Subgroups — C₆₁×#18062 #29521495

Order 15616.

618 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	C₆₁×#2984	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1593	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1584	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1579	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1568	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1491	C₂	1	2	yes	maximal
8	C₆₁×#1451	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#446	—	2	4	no	—
12	C₆₁×#446	C₂²	1	4	yes	—
13	C₈×Q₈×C₆₁	C₂²	1	4	yes	—
14	C₄×Q₁₆×C₆₁	C₄	1	2	yes	—
15	C₄×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
16	C₄×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
17	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	C₆₁×#432	—	2	2	no	—
21	C₆₁×#432	C₂²	1	2	yes	—
22	C₆₁×#431	—	2	2	no	—
23	C₆₁×#431	C₂²	1	1	yes	characteristic
24	C₂×#107×C₆₁	C₄	1	1	yes	characteristic
25	C₂×#103×C₆₁	C₄	1	2	yes	—
26	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#335	C₂²	1	2	yes	—
29	C₆₁×#334	C₂²	1	2	yes	—
30	C₆₁×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₆₁×Q₈⋊C₈	C₂²	1	4	yes	—
32	C₆₁×#322	C₂²	1	2	yes	—
33	C₆₁×#322	C₂²	1	2	yes	—
34	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₂×Q₁₆×C₆₁	—	2	4	no	—
36	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₆₁	—	2	1	no	—
40	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₆₁	—	2	2	no	—
43	C₄×Q₈×C₆₁	—	2	2	no	—
44	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
47	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
52	#107×C₆₁	C₂×C₄	1	2	yes	—
53	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
54	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₈×C₆₁	—	2	4	no	—
56	C₄⋊C₈×C₆₁	—	2	4	no	—
57	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
58	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
59	#103×C₆₁	C₂×C₄	1	2	yes	—
60	#103×C₆₁	C₂×C₄	1	2	yes	—
61	#103×C₆₁	C₂×C₄	1	2	yes	—
62	#103×C₆₁	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₆₁	—	2	2	no	—
64	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
65	#97×C₆₁	—	2	4	no	—
66	#97×C₆₁	—	2	2	no	—
67	#97×C₆₁	—	2	1	no	—
68	C₄×C₈×C₆₁	—	2	4	no	—
69	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₆₁	C₂³	1	2	yes	—
72	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₁	C₂³	1	2	yes	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	2	4	no	—
77	C₂×Q₈×C₆₁	—	4	2	no	—
78	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₁	—	2	2	no	—
81	C₂²×C₄×C₆₁	—	2	2	no	—
82	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
85	Q₁₆×C₆₁	—	2	4	no	—
86	Q₁₆×C₆₁	—	2	2	no	—
87	Q₁₆×C₆₁	M₁₆	1	4	yes	complemented
88	M₁₆×C₆₁	—	4	2	no	—
89	C₂×C₈×C₆₁	—	2	2	no	—
90	C₂×C₈×C₆₁	—	2	4	no	—
91	C₂×C₈×C₆₁	—	2	2	no	—
92	C₂×C₈×C₆₁	—	4	1	no	—
93	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₄⋊C₄×C₆₁	—	2	2	no	—
101	C₄⋊C₄×C₆₁	—	2	2	no	—
102	C₄⋊C₄×C₆₁	—	2	2	no	—
103	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
104	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
105	C₄²×C₆₁	—	2	2	no	—
106	C₄²×C₆₁	—	2	2	no	—
107	C₄²×C₆₁	—	2	2	no	—
108	C₄²×C₆₁	—	2	2	no	—
109	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₁	#41	1	1	yes	characteristic
112	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
113	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
114	Q₈×C₆₁	—	2	2	no	—
115	Q₈×C₆₁	—	2	2	no	—
116	Q₈×C₆₁	—	2	2	no	—
117	Q₈×C₆₁	—	4	4	no	—
118	Q₈×C₆₁	C₂×M₁₆	1	4	yes	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	2	1	no	—
122	C₂×C₄×C₆₁	—	2	1	no	—
123	C₂×C₄×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	4	4	no	—
128	C₂×C₄×C₆₁	—	4	4	no	—
129	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
135	C₈×C₆₁	—	2	1	no	—
136	C₈×C₆₁	—	2	1	no	—
137	C₈×C₆₁	—	2	1	no	—
138	C₈×C₆₁	—	2	4	no	—
139	C₈×C₆₁	—	4	2	no	—
140	C₈×C₆₁	—	4	4	no	—
141	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
142	#18062	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₆₁	—	2	1	no	—
144	C₂²×C₆₁	—	2	1	no	—
145	C₂²×C₆₁	#444	1	2	yes	—
146	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	4	no	—
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	2	1	no	—
151	C₄×C₆₁	—	2	1	no	—
152	C₄×C₆₁	—	4	2	no	—
153	C₄×C₆₁	—	4	2	no	—
154	C₄×C₆₁	—	4	2	no	—
155	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
156	C₄×C₆₁	#568	1	1	yes	characteristic
157	C₄×C₆₁	#578	1	2	yes	—
158	#2984	C₂×C₆₁	1	2	yes	—
159	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic
160	#1593	C₂×C₆₁	1	2	yes	complemented
161	#1584	C₂×C₆₁	1	4	yes	complemented
162	#1579	C₂×C₆₁	1	2	yes	complemented
163	#1568	C₂×C₆₁	1	1	yes	characteristic
164	#1491	C₂×C₆₁	1	2	yes	—
165	#1451	C₂×C₆₁	1	1	yes	characteristic
166	C₂×C₆₁	—	2	2	no	—
167	C₂×C₆₁	D₄×M₁₆	1	1	yes	characteristic
168	C₂×C₆₁	#2950	1	1	yes	characteristic
169	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
170	C₂²×Q₁₆	C₄×C₆₁	1	1	yes	characteristic
171	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
172	#446	—	2	4	no	—
173	#446	C₂²×C₆₁	1	4	yes	—
174	C₈×Q₈	C₂²×C₆₁	1	4	yes	—
175	C₄×Q₁₆	C₄×C₆₁	1	2	yes	—
176	C₄×Q₁₆	C₄×C₆₁	1	1	yes	characteristic
177	C₄×Q₁₆	C₄×C₆₁	1	1	yes	characteristic
178	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
179	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
180	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
181	#432	—	2	2	no	—
182	#432	C₂²×C₆₁	1	2	yes	—
183	#431	—	2	2	no	—
184	#431	C₂²×C₆₁	1	1	yes	characteristic
185	C₂×#107	C₄×C₆₁	1	1	yes	characteristic
186	C₂×#103	C₄×C₆₁	1	2	yes	—
187	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
189	#335	C₂²×C₆₁	1	2	yes	—
190	#334	C₂²×C₆₁	1	2	yes	—
191	Q₈⋊C₈	C₂²×C₆₁	1	4	yes	—
192	Q₈⋊C₈	C₂²×C₆₁	1	4	yes	—
193	#322	C₂²×C₆₁	1	2	yes	—
194	#322	C₂²×C₆₁	1	2	yes	—
195	C₆₁	#18062	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²×Q₈	C₂×C₄×C₆₁	1	2	yes	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	C₂×C₄×C₆₁	1	2	yes	—
199	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
200	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
201	C₂×M₁₆	—	2	1	no	—
202	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
203	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
207	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
208	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
209	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
214	#107	C₂×C₄×C₆₁	1	2	yes	—
215	#107	C₂×C₄×C₆₁	1	1	yes	characteristic
216	#107	C₂×C₄×C₆₁	1	1	yes	characteristic
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
220	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
221	#103	C₂×C₄×C₆₁	1	2	yes	—
222	#103	C₂×C₄×C₆₁	1	2	yes	—
223	#103	C₂×C₄×C₆₁	1	2	yes	—
224	#103	C₂×C₄×C₆₁	1	2	yes	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
227	#97	—	2	4	no	—
228	#97	—	2	2	no	—
229	#97	—	2	1	no	—
230	C₄×C₈	—	2	4	no	—
231	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
232	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₆₁	1	2	yes	—
234	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₆₁	1	2	yes	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
241	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
246	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	M₁₆×C₆₁	1	4	yes	complemented
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
257	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
266	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
273	C₄²	#41×C₆₁	1	1	yes	characteristic
274	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
275	C₂³	C₄×D₄×C₆₁	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	2	2	no	—
279	Q₈	—	4	4	no	—
280	Q₈	C₂×M₁₆×C₆₁	1	4	yes	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
292	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
293	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
294	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	4	no	—
301	C₈	—	4	2	no	—
302	C₈	—	4	4	no	—
303	C₈	C₂×M₁₆×C₆₁	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	C₆₁×#444	1	2	yes	—
307	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	4	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	C₂²×M₁₆×C₆₁	1	1	yes	characteristic
317	C₄	C₆₁×#568	1	1	yes	characteristic
318	C₄	C₆₁×#578	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	D₄×M₁₆×C₆₁	1	1	yes	characteristic
321	C₂	C₆₁×#2950	1	1	yes	characteristic
322	C₂	C₂×C₆₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₄²	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₆₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1593	C₂	semidirect
4. C₆₁×#1584	C₂	semidirect
5. C₆₁×#1579	C₂	semidirect
87. Q₁₆×C₆₁	M₁₆	semidirect
142. #18062	C₆₁	direct (N × H)
160. #1593	C₂×C₆₁	semidirect
161. #1584	C₂×C₆₁	semidirect
162. #1579	C₂×C₆₁	semidirect
195. C₆₁	#18062	direct (N × H)
249. Q₁₆	M₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2984	2
C₂×C₄×Q₁₆×C₆₁	2
C₆₁×#1593	2
C₆₁×#1584	2
C₆₁×#1579	2
C₆₁×#1568	2
C₆₁×#1491	2
C₆₁×#1451	2
#18062	61

Subgroups — C61×#18062 #29521495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18062 #29521495