Subgroups — C₂×C₆₁×#1971 #29524658

Order 15616.

1574 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#103×C₆₁	C₂	1	2	yes	maximal
4	C₂²×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#391	C₂	1	2	yes	maximal
6	C₆₁×#1971	C₂	1	8	yes	maximal, direct product factor
7	C₂³×Q₈×C₆₁	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₆₁	—	2	4	no	—
9	C₂²×Q₁₆×C₆₁	—	2	4	no	—
10	C₂²×Q₁₆×C₆₁	C₄	1	8	yes	complemented
11	C₂²×Q₁₆×C₆₁	C₂²	1	4	yes	—
12	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#106×C₆₁	—	2	4	no	—
17	C₂×#106×C₆₁	C₂²	1	4	yes	—
18	C₂×#103×C₆₁	—	2	8	no	—
19	C₂×#103×C₆₁	C₂²	1	8	yes	—
20	C₂×#97×C₆₁	—	2	4	no	—
21	C₂×#97×C₆₁	C₂²	1	4	yes	—
22	C₆₁×#391	—	2	4	no	—
23	C₆₁×#391	C₂²	1	8	yes	—
24	C₂×#95×C₆₁	—	2	4	no	—
25	C₂²×Q₈×C₆₁	—	2	2	no	—
26	C₂²×Q₈×C₆₁	—	2	4	no	—
27	C₂²×Q₈×C₆₁	—	2	4	no	—
28	C₂²×Q₈×C₆₁	—	2	4	no	—
29	C₂²×Q₈×C₆₁	—	2	4	no	—
30	C₂²×Q₈×C₆₁	—	2	4	no	—
31	C₂²×Q₈×C₆₁	C₂×C₄	1	8	yes	—
32	C₂²×Q₈×C₆₁	C₂³	1	4	yes	—
33	C₂³×C₄×C₆₁	—	2	2	no	—
34	C₂³×C₄×C₆₁	—	2	2	no	—
35	C₂³×C₄×C₆₁	D₄	1	2	yes	—
36	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₆₁	—	2	32	no	—
38	C₂×Q₁₆×C₆₁	—	2	8	no	—
39	C₂×Q₁₆×C₆₁	—	2	8	no	—
40	C₂×Q₁₆×C₆₁	C₂×C₄	1	16	yes	complemented
41	C₂²×C₈×C₆₁	—	2	2	no	—
42	C₂²×C₈×C₆₁	—	2	2	no	—
43	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₆₁	D₄	1	4	yes	—
45	C₂²×C₈×C₆₁	D₄	1	4	yes	—
46	C₂²×C₈×C₆₁	D₄	1	4	yes	—
47	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
50	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
51	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
53	C₂×C₄²×C₆₁	—	2	2	no	—
54	C₂×C₄²×C₆₁	—	2	2	no	—
55	C₂×C₄²×C₆₁	D₄	1	2	yes	—
56	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
57	#106×C₆₁	—	2	8	no	—
58	#103×C₆₁	—	2	16	no	—
59	#97×C₆₁	—	2	8	no	—
60	#95×C₆₁	—	2	16	no	—
61	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₆₁	—	2	8	no	—
63	C₂×Q₈×C₆₁	—	2	4	no	—
64	C₂×Q₈×C₆₁	—	2	16	no	—
65	C₂×Q₈×C₆₁	—	2	16	no	—
66	C₂×Q₈×C₆₁	—	2	4	no	—
67	C₂×Q₈×C₆₁	—	2	4	no	—
68	C₂×Q₈×C₆₁	—	2	4	no	—
69	C₂×Q₈×C₆₁	—	2	4	no	—
70	C₂×Q₈×C₆₁	—	4	4	no	—
71	C₂×Q₈×C₆₁	—	4	8	no	—
72	C₂×Q₈×C₆₁	—	4	8	no	—
73	C₂×Q₈×C₆₁	—	4	4	no	—
74	C₂×Q₈×C₆₁	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₆₁	—	2	8	no	—
76	C₂²×C₄×C₆₁	—	2	4	no	—
77	C₂²×C₄×C₆₁	—	2	8	no	—
78	C₂²×C₄×C₆₁	—	2	4	no	—
79	C₂²×C₄×C₆₁	—	2	4	no	—
80	C₂²×C₄×C₆₁	—	4	4	no	—
81	C₂²×C₄×C₆₁	—	4	4	no	—
82	C₂²×C₄×C₆₁	—	4	4	no	—
83	C₂²×C₄×C₆₁	—	4	8	no	—
84	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₁	#41	1	2	yes	—
91	C₂²×C₄×C₆₁	#41	1	2	yes	—
92	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₆₁	—	2	32	no	—
94	C₂×C₈×C₆₁	—	2	8	no	—
95	C₂×C₈×C₆₁	—	2	2	no	—
96	C₂×C₈×C₆₁	—	2	2	no	—
97	C₂×C₈×C₆₁	C₂²⋊C₄	1	16	yes	—
98	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
102	C₄⋊C₄×C₆₁	—	2	4	no	—
103	C₄⋊C₄×C₆₁	—	2	4	no	—
104	C₄⋊C₄×C₆₁	—	4	4	no	—
105	C₄²×C₆₁	—	2	4	no	—
106	C₄²×C₆₁	—	2	2	no	—
107	C₄²×C₆₁	—	2	2	no	—
108	C₂³×C₆₁	C₄×D₄	1	2	yes	—
109	C₂³×C₆₁	#121	1	2	yes	—
110	C₂³×C₆₁	#127	1	2	yes	—
111	C₂³×C₆₁	#137	1	8	yes	—
112	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₆₁	—	2	8	no	—
114	Q₈×C₆₁	—	2	8	no	—
115	Q₈×C₆₁	—	4	4	no	—
116	Q₈×C₆₁	—	4	4	no	—
117	Q₈×C₆₁	—	4	16	no	—
118	C₂×C₄×C₆₁	—	2	8	no	—
119	C₂×C₄×C₆₁	—	2	4	no	—
120	C₂×C₄×C₆₁	—	2	8	no	—
121	C₂×C₄×C₆₁	—	4	4	no	—
122	C₂×C₄×C₆₁	—	4	4	no	—
123	C₂×C₄×C₆₁	—	4	16	no	—
124	C₂×C₄×C₆₁	—	4	8	no	—
125	C₂×C₄×C₆₁	—	4	4	no	—
126	C₂×C₄×C₆₁	—	4	8	no	—
127	C₂×C₄×C₆₁	—	8	8	no	—
128	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
129	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
130	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₆₁	#121	1	4	yes	—
132	C₂×C₄×C₆₁	#121	1	2	yes	—
133	C₂×C₄×C₆₁	#124	1	4	yes	—
134	C₂×C₄×C₆₁	#127	1	2	yes	—
135	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₆₁	—	2	4	no	—
142	C₈×C₆₁	C₂×C₂²⋊C₄	1	8	yes	—
143	C₂×#1971	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₆₁	#390	1	4	yes	—
145	C₂²×C₆₁	#441	1	8	yes	—
146	C₂²×C₆₁	#470	1	8	yes	—
147	C₂²×C₆₁	#497	1	8	yes	—
148	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
150	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
151	C₂²×C₆₁	C₂×#137	1	4	yes	—
152	C₄×C₆₁	—	4	4	no	—
153	C₄×C₆₁	—	4	8	no	—
154	C₄×C₆₁	—	8	4	no	—
155	C₄×C₆₁	#390	1	4	yes	—
156	C₄×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
159	C₄×C₆₁	C₂×#124	1	1	yes	characteristic
160	C₂³×Q₁₆	C₂×C₆₁	1	1	yes	characteristic
161	C₂²×#106	C₂×C₆₁	1	1	yes	characteristic
162	C₂²×#103	C₂×C₆₁	1	2	yes	—
163	C₂²×#97	C₂×C₆₁	1	1	yes	characteristic
164	C₂×#391	C₂×C₆₁	1	2	yes	—
165	#1971	C₂×C₆₁	1	8	yes	direct product factor
166	C₂×C₆₁	#1971	1	8	yes	direct product factor
167	C₂×C₆₁	C₂×#390	1	1	yes	characteristic
168	C₂×C₆₁	C₂×#441	1	2	yes	—
169	C₂×C₆₁	C₂×#470	1	2	yes	—
170	C₂×C₆₁	C₂×#497	1	2	yes	—
171	C₂³×Q₈	C₂²×C₆₁	1	2	yes	—
172	C₂²×Q₁₆	—	2	4	no	—
173	C₂²×Q₁₆	—	2	4	no	—
174	C₂²×Q₁₆	C₄×C₆₁	1	8	yes	complemented
175	C₂²×Q₁₆	C₂²×C₆₁	1	4	yes	—
176	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
177	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
178	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
179	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
180	C₂×#106	—	2	4	no	—
181	C₂×#106	C₂²×C₆₁	1	4	yes	—
182	C₂×#103	—	2	8	no	—
183	C₂×#103	C₂²×C₆₁	1	8	yes	—
184	C₂×#97	—	2	4	no	—
185	C₂×#97	C₂²×C₆₁	1	4	yes	—
186	#391	—	2	4	no	—
187	#391	C₂²×C₆₁	1	8	yes	—
188	C₂×#95	—	2	4	no	—
189	C₆₁	C₂×#1971	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×Q₈	—	2	2	no	—
191	C₂²×Q₈	—	2	4	no	—
192	C₂²×Q₈	—	2	4	no	—
193	C₂²×Q₈	—	2	4	no	—
194	C₂²×Q₈	—	2	4	no	—
195	C₂²×Q₈	—	2	4	no	—
196	C₂²×Q₈	C₂×C₄×C₆₁	1	8	yes	—
197	C₂²×Q₈	C₂³×C₆₁	1	4	yes	—
198	C₂³×C₄	—	2	2	no	—
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	D₄×C₆₁	1	2	yes	—
201	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
202	C₂×Q₁₆	—	2	32	no	—
203	C₂×Q₁₆	—	2	8	no	—
204	C₂×Q₁₆	—	2	8	no	—
205	C₂×Q₁₆	C₂×C₄×C₆₁	1	16	yes	complemented
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	C₂×C₄×C₆₁	1	4	yes	—
209	C₂²×C₈	D₄×C₆₁	1	4	yes	—
210	C₂²×C₈	D₄×C₆₁	1	4	yes	—
211	C₂²×C₈	D₄×C₆₁	1	4	yes	—
212	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
213	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	D₄×C₆₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
222	#106	—	2	8	no	—
223	#103	—	2	16	no	—
224	#97	—	2	8	no	—
225	#95	—	2	16	no	—
226	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	16	no	—
230	C₂×Q₈	—	2	16	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	8	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	C₂²×C₄×C₆₁	1	8	yes	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	—	4	8	no	—
249	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
251	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
255	C₂²×C₄	#41×C₆₁	1	2	yes	—
256	C₂²×C₄	#41×C₆₁	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
258	Q₁₆	—	2	32	no	—
259	C₂×C₈	—	2	8	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂²⋊C₄×C₆₁	1	16	yes	—
263	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
264	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
266	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₂³	C₄×D₄×C₆₁	1	2	yes	—
274	C₂³	#121×C₆₁	1	2	yes	—
275	C₂³	#127×C₆₁	1	2	yes	—
276	C₂³	#137×C₆₁	1	8	yes	—
277	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	2	8	no	—
280	Q₈	—	4	16	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	4	4	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	16	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	8	8	no	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
294	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
295	C₂×C₄	C₄×D₄×C₆₁	1	4	yes	—
296	C₂×C₄	#121×C₆₁	1	2	yes	—
297	C₂×C₄	#121×C₆₁	1	4	yes	—
298	C₂×C₄	#124×C₆₁	1	4	yes	—
299	C₂×C₄	#127×C₆₁	1	2	yes	—
300	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
306	C₈	—	2	4	no	—
307	C₈	C₂×C₂²⋊C₄×C₆₁	1	8	yes	—
308	C₂²	C₆₁×#390	1	4	yes	—
309	C₂²	C₆₁×#441	1	8	yes	—
310	C₂²	C₆₁×#470	1	8	yes	—
311	C₂²	C₆₁×#497	1	8	yes	—
312	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
313	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
314	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
315	C₂²	C₂×#137×C₆₁	1	4	yes	—
316	C₄	—	4	4	no	—
317	C₄	—	4	8	no	—
318	C₄	—	8	4	no	—
319	C₄	C₆₁×#390	1	4	yes	—
320	C₄	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
321	C₄	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
322	C₄	C₂×#121×C₆₁	1	1	yes	characteristic
323	C₄	C₂×#124×C₆₁	1	1	yes	characteristic
324	C₂	C₆₁×#1971	1	8	yes	direct product factor
325	C₂	C₂×C₆₁×#390	1	1	yes	characteristic
326	C₂	C₂×C₆₁×#441	1	2	yes	—
327	C₂	C₂×C₆₁×#470	1	2	yes	—
328	C₂	C₂×C₆₁×#497	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
6. C₆₁×#1971	C₂	direct (N × H)
10. C₂²×Q₁₆×C₆₁	C₄	semidirect
40. C₂×Q₁₆×C₆₁	C₂×C₄	semidirect
143. C₂×#1971	C₆₁	direct (N × H)
165. #1971	C₂×C₆₁	direct (N × H)
166. C₂×C₆₁	#1971	direct (N × H)
174. C₂²×Q₁₆	C₄×C₆₁	semidirect
189. C₆₁	C₂×#1971	direct (N × H)
205. C₂×Q₁₆	C₂×C₄×C₆₁	semidirect
324. C₂	C₆₁×#1971	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₆₁	2
C₂²×#106×C₆₁	2
C₂²×#103×C₆₁	2
C₂²×#97×C₆₁	2
C₂×C₆₁×#391	2
C₆₁×#1971	2
C₂×#1971	61

Subgroups — C2×C61×#1971 #29524658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1971 #29524658