Subgroups — C₆₁×#12977 #29516410

Order 15616.

554 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1374	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₆₁	—	2	2	no	—
10	C₂×M₃₂×C₆₁	—	2	4	no	—
11	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
13	C₈×D₄×C₆₁	—	2	2	no	—
14	C₈×D₄×C₆₁	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₆₁	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₆₁	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×C₂²⋊C₁₆	—	2	2	no	—
20	C₆₁×C₂²⋊C₁₆	—	2	4	no	—
21	C₂²×D₄×C₆₁	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₂²×C₈×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	—	2	1	no	—
28	C₂²×C₈×C₆₁	—	2	1	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	D₄	1	2	yes	—
34	C₂²×C₈×C₆₁	D₄	1	2	yes	—
35	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₆₁	—	2	2	no	—
37	C₄×D₄×C₆₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₆₁	C₈	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₆₁	C₈	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₆₁	—	2	8	no	—
43	C₂×C₁₆×C₆₁	—	2	2	no	—
44	C₂×C₁₆×C₆₁	—	2	2	no	—
45	C₂×C₁₆×C₆₁	—	2	2	no	—
46	C₂×C₁₆×C₆₁	—	2	2	no	—
47	C₂×C₁₆×C₆₁	—	2	2	no	—
48	C₂×C₁₆×C₆₁	D₄	1	4	yes	complemented
49	C₄⋊C₈×C₆₁	—	2	2	no	—
50	C₂²⋊C₈×C₆₁	—	2	2	no	—
51	C₂²⋊C₈×C₆₁	—	2	2	no	—
52	C₄×C₈×C₆₁	—	2	1	no	—
53	C₄×C₈×C₆₁	—	2	1	no	—
54	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
55	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
56	C₂×D₄×C₆₁	—	2	2	no	—
57	C₂×D₄×C₆₁	—	2	2	no	—
58	C₂×D₄×C₆₁	—	2	2	no	—
59	C₂²×C₄×C₆₁	—	2	2	no	—
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	—	2	1	no	—
62	C₂²×C₄×C₆₁	—	2	2	no	—
63	C₂²×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	—	2	2	no	—
65	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
71	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	4	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	1	no	—
77	C₂×C₈×C₆₁	—	2	1	no	—
78	C₂×C₈×C₆₁	—	2	2	no	—
79	C₂×C₈×C₆₁	—	2	2	no	—
80	C₂×C₈×C₆₁	—	4	1	no	—
81	C₂×C₈×C₆₁	—	4	2	no	—
82	C₂×C₈×C₆₁	—	4	1	no	—
83	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₆₁	#41	1	2	yes	—
88	C₄⋊C₄×C₆₁	—	2	2	no	—
89	C₂²⋊C₄×C₆₁	—	2	2	no	—
90	C₂²⋊C₄×C₆₁	—	2	2	no	—
91	C₄²×C₆₁	—	2	1	no	—
92	C₄²×C₆₁	—	2	1	no	—
93	C₁₆×C₆₁	—	2	4	no	—
94	C₁₆×C₆₁	—	4	2	no	—
95	C₂³×C₆₁	—	2	1	no	—
96	C₂³×C₆₁	—	2	2	no	—
97	C₂³×C₆₁	—	2	2	no	—
98	C₂³×C₆₁	—	2	1	no	—
99	C₂³×C₆₁	—	2	1	no	—
100	C₂³×C₆₁	—	2	1	no	—
101	C₂³×C₆₁	—	2	2	no	—
102	C₂³×C₆₁	—	2	2	no	—
103	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
104	C₂³×C₆₁	M₃₂	1	2	yes	complemented
105	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
106	D₄×C₆₁	—	2	8	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	2	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	2	2	no	—
112	C₂×C₄×C₆₁	—	2	4	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	1	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	4	1	no	—
118	C₂×C₄×C₆₁	—	4	1	no	—
119	C₂×C₄×C₆₁	—	4	2	no	—
120	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₆₁	M₃₂	1	2	yes	complemented
122	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#131	1	2	yes	—
126	C₈×C₆₁	—	2	2	no	—
127	C₈×C₆₁	—	4	1	no	—
128	C₈×C₆₁	—	4	1	no	—
129	C₈×C₆₁	#120	1	1	yes	characteristic
130	C₈×C₆₁	#121	1	2	yes	—
131	C₈×C₆₁	#123	1	1	yes	characteristic
132	#12977	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₆₁	—	2	2	no	—
134	C₂²×C₆₁	—	2	2	no	—
135	C₂²×C₆₁	—	2	2	no	—
136	C₂²×C₆₁	—	2	2	no	—
137	C₂²×C₆₁	—	2	4	no	—
138	C₂²×C₆₁	—	4	1	no	—
139	C₂²×C₆₁	—	4	1	no	—
140	C₂²×C₆₁	—	4	2	no	—
141	C₂²×C₆₁	—	4	2	no	—
142	C₂²×C₆₁	—	4	2	no	—
143	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₆₁	C₈×D₄	1	2	yes	—
145	C₂²×C₆₁	C₂×M₃₂	1	1	yes	characteristic
146	C₂²×C₆₁	C₂×M₃₂	1	1	yes	characteristic
147	C₂²×C₆₁	#504	1	2	yes	—
148	C₄×C₆₁	—	2	2	no	—
149	C₄×C₆₁	—	2	2	no	—
150	C₄×C₆₁	—	4	1	no	—
151	C₄×C₆₁	—	4	1	no	—
152	C₄×C₆₁	#386	1	1	yes	characteristic
153	C₄×C₆₁	#408	1	1	yes	characteristic
154	C₄×C₆₁	#435	1	2	yes	—
155	C₂²×M₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
156	C₂×C₈×D₄	C₂×C₆₁	1	1	yes	characteristic
157	C₂×C₂²⋊C₁₆	C₂×C₆₁	1	2	yes	complemented
158	C₂×C₂²⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
159	#1374	C₂×C₆₁	1	2	yes	complemented
160	C₂×C₆₁	—	4	2	no	—
161	C₂×C₆₁	—	4	2	no	—
162	C₂×C₆₁	#1869	1	1	yes	characteristic
163	C₂×C₆₁	#2112	1	1	yes	characteristic
164	C₂×C₆₁	#2113	1	1	yes	characteristic
165	C₂×C₆₁	#2168	1	2	yes	—
166	C₂×C₆₁	#2169	1	2	yes	—
167	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
168	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
169	C₂×C₄×D₄	C₄×C₆₁	1	1	yes	characteristic
170	C₂×M₃₂	—	2	2	no	—
171	C₂×M₃₂	—	2	4	no	—
172	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
173	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
174	C₈×D₄	—	2	2	no	—
175	C₈×D₄	—	2	2	no	—
176	C₂×C₄⋊C₈	C₄×C₆₁	1	1	yes	characteristic
177	C₂×C₂²⋊C₈	C₄×C₆₁	1	1	yes	characteristic
178	C₂×C₂²⋊C₈	C₄×C₆₁	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
180	C₂²⋊C₁₆	—	2	2	no	—
181	C₂²⋊C₁₆	—	2	4	no	—
182	C₆₁	#12977	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×D₄	C₈×C₆₁	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
185	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
193	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
194	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
195	C₂²×C₈	D₄×C₆₁	1	2	yes	—
196	C₂²×C₈	D₄×C₆₁	1	2	yes	—
197	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₈×C₆₁	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	C₈×C₆₁	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	C₈×C₆₁	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
204	M₃₂	—	2	8	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	—	2	2	no	—
209	C₂×C₁₆	—	2	2	no	—
210	C₂×C₁₆	D₄×C₆₁	1	4	yes	complemented
211	C₄⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	2	no	—
213	C₂²⋊C₈	—	2	2	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₂⁴	C₂×C₈×C₆₁	1	1	yes	characteristic
217	C₂⁴	C₂×C₈×C₆₁	1	1	yes	characteristic
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
228	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
232	C₂²×C₄	M₁₆×C₆₁	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₂×C₈	#41×C₆₁	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₁₆	—	2	4	no	—
256	C₁₆	—	4	2	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	C₂²⋊C₈×C₆₁	1	2	yes	—
266	C₂³	M₃₂×C₆₁	1	2	yes	complemented
267	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
268	D₄	—	2	8	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
283	C₂×C₄	M₃₂×C₆₁	1	2	yes	complemented
284	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
286	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
287	C₂×C₄	#131×C₆₁	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	4	1	no	—
290	C₈	—	4	1	no	—
291	C₈	#120×C₆₁	1	1	yes	characteristic
292	C₈	#121×C₆₁	1	2	yes	—
293	C₈	#123×C₆₁	1	1	yes	characteristic
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
305	C₂²	C₈×D₄×C₆₁	1	2	yes	—
306	C₂²	C₂×M₃₂×C₆₁	1	1	yes	characteristic
307	C₂²	C₂×M₃₂×C₆₁	1	1	yes	characteristic
308	C₂²	C₆₁×#504	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	C₆₁×#386	1	1	yes	characteristic
314	C₄	C₆₁×#408	1	1	yes	characteristic
315	C₄	C₆₁×#435	1	2	yes	—
316	C₂	—	4	2	no	—
317	C₂	—	4	2	no	—
318	C₂	C₆₁×#1869	1	1	yes	characteristic
319	C₂	C₆₁×#2112	1	1	yes	characteristic
320	C₂	C₆₁×#2113	1	1	yes	characteristic
321	C₂	C₆₁×#2168	1	2	yes	—
322	C₂	C₆₁×#2169	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₆₁	C₂³
commutator	C₂³	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂²×D₄×C₆₁	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₃₂×C₆₁	C₂	semidirect
3. C₂×C₆₁×C₂²⋊C₁₆	C₂	semidirect
4. C₂×C₆₁×C₂²⋊C₁₆	C₂	semidirect
5. C₆₁×#1374	C₂	semidirect
48. C₂×C₁₆×C₆₁	D₄	semidirect
104. C₂³×C₆₁	M₃₂	semidirect
121. C₂×C₄×C₆₁	M₃₂	semidirect
132. #12977	C₆₁	direct (N × H)
155. C₂²×M₃₂	C₂×C₆₁	semidirect
157. C₂×C₂²⋊C₁₆	C₂×C₆₁	semidirect
158. C₂×C₂²⋊C₁₆	C₂×C₆₁	semidirect
159. #1374	C₂×C₆₁	semidirect
182. C₆₁	#12977	direct (N × H)
210. C₂×C₁₆	D₄×C₆₁	semidirect
266. C₂³	M₃₂×C₆₁	semidirect
283. C₂×C₄	M₃₂×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₆₁	2
C₂×C₈×D₄×C₆₁	2
C₂×C₆₁×C₂²⋊C₁₆	2
C₆₁×#1374	2
#12977	61

Subgroups — C61×#12977 #29516410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12977 #29516410