Subgroups — C₆₁×#20386 #29523819

Order 15616.

1630 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#407	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#405	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2280	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1753	C₂	1	8	yes	maximal, complemented
6	C₆₁×#1729	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₁	—	2	8	no	—
9	C₂²×M₁₆×C₆₁	C₄	1	8	yes	complemented
10	C₂²×M₁₆×C₆₁	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
12	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#407	C₄	1	8	yes	complemented
16	C₂×C₂²⋊C₈×C₆₁	C₂²	1	4	yes	—
17	C₆₁×#405	—	2	8	no	—
18	C₂×#97×C₆₁	C₂²	1	4	yes	complemented
19	C₂×C₄×C₈×C₆₁	C₂²	1	4	yes	complemented
20	C₆₁×#381	C₄	1	2	yes	—
21	C₆₁×#380	C₄	1	4	yes	—
22	C₆₁×#379	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
24	C₆₁×#336	C₂²	1	8	yes	complemented
25	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₁	—	2	4	no	—
27	C₂³×C₄×C₆₁	—	2	4	no	—
28	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₁	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₆₁	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₆₁	—	2	4	no	—
35	C₂×M₁₆×C₆₁	—	2	4	no	—
36	C₂×M₁₆×C₆₁	—	2	32	no	—
37	C₂×M₁₆×C₆₁	—	2	8	no	—
38	C₂×M₁₆×C₆₁	—	2	8	no	—
39	C₂×M₁₆×C₆₁	C₂×C₄	1	16	yes	—
40	C₂×M₁₆×C₆₁	C₂×C₄	1	8	yes	—
41	C₂²×C₈×C₆₁	—	2	4	no	—
42	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₆₁	D₄	1	8	yes	—
45	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
47	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
48	#117×C₆₁	—	2	4	no	—
49	#117×C₆₁	—	2	4	no	—
50	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
51	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
53	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
54	C₂²⋊C₈×C₆₁	C₂×C₄	1	16	yes	—
55	#97×C₆₁	—	2	8	no	—
56	C₄×C₈×C₆₁	—	2	8	no	—
57	#95×C₆₁	C₂×C₄	1	8	yes	—
58	C₂⁴×C₆₁	—	2	4	no	—
59	C₂⁴×C₆₁	—	4	4	no	—
60	C₂⁴×C₆₁	C₄²	1	4	yes	—
61	C₂⁴×C₆₁	C₂²×C₄	1	2	yes	—
62	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	—	2	4	no	—
64	C₂²×C₄×C₆₁	—	2	8	no	—
65	C₂²×C₄×C₆₁	—	2	8	no	—
66	C₂²×C₄×C₆₁	—	2	8	no	—
67	C₂²×C₄×C₆₁	—	2	16	no	—
68	C₂²×C₄×C₆₁	—	2	4	no	—
69	C₂²×C₄×C₆₁	—	2	4	no	—
70	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
71	C₂²×C₄×C₆₁	C₄²	1	8	yes	—
72	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₆₁	#41	1	4	yes	—
81	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₆₁	—	2	16	no	—
83	M₁₆×C₆₁	—	2	32	no	—
84	C₂×C₈×C₆₁	—	2	8	no	—
85	C₂×C₈×C₆₁	—	2	4	no	—
86	C₂×C₈×C₆₁	—	2	4	no	—
87	C₂×C₈×C₆₁	—	2	4	no	—
88	C₂×C₈×C₆₁	C₂²⋊C₄	1	16	yes	complemented
89	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
90	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
94	C₄⋊C₄×C₆₁	—	2	8	no	—
95	C₂²⋊C₄×C₆₁	—	2	8	no	—
96	C₂²⋊C₄×C₆₁	—	4	8	no	—
97	C₄²×C₆₁	—	2	4	no	—
98	C₄²×C₆₁	—	2	4	no	—
99	C₂³×C₆₁	—	2	4	no	—
100	C₂³×C₆₁	—	2	4	no	—
101	C₂³×C₆₁	—	2	4	no	—
102	C₂³×C₆₁	—	2	4	no	—
103	C₂³×C₆₁	—	2	8	no	—
104	C₂³×C₆₁	—	4	8	no	—
105	C₂³×C₆₁	—	4	16	no	—
106	C₂³×C₆₁	C₂×C₄²	1	4	yes	—
107	C₂³×C₆₁	C₂×C₄²	1	2	yes	—
108	C₂³×C₆₁	C₄×D₄	1	4	yes	—
109	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	—	2	4	no	—
111	C₂×C₄×C₆₁	—	2	8	no	—
112	C₂×C₄×C₆₁	—	2	4	no	—
113	C₂×C₄×C₆₁	—	2	4	no	—
114	C₂×C₄×C₆₁	—	2	8	no	—
115	C₂×C₄×C₆₁	—	2	16	no	—
116	C₂×C₄×C₆₁	—	2	8	no	—
117	C₂×C₄×C₆₁	—	2	16	no	—
118	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
119	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
121	C₂×C₄×C₆₁	#117	1	4	yes	—
122	C₂×C₄×C₆₁	C₄×D₄	1	8	yes	—
123	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
124	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
128	C₈×C₆₁	—	2	4	no	—
129	C₈×C₆₁	—	2	8	no	—
130	C₈×C₆₁	C₂×C₂²⋊C₄	1	8	yes	—
131	#20386	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₆₁	—	2	4	no	—
133	C₂²×C₆₁	—	2	8	no	—
134	C₂²×C₆₁	—	2	4	no	—
135	C₂²×C₆₁	—	2	4	no	—
136	C₂²×C₆₁	—	2	4	no	—
137	C₂²×C₆₁	—	4	16	no	—
138	C₂²×C₆₁	—	4	8	no	—
139	C₂²×C₆₁	C₄×C₂²⋊C₄	1	4	yes	—
140	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
141	C₂²×C₆₁	C₂×C₄×D₄	1	2	yes	—
142	C₂²×C₆₁	#568	1	4	yes	—
143	C₄×C₆₁	—	2	4	no	—
144	C₄×C₆₁	—	4	8	no	—
145	C₄×C₆₁	C₄×C₂²⋊C₄	1	4	yes	—
146	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
147	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
148	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
149	C₂³×M₁₆	C₂×C₆₁	1	1	yes	characteristic
150	C₂×#407	C₂×C₆₁	1	1	yes	characteristic
151	C₂×#405	C₂×C₆₁	1	2	yes	complemented
152	#2280	C₂×C₆₁	1	1	yes	characteristic
153	#1753	C₂×C₆₁	1	8	yes	complemented
154	#1729	C₂×C₆₁	1	2	yes	complemented
155	C₂×C₆₁	—	2	4	no	—
156	C₂×C₆₁	—	4	4	no	—
157	C₂×C₆₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₆₁	#2873	1	2	yes	—
159	C₂×C₆₁	#2930	1	4	yes	—
160	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
161	C₂²×M₁₆	—	2	8	no	—
162	C₂²×M₁₆	C₄×C₆₁	1	8	yes	complemented
163	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₆₁	1	4	yes	—
165	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
166	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
167	C₂×#117	C₂²×C₆₁	1	2	yes	—
168	#407	C₄×C₆₁	1	8	yes	complemented
169	C₂×C₂²⋊C₈	C₂²×C₆₁	1	4	yes	—
170	#405	—	2	8	no	—
171	C₂×#97	C₂²×C₆₁	1	4	yes	complemented
172	C₂×C₄×C₈	C₂²×C₆₁	1	4	yes	complemented
173	#381	C₄×C₆₁	1	2	yes	—
174	#380	C₄×C₆₁	1	4	yes	—
175	#379	C₄×C₆₁	1	2	yes	—
176	C₄×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
177	#336	C₂²×C₆₁	1	8	yes	complemented
178	C₆₁	#20386	1	1	yes	characteristic, direct product factor, Sylow
179	C₂⁵	C₂×C₄×C₆₁	1	1	yes	characteristic
180	C₂³×C₄	—	2	4	no	—
181	C₂³×C₄	—	2	4	no	—
182	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
183	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₆₁	1	4	yes	—
185	C₂³×C₄	C₂×C₄×C₆₁	1	4	yes	—
186	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
187	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	—	2	32	no	—
191	C₂×M₁₆	—	2	8	no	—
192	C₂×M₁₆	—	2	8	no	—
193	C₂×M₁₆	C₂×C₄×C₆₁	1	16	yes	—
194	C₂×M₁₆	C₂×C₄×C₆₁	1	8	yes	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
197	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
198	C₂²×C₈	D₄×C₆₁	1	8	yes	—
199	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
200	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
201	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
202	#117	—	2	4	no	—
203	#117	—	2	4	no	—
204	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
205	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
206	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	8	yes	—
207	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
208	C₂²⋊C₈	C₂×C₄×C₆₁	1	16	yes	—
209	#97	—	2	8	no	—
210	C₄×C₈	—	2	8	no	—
211	#95	C₂×C₄×C₆₁	1	8	yes	—
212	C₂⁴	—	2	4	no	—
213	C₂⁴	—	4	4	no	—
214	C₂⁴	C₄²×C₆₁	1	4	yes	—
215	C₂⁴	C₂²×C₄×C₆₁	1	2	yes	—
216	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	16	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	C₄²×C₆₁	1	4	yes	—
225	C₂²×C₄	C₄²×C₆₁	1	8	yes	—
226	C₂²×C₄	C₂²×C₄×C₆₁	1	8	yes	—
227	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
234	C₂²×C₄	#41×C₆₁	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
236	M₁₆	—	2	16	no	—
237	M₁₆	—	2	32	no	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂²⋊C₄×C₆₁	1	16	yes	complemented
243	C₂×C₈	C₂²×C₄×C₆₁	1	8	yes	—
244	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
245	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
246	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
247	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
248	C₄⋊C₄	—	2	8	no	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	4	8	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	8	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	4	8	no	—
259	C₂³	—	4	16	no	—
260	C₂³	C₂×C₄²×C₆₁	1	4	yes	—
261	C₂³	C₂×C₄²×C₆₁	1	2	yes	—
262	C₂³	C₄×D₄×C₆₁	1	4	yes	—
263	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	16	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	16	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	C₂×C₄²×C₆₁	1	4	yes	—
273	C₂×C₄	C₂×C₄²×C₆₁	1	2	yes	—
274	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
275	C₂×C₄	#117×C₆₁	1	4	yes	—
276	C₂×C₄	C₄×D₄×C₆₁	1	4	yes	—
277	C₂×C₄	C₄×D₄×C₆₁	1	8	yes	—
278	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
279	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
281	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	2	8	no	—
284	C₈	C₂×C₂²⋊C₄×C₆₁	1	8	yes	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	8	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	16	no	—
291	C₂²	—	4	8	no	—
292	C₂²	C₄×C₂²⋊C₄×C₆₁	1	4	yes	—
293	C₂²	C₂²×C₄²×C₆₁	1	1	yes	characteristic
294	C₂²	C₂×C₄×D₄×C₆₁	1	2	yes	—
295	C₂²	C₆₁×#568	1	4	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	4	8	no	—
298	C₄	C₄×C₂²⋊C₄×C₆₁	1	4	yes	—
299	C₄	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
300	C₄	C₂×#117×C₆₁	1	1	yes	characteristic
301	C₄	C₂×C₄×D₄×C₆₁	1	2	yes	—
302	C₂	—	2	4	no	—
303	C₂	—	4	4	no	—
304	C₂	C₂×C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
305	C₂	C₆₁×#2873	1	2	yes	—
306	C₂	C₆₁×#2930	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	C₂⁵×C₆₁	C₂×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
3. C₂×C₆₁×#405	C₂	semidirect
5. C₆₁×#1753	C₂	semidirect
6. C₆₁×#1729	C₂	semidirect
9. C₂²×M₁₆×C₆₁	C₄	semidirect
15. C₆₁×#407	C₄	semidirect
18. C₂×#97×C₆₁	C₂²	semidirect
19. C₂×C₄×C₈×C₆₁	C₂²	semidirect
24. C₆₁×#336	C₂²	semidirect
88. C₂×C₈×C₆₁	C₂²⋊C₄	semidirect
131. #20386	C₆₁	direct (N × H)
151. C₂×#405	C₂×C₆₁	semidirect
153. #1753	C₂×C₆₁	semidirect
154. #1729	C₂×C₆₁	semidirect
162. C₂²×M₁₆	C₄×C₆₁	semidirect
168. #407	C₄×C₆₁	semidirect
171. C₂×#97	C₂²×C₆₁	semidirect
172. C₂×C₄×C₈	C₂²×C₆₁	semidirect
177. #336	C₂²×C₆₁	semidirect
178. C₆₁	#20386	direct (N × H)
242. C₂×C₈	C₂²⋊C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₆₁	2
C₂×C₆₁×#407	2
C₂×C₆₁×#405	2
C₆₁×#2280	2
C₆₁×#1753	2
C₆₁×#1729	2
#20386	61

Subgroups — C61×#20386 #29523819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#20386 #29523819