Subgroups — C₆₁×#20514 #29523947

Order 15616.

998 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×#443	C₂	1	2	yes	maximal
2	C₄×#135×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2875	C₂	1	2	yes	maximal, complemented
4	C₄×#108×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1772	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1765	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1759	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₁	C₂²	1	4	yes	—
11	C₄×#41×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#443	C₄	1	16	yes	complemented
13	C₄×Q₁₆×C₆₁	C₄	1	2	yes	complemented
14	C₄×SD₁₆×C₆₁	C₄	1	4	yes	complemented
15	C₄×D₈×C₆₁	C₄	1	2	yes	complemented
16	C₂×#108×C₆₁	C₂²	1	2	yes	—
17	C₆₁×#427	C₂²	1	1	yes	characteristic
18	C₂×#104×C₆₁	C₂²	1	4	yes	—
19	C₆₁×#416	C₂²	1	2	yes	—
20	C₆₁×#405	—	2	4	no	—
21	C₆₁×#405	C₂²	1	4	yes	—
22	C₄×M₁₆×C₆₁	—	2	2	no	—
23	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
24	C₂×#97×C₆₁	—	2	2	no	—
25	C₂×#97×C₆₁	C₂²	1	2	yes	—
26	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
29	C₆₁×#376	C₂²	1	1	yes	characteristic
30	C₆₁×#341	C₂²	1	2	yes	—
31	C₆₁×#339	C₂²	1	4	yes	—
32	C₂×#41×C₆₁	C₂³	1	2	yes	—
33	#135×C₆₁	C₂×C₄	1	8	yes	—
34	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×SD₁₆×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	#131×C₆₁	—	2	8	no	—
38	#131×C₆₁	C₂×C₄	1	16	yes	complemented
39	C₂×M₁₆×C₆₁	—	2	4	no	—
40	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
41	C₂²×C₈×C₆₁	—	2	4	no	—
42	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
45	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₆₁	—	2	4	no	—
47	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
48	#117×C₆₁	—	2	2	no	—
49	#117×C₆₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
52	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
53	#108×C₆₁	C₂×C₄	1	8	yes	complemented
54	#107×C₆₁	C₂×C₄	1	2	yes	—
55	#106×C₆₁	C₂×C₄	1	2	yes	—
56	#104×C₆₁	C₂×C₄	1	16	yes	complemented
57	#103×C₆₁	C₂×C₄	1	4	yes	—
58	#102×C₆₁	C₂×C₄	1	4	yes	—
59	#97×C₆₁	—	2	4	no	—
60	#97×C₆₁	—	2	4	no	—
61	#97×C₆₁	—	2	2	no	—
62	#97×C₆₁	D₄	1	1	yes	characteristic
63	#97×C₆₁	D₄	1	1	yes	characteristic
64	#97×C₆₁	D₄	1	2	yes	—
65	C₄×C₈×C₆₁	—	2	4	no	—
66	C₄×C₈×C₆₁	—	2	2	no	—
67	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	complemented
70	#95×C₆₁	—	2	2	no	—
71	#41×C₆₁	—	2	4	no	—
72	#41×C₆₁	C₂²×C₄	1	8	yes	—
73	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₆₁	—	2	2	no	—
75	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₁	—	2	2	no	—
77	C₂²×C₄×C₆₁	—	2	2	no	—
78	C₂²×C₄×C₆₁	—	2	4	no	—
79	C₂²×C₄×C₆₁	#41	1	2	yes	—
80	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₆₁	C₄²	1	4	yes	—
82	SD₁₆×C₆₁	C₄²	1	8	yes	—
83	D₈×C₆₁	C₄²	1	4	yes	—
84	M₁₆×C₆₁	—	2	4	no	—
85	M₁₆×C₆₁	—	2	16	no	—
86	M₁₆×C₆₁	C₂²×C₄	1	8	yes	—
87	C₂×C₈×C₆₁	—	2	4	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	—	2	16	no	—
90	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₁	#41	1	4	yes	—
98	C₄⋊C₄×C₆₁	—	2	4	no	—
99	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄×C₆₁	—	2	2	no	—
101	C₂²⋊C₄×C₆₁	—	2	4	no	—
102	C₄²×C₆₁	—	2	4	no	—
103	C₄²×C₆₁	—	2	2	no	—
104	C₄²×C₆₁	—	2	2	no	—
105	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
106	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
107	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₆₁	—	2	2	no	—
110	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
111	Q₈×C₆₁	—	2	2	no	—
112	Q₈×C₆₁	C₂×C₄²	1	4	yes	—
113	D₄×C₆₁	—	2	8	no	—
114	D₄×C₆₁	—	2	2	no	—
115	D₄×C₆₁	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	8	no	—
118	C₂×C₄×C₆₁	—	2	4	no	—
119	C₂×C₄×C₆₁	—	2	4	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	—	4	4	no	—
122	C₂×C₄×C₆₁	—	8	4	no	—
123	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
126	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
129	C₈×C₆₁	—	2	8	no	—
130	C₈×C₆₁	—	2	2	no	—
131	C₈×C₆₁	C₂×C₄²	1	2	yes	—
132	C₈×C₆₁	C₂×C₄²	1	2	yes	—
133	C₈×C₆₁	C₄×D₄	1	4	yes	—
134	C₈×C₆₁	C₄×D₄	1	4	yes	—
135	#20514	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₆₁	—	2	2	no	—
137	C₂²×C₆₁	—	2	4	no	—
138	C₂²×C₆₁	—	4	4	no	—
139	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₆₁	C₄×#41	1	2	yes	—
141	C₄×C₆₁	—	2	2	no	—
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	—	2	4	no	—
144	C₄×C₆₁	—	4	4	no	—
145	C₄×C₆₁	—	4	4	no	—
146	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
148	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
149	C₂×#443	C₂×C₆₁	1	2	yes	—
150	C₄×#135	C₂×C₆₁	1	1	yes	characteristic
151	#2875	C₂×C₆₁	1	2	yes	complemented
152	C₄×#108	C₂×C₆₁	1	1	yes	characteristic, complemented
153	#1772	C₂×C₆₁	1	2	yes	complemented
154	#1765	C₂×C₆₁	1	4	yes	complemented
155	#1759	C₂×C₆₁	1	2	yes	complemented
156	#1726	C₂×C₆₁	1	1	yes	characteristic, complemented
157	C₂×C₆₁	—	2	2	no	—
158	C₂×C₆₁	—	4	4	no	—
159	C₂×C₆₁	C₄²×D₄	1	1	yes	characteristic
160	C₂×C₆₁	#2957	1	2	yes	—
161	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
162	C₂×#131	C₂²×C₆₁	1	4	yes	—
163	C₄×#41	C₂²×C₆₁	1	2	yes	—
164	#443	C₄×C₆₁	1	16	yes	complemented
165	C₄×Q₁₆	C₄×C₆₁	1	2	yes	complemented
166	C₄×SD₁₆	C₄×C₆₁	1	4	yes	complemented
167	C₄×D₈	C₄×C₆₁	1	2	yes	complemented
168	C₂×#108	C₂²×C₆₁	1	2	yes	—
169	#427	C₂²×C₆₁	1	1	yes	characteristic
170	C₂×#104	C₂²×C₆₁	1	4	yes	—
171	#416	C₂²×C₆₁	1	2	yes	—
172	#405	—	2	4	no	—
173	#405	C₂²×C₆₁	1	4	yes	—
174	C₄×M₁₆	—	2	2	no	—
175	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
176	C₂×#97	—	2	2	no	—
177	C₂×#97	C₂²×C₆₁	1	2	yes	—
178	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
181	#376	C₂²×C₆₁	1	1	yes	characteristic
182	#341	C₂²×C₆₁	1	2	yes	—
183	#339	C₂²×C₆₁	1	4	yes	—
184	C₆₁	#20514	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×#41	C₂³×C₆₁	1	2	yes	—
186	#135	C₂×C₄×C₆₁	1	8	yes	—
187	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
188	C₂×SD₁₆	C₂×C₄×C₆₁	1	2	yes	—
189	C₂×D₈	C₂×C₄×C₆₁	1	1	yes	characteristic
190	#131	—	2	8	no	—
191	#131	C₂×C₄×C₆₁	1	16	yes	complemented
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
198	C₄×Q₈	C₂×C₄×C₆₁	1	4	yes	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
201	#117	—	2	2	no	—
202	#117	C₂³×C₆₁	1	2	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
205	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
206	#108	C₂×C₄×C₆₁	1	8	yes	complemented
207	#107	C₂×C₄×C₆₁	1	2	yes	—
208	#106	C₂×C₄×C₆₁	1	2	yes	—
209	#104	C₂×C₄×C₆₁	1	16	yes	complemented
210	#103	C₂×C₄×C₆₁	1	4	yes	—
211	#102	C₂×C₄×C₆₁	1	4	yes	—
212	#97	—	2	4	no	—
213	#97	—	2	4	no	—
214	#97	—	2	2	no	—
215	#97	D₄×C₆₁	1	1	yes	characteristic
216	#97	D₄×C₆₁	1	1	yes	characteristic
217	#97	D₄×C₆₁	1	2	yes	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
221	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
222	C₄×C₈	C₂×C₄×C₆₁	1	8	yes	complemented
223	#95	—	2	2	no	—
224	#41	—	2	4	no	—
225	#41	C₂²×C₄×C₆₁	1	8	yes	—
226	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	#41×C₆₁	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
234	Q₁₆	C₄²×C₆₁	1	4	yes	—
235	SD₁₆	C₄²×C₆₁	1	8	yes	—
236	D₈	C₄²×C₆₁	1	4	yes	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	16	no	—
239	M₁₆	C₂²×C₄×C₆₁	1	8	yes	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	16	no	—
243	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
244	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
245	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
247	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
248	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
250	C₂×C₈	#41×C₆₁	1	4	yes	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
259	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
260	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	C₂×C₄²×C₆₁	1	4	yes	—
266	D₄	—	2	8	no	—
267	D₄	—	2	2	no	—
268	D₄	C₂×C₄²×C₆₁	1	4	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
277	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
278	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
279	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
282	C₈	—	2	8	no	—
283	C₈	—	2	2	no	—
284	C₈	C₂×C₄²×C₆₁	1	2	yes	—
285	C₈	C₂×C₄²×C₆₁	1	2	yes	—
286	C₈	C₄×D₄×C₆₁	1	4	yes	—
287	C₈	C₄×D₄×C₆₁	1	4	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
292	C₂²	C₄×#41×C₆₁	1	2	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	4	4	no	—
298	C₄	C₂²×C₄²×C₆₁	1	1	yes	characteristic
299	C₄	C₂×C₄×D₄×C₆₁	1	2	yes	—
300	C₄	C₄×#41×C₆₁	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	4	no	—
303	C₂	C₄²×D₄×C₆₁	1	1	yes	characteristic
304	C₂	C₆₁×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#135×C₆₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×#97×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#2875	C₂	semidirect
4. C₄×#108×C₆₁	C₂	semidirect
5. C₆₁×#1772	C₂	semidirect
6. C₆₁×#1765	C₂	semidirect
7. C₆₁×#1759	C₂	semidirect
8. C₆₁×#1726	C₂	semidirect
12. C₆₁×#443	C₄	semidirect
13. C₄×Q₁₆×C₆₁	C₄	semidirect
14. C₄×SD₁₆×C₆₁	C₄	semidirect
15. C₄×D₈×C₆₁	C₄	semidirect
38. #131×C₆₁	C₂×C₄	semidirect
53. #108×C₆₁	C₂×C₄	semidirect
56. #104×C₆₁	C₂×C₄	semidirect
69. C₄×C₈×C₆₁	C₂×C₄	semidirect
135. #20514	C₆₁	direct (N × H)
151. #2875	C₂×C₆₁	semidirect
152. C₄×#108	C₂×C₆₁	semidirect
153. #1772	C₂×C₆₁	semidirect
154. #1765	C₂×C₆₁	semidirect
155. #1759	C₂×C₆₁	semidirect
156. #1726	C₂×C₆₁	semidirect
164. #443	C₄×C₆₁	semidirect
165. C₄×Q₁₆	C₄×C₆₁	semidirect
166. C₄×SD₁₆	C₄×C₆₁	semidirect
167. C₄×D₈	C₄×C₆₁	semidirect
184. C₆₁	#20514	direct (N × H)
191. #131	C₂×C₄×C₆₁	semidirect
206. #108	C₂×C₄×C₆₁	semidirect
209. #104	C₂×C₄×C₆₁	semidirect
222. C₄×C₈	C₂×C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#443	2
C₄×#135×C₆₁	2
C₆₁×#2875	2
C₄×#108×C₆₁	2
C₆₁×#1772	2
C₆₁×#1765	2
C₆₁×#1759	2
C₆₁×#1726	2
#20514	61

Subgroups — C61×#20514 #29523947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#20514 #29523947