1	4	yes	—
214	#95	C₂×C₄×C₆₁	1	4	yes	—
215	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₄²×C₆₁	1	4	yes	—
221	C₂²×C₄	C₄²×C₆₁	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₆₁	1	8	yes	—
223	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	8	no	—
238	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
239	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
242	C₄⋊C₄	—	2	8	no	—
243	C₂²⋊C₄	—	2	8	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	C₄²×C₆₁	1	4	yes	—
247	C₄²	C₄²×C₆₁	1	4	yes	—
248	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
249	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
250	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
251	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄²×C₆₁	1	2	yes	—
255	C₂³	#131×C₆₁	1	4	yes	—
256	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
270	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
271	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
273	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
274	C₂×C₄	#131×C₆₁	1	4	yes	—
275	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
276	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
277	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
278	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
279	C₈	—	2	4	no	—
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	8	no	—
286	C₂²	C₆₁×#405	1	4	yes	—
287	C₂²	C₂²×C₄²×C₆₁	1	1	yes	characteristic
288	C₂²	C₆₁×#520	1	2	yes	—
289	C₂²	C₂×#131×C₆₁	1	2	yes	—
290	C₂²	C₂×#131×C₆₁	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	8	no	—
296	C₄	C₆₁×#405	1	4	yes	—
297	C₄	C₆₁×#518	1	1	yes	characteristic
298	C₄	C₆₁×#519	1	1	yes	characteristic
299	C₄	C₆₁×#520	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	C₆₁×#2274	1	1	yes	characteristic
302	C₂	C₂×C₆₁×#405	1	1	yes	characteristic
303	C₂	C₂×C₆₁×#405	1	1	yes	characteristic
304	C₂	C₆₁×#2994	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1770	C₂	semidirect
5. C₈×C₄⋊C₄×C₆₁	C₂	semidirect
6. C₆₁×#1768	C₂	semidirect
11. C₆₁×#431	C₄	semidirect
12. C₂×C₄⋊C₈×C₆₁	C₄	semidirect
130. #20545	C₆₁	direct (N × H)
151. #1770	C₂×C₆₁	semidirect
152. C₈×C₄⋊C₄	C₂×C₆₁	semidirect
153. #1768	C₂×C₆₁	semidirect
163. #431	C₄×C₆₁	semidirect
164. C₂×C₄⋊C₈	C₄×C₆₁	semidirect
183. C₆₁	#20545	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₆₁×#431	2
C₂²×C₄⋊C₈×C₆₁	2
C₆₁×#2283	2
C₆₁×#1770	2
C₈×C₄⋊C₄×C₆₁	2
C₆₁×#1768	2
C₆₁×#1752	2
C₈×C₂²⋊C₄×C₆₁	2
#20545	61

Subgroups — C61×#20545 #29523978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#20545 #29523978