Subgroups — C₆₁×#19717 #29523150

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3231	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3208	C₂	1	2	yes	maximal
3	C₆₁×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3138	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1720	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1719	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1714	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1708	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#533	C₂²	1	1	yes	characteristic
11	C₆₁×#533	C₂²	1	2	yes	—
12	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#500	C₂²	1	2	yes	complemented
14	C₆₁×#498	C₂²	1	1	yes	characteristic, complemented
15	C₆₁×#498	C₂²	1	2	yes	complemented
16	C₆₁×#494	C₂²	1	1	yes	characteristic, complemented
17	C₆₁×#492	C₂²	1	2	yes	—
18	C₆₁×#492	C₂²	1	2	yes	—
19	C₆₁×#490	—	2	2	no	—
20	C₆₁×#487	C₂²	1	2	yes	complemented
21	C₆₁×#487	C₂²	1	2	yes	complemented
22	C₆₁×#487	C₂²	1	2	yes	complemented
23	C₆₁×#486	C₂²	1	2	yes	—
24	C₆₁×#486	C₂²	1	2	yes	—
25	C₆₁×#486	C₂²	1	2	yes	—
26	C₆₁×#486	C₂²	1	2	yes	—
27	C₆₁×#485	—	2	4	no	—
28	C₆₁×#479	—	2	4	no	—
29	C₆₁×#470	—	2	2	no	—
30	C₆₁×#460	—	2	4	no	—
31	C₆₁×#450	—	2	4	no	—
32	C₆₁×#441	—	2	2	no	—
33	C₆₁×#431	C₂²	1	2	yes	—
34	C₆₁×#418	—	2	2	no	—
35	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#322	C₂²	1	2	yes	complemented
37	C₆₁×#322	C₂²	1	1	yes	characteristic, complemented
38	C₈²×C₆₁	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₆₁	—	2	1	no	—
40	C₂²×D₄×C₆₁	—	2	2	no	—
41	C₂×Q₁₆×C₆₁	—	2	2	no	—
42	C₂×SD₁₆×C₆₁	—	2	4	no	—
43	C₂×SD₁₆×C₆₁	—	2	4	no	—
44	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
45	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
48	#127×C₆₁	—	2	2	no	—
49	#127×C₆₁	—	2	1	no	—
50	#127×C₆₁	C₂³	1	2	yes	—
51	#125×C₆₁	—	2	2	no	—
52	#125×C₆₁	—	2	1	no	—
53	C₂²⋊Q₈×C₆₁	—	2	2	no	—
54	C₂²⋊Q₈×C₆₁	—	2	2	no	—
55	C₂²⋊Q₈×C₆₁	—	2	2	no	—
56	C₄×Q₈×C₆₁	—	2	2	no	—
57	C₄×Q₈×C₆₁	—	2	2	no	—
58	C₄×Q₈×C₆₁	—	2	2	no	—
59	#117×C₆₁	—	2	1	no	—
60	#117×C₆₁	—	2	2	no	—
61	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
62	#107×C₆₁	—	2	4	no	—
63	#106×C₆₁	—	2	2	no	—
64	#106×C₆₁	—	2	4	no	—
65	C₄⋊C₈×C₆₁	D₄	1	4	yes	—
66	#103×C₆₁	—	2	4	no	—
67	#103×C₆₁	—	2	4	no	—
68	#103×C₆₁	—	2	4	no	—
69	#102×C₆₁	—	2	4	no	—
70	#102×C₆₁	—	2	4	no	—
71	#102×C₆₁	—	2	4	no	—
72	#102×C₆₁	—	2	4	no	—
73	C₂²⋊C₈×C₆₁	D₄	1	4	yes	—
74	#97×C₆₁	D₄	1	2	yes	—
75	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₆₁	C₂³	1	2	yes	—
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂³	1	2	yes	—
79	C₂⁴×C₆₁	—	4	1	no	—
80	C₂×Q₈×C₆₁	—	2	2	no	—
81	C₂×Q₈×C₆₁	—	2	2	no	—
82	C₂×Q₈×C₆₁	—	2	2	no	—
83	C₂×D₄×C₆₁	—	2	2	no	—
84	C₂×D₄×C₆₁	—	2	2	no	—
85	C₂×D₄×C₆₁	—	2	2	no	—
86	C₂×D₄×C₆₁	—	4	4	no	—
87	C₂×D₄×C₆₁	—	4	1	no	—
88	C₂×D₄×C₆₁	—	4	2	no	—
89	C₂×D₄×C₆₁	—	8	4	no	—
90	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₆₁	—	4	4	no	—
93	SD₁₆×C₆₁	—	8	4	no	—
94	SD₁₆×C₆₁	—	8	4	no	—
95	M₁₆×C₆₁	—	4	2	no	—
96	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
99	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
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	—	2	2	no	—
104	C₄⋊C₄×C₆₁	—	2	2	no	—
105	C₄⋊C₄×C₆₁	—	2	2	no	—
106	C₄⋊C₄×C₆₁	—	4	2	no	—
107	C₄⋊C₄×C₆₁	—	4	2	no	—
108	C₄⋊C₄×C₆₁	—	4	2	no	—
109	C₂²⋊C₄×C₆₁	—	4	1	no	—
110	C₂²⋊C₄×C₆₁	—	4	2	no	—
111	C₄²×C₆₁	—	4	1	no	—
112	C₄²×C₆₁	—	4	2	no	—
113	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₁	C₂×D₄	1	2	yes	—
115	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₁	—	4	2	no	—
117	C₂³×C₆₁	—	8	2	no	—
118	C₂³×C₆₁	—	8	4	no	—
119	C₂³×C₆₁	#127	1	1	yes	characteristic
120	Q₈×C₆₁	—	4	4	no	—
121	Q₈×C₆₁	—	4	4	no	—
122	Q₈×C₆₁	—	4	4	no	—
123	D₄×C₆₁	—	4	4	no	—
124	D₄×C₆₁	—	4	4	no	—
125	D₄×C₆₁	—	4	4	no	—
126	D₄×C₆₁	—	8	2	no	—
127	D₄×C₆₁	—	8	4	no	—
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	2	4	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
134	C₂×C₄×C₆₁	#127	1	2	yes	—
135	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
137	C₈×C₆₁	—	2	2	no	—
138	C₈×C₆₁	—	2	2	no	—
139	C₈×C₆₁	—	2	4	no	—
140	C₈×C₆₁	—	2	4	no	—
141	#19717	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₆₁	—	2	1	no	—
143	C₂²×C₆₁	—	2	2	no	—
144	C₂²×C₆₁	—	8	2	no	—
145	C₂²×C₆₁	—	8	2	no	—
146	C₂²×C₆₁	—	8	2	no	—
147	C₂²×C₆₁	—	16	4	no	—
148	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	2	2	no	—
151	C₄×C₆₁	—	8	2	no	—
152	C₄×C₆₁	—	8	2	no	—
153	C₄×C₆₁	—	8	2	no	—
154	C₄×C₆₁	#575	1	2	yes	—
155	C₄×C₆₁	#577	1	4	yes	—
156	#3231	C₂×C₆₁	1	2	yes	complemented
157	#3208	C₂×C₆₁	1	2	yes	—
158	#3185	C₂×C₆₁	1	1	yes	characteristic, complemented
159	#3138	C₂×C₆₁	1	1	yes	characteristic
160	#1720	C₂×C₆₁	1	2	yes	complemented
161	#1719	C₂×C₆₁	1	2	yes	complemented
162	#1714	C₂×C₆₁	1	2	yes	complemented
163	#1708	C₂×C₆₁	1	2	yes	complemented
164	#1453	C₂×C₆₁	1	1	yes	characteristic, complemented
165	C₂×C₆₁	—	4	1	no	—
166	C₂×C₆₁	—	16	2	no	—
167	C₂×C₆₁	#3153	1	1	yes	characteristic
168	C₂×C₆₁	#3154	1	2	yes	—
169	#533	C₂²×C₆₁	1	1	yes	characteristic
170	#533	C₂²×C₆₁	1	2	yes	—
171	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
172	#500	C₂²×C₆₁	1	2	yes	complemented
173	#498	C₂²×C₆₁	1	1	yes	characteristic, complemented
174	#498	C₂²×C₆₁	1	2	yes	complemented
175	#494	C₂²×C₆₁	1	1	yes	characteristic, complemented
176	#492	C₂²×C₆₁	1	2	yes	—
177	#492	C₂²×C₆₁	1	2	yes	—
178	#490	—	2	2	no	—
179	#487	C₂²×C₆₁	1	2	yes	complemented
180	#487	C₂²×C₆₁	1	2	yes	complemented
181	#487	C₂²×C₆₁	1	2	yes	complemented
182	#486	C₂²×C₆₁	1	2	yes	—
183	#486	C₂²×C₆₁	1	2	yes	—
184	#486	C₂²×C₆₁	1	2	yes	—
185	#486	C₂²×C₆₁	1	2	yes	—
186	#485	—	2	4	no	—
187	#479	—	2	4	no	—
188	#470	—	2	2	no	—
189	#460	—	2	4	no	—
190	#450	—	2	4	no	—
191	#441	—	2	2	no	—
192	#431	C₂²×C₆₁	1	2	yes	—
193	#418	—	2	2	no	—
194	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
195	#322	C₂²×C₆₁	1	2	yes	complemented
196	#322	C₂²×C₆₁	1	1	yes	characteristic, complemented
197	C₈²	C₂²×C₆₁	1	1	yes	characteristic, complemented
198	C₆₁	#19717	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×D₄	—	2	1	no	—
200	C₂²×D₄	—	2	2	no	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×SD₁₆	—	2	4	no	—
204	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
208	#127	—	2	2	no	—
209	#127	—	2	1	no	—
210	#127	C₂³×C₆₁	1	2	yes	—
211	#125	—	2	2	no	—
212	#125	—	2	1	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	#117	—	2	1	no	—
220	#117	—	2	2	no	—
221	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
222	#107	—	2	4	no	—
223	#106	—	2	2	no	—
224	#106	—	2	4	no	—
225	C₄⋊C₈	D₄×C₆₁	1	4	yes	—
226	#103	—	2	4	no	—
227	#103	—	2	4	no	—
228	#103	—	2	4	no	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	#102	—	2	4	no	—
233	C₂²⋊C₈	D₄×C₆₁	1	4	yes	—
234	#97	D₄×C₆₁	1	2	yes	—
235	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₆₁	1	2	yes	—
237	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₆₁	1	2	yes	—
239	C₂⁴	—	4	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	8	4	no	—
250	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
252	Q₁₆	—	4	4	no	—
253	SD₁₆	—	8	4	no	—
254	SD₁₆	—	8	4	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
257	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
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	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₆₁	1	2	yes	—
275	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
276	C₂³	—	4	2	no	—
277	C₂³	—	8	2	no	—
278	C₂³	—	8	4	no	—
279	C₂³	#127×C₆₁	1	1	yes	characteristic
280	Q₈	—	4	4	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	4	no	—
286	D₄	—	8	2	no	—
287	D₄	—	8	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
294	C₂×C₄	#127×C₆₁	1	2	yes	—
295	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	4	no	—
300	C₈	—	2	4	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	16	4	no	—
307	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	C₆₁×#575	1	2	yes	—
314	C₄	C₆₁×#577	1	4	yes	—
315	C₂	—	4	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₆₁×#3153	1	1	yes	characteristic
318	C₂	C₆₁×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₂×#127×C₆₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3231	C₂	semidirect
3. C₆₁×#3185	C₂	semidirect
5. C₆₁×#1720	C₂	semidirect
6. C₆₁×#1719	C₂	semidirect
7. C₆₁×#1714	C₂	semidirect
8. C₆₁×#1708	C₂	semidirect
9. C₆₁×#1453	C₂	semidirect
13. C₆₁×#500	C₂²	semidirect
14. C₆₁×#498	C₂²	semidirect
15. C₆₁×#498	C₂²	semidirect
16. C₆₁×#494	C₂²	semidirect
20. C₆₁×#487	C₂²	semidirect
21. C₆₁×#487	C₂²	semidirect
22. C₆₁×#487	C₂²	semidirect
36. C₆₁×#322	C₂²	semidirect
37. C₆₁×#322	C₂²	semidirect
38. C₈²×C₆₁	C₂²	semidirect
141. #19717	C₆₁	direct (N × H)
156. #3231	C₂×C₆₁	semidirect
158. #3185	C₂×C₆₁	semidirect
160. #1720	C₂×C₆₁	semidirect
161. #1719	C₂×C₆₁	semidirect
162. #1714	C₂×C₆₁	semidirect
163. #1708	C₂×C₆₁	semidirect
164. #1453	C₂×C₆₁	semidirect
172. #500	C₂²×C₆₁	semidirect
173. #498	C₂²×C₆₁	semidirect
174. #498	C₂²×C₆₁	semidirect
175. #494	C₂²×C₆₁	semidirect
179. #487	C₂²×C₆₁	semidirect
180. #487	C₂²×C₆₁	semidirect
181. #487	C₂²×C₆₁	semidirect
195. #322	C₂²×C₆₁	semidirect
196. #322	C₂²×C₆₁	semidirect
197. C₈²	C₂²×C₆₁	semidirect
198. C₆₁	#19717	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3231	2
C₆₁×#3208	2
C₆₁×#3185	2
C₆₁×#3138	2
C₆₁×#1720	2
C₆₁×#1719	2
C₆₁×#1714	2
C₆₁×#1708	2
C₆₁×#1453	2
#19717	61

Subgroups — C61×#19717 #29523150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19717 #29523150