Subgroups — C₆₁×#19729 #29523162

Order 15616.

788 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3210	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3205	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1721	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1720	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1717	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1709	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×#532	C₂²	1	2	yes	—
12	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#498	C₂²	1	1	yes	characteristic, complemented
14	C₆₁×#494	C₂²	1	2	yes	complemented
15	C₆₁×#493	C₂²	1	2	yes	—
16	C₆₁×#493	C₂²	1	2	yes	—
17	C₆₁×#493	C₂²	1	1	yes	characteristic, complemented
18	C₆₁×#492	C₂²	1	2	yes	complemented
19	C₆₁×#490	—	2	2	no	—
20	C₆₁×#487	C₂²	1	2	yes	complemented
21	C₆₁×#486	C₂²	1	2	yes	—
22	C₆₁×#486	C₂²	1	2	yes	—
23	C₆₁×#486	C₂²	1	2	yes	—
24	C₆₁×#486	C₂²	1	2	yes	—
25	C₆₁×#486	C₂²	1	2	yes	complemented
26	C₆₁×#486	C₂²	1	2	yes	complemented
27	C₆₁×#469	—	2	2	no	—
28	C₆₁×#464	—	2	4	no	—
29	C₆₁×#459	—	2	4	no	—
30	C₆₁×#452	—	2	4	no	—
31	C₆₁×#447	—	2	4	no	—
32	C₆₁×#442	—	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	1	yes	characteristic, complemented
37	C₆₁×#322	C₂²	1	2	yes	complemented
38	C₈²×C₆₁	C₂²	1	1	yes	characteristic, complemented
39	C₂×#41×C₆₁	—	2	2	no	—
40	C₂²×D₄×C₆₁	—	2	1	no	—
41	C₂²×D₄×C₆₁	—	2	2	no	—
42	C₂×Q₁₆×C₆₁	—	2	4	no	—
43	C₂×SD₁₆×C₆₁	—	2	4	no	—
44	C₂×D₈×C₆₁	—	2	2	no	—
45	C₂×D₈×C₆₁	—	2	4	no	—
46	C₂×D₈×C₆₁	—	2	4	no	—
47	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
48	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
50	#127×C₆₁	—	2	2	no	—
51	#127×C₆₁	—	2	1	no	—
52	#127×C₆₁	C₂³	1	2	yes	—
53	#127×C₆₁	C₂³	1	2	yes	—
54	#125×C₆₁	—	2	1	no	—
55	#124×C₆₁	—	2	2	no	—
56	C₂²⋊Q₈×C₆₁	—	2	2	no	—
57	#121×C₆₁	—	2	2	no	—
58	#121×C₆₁	—	2	2	no	—
59	C₄×Q₈×C₆₁	—	2	2	no	—
60	C₄×D₄×C₆₁	—	2	2	no	—
61	C₄×D₄×C₆₁	—	2	2	no	—
62	#117×C₆₁	—	2	1	no	—
63	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
64	#106×C₆₁	—	2	2	no	—
65	C₄⋊C₈×C₆₁	D₄	1	4	yes	—
66	#103×C₆₁	—	2	4	no	—
67	#102×C₆₁	—	2	4	no	—
68	#102×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	#41×C₆₁	—	8	4	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	—	2	2	no	—
87	C₂×D₄×C₆₁	—	2	2	no	—
88	C₂×D₄×C₆₁	—	2	2	no	—
89	C₂×D₄×C₆₁	—	4	4	no	—
90	C₂×D₄×C₆₁	—	4	2	no	—
91	C₂×D₄×C₆₁	—	4	1	no	—
92	C₂×D₄×C₆₁	—	4	2	no	—
93	C₂×D₄×C₆₁	—	8	4	no	—
94	C₂²×C₄×C₆₁	—	4	2	no	—
95	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
97	Q₁₆×C₆₁	—	8	4	no	—
98	SD₁₆×C₆₁	—	8	4	no	—
99	D₈×C₆₁	—	4	4	no	—
100	D₈×C₆₁	—	8	4	no	—
101	D₈×C₆₁	—	8	4	no	—
102	M₁₆×C₆₁	—	4	2	no	—
103	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
106	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
107	C₄⋊C₄×C₆₁	—	2	2	no	—
108	C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	—	2	2	no	—
111	C₄⋊C₄×C₆₁	—	4	2	no	—
112	C₂²⋊C₄×C₆₁	—	4	1	no	—
113	C₂²⋊C₄×C₆₁	—	4	2	no	—
114	C₂²⋊C₄×C₆₁	—	4	2	no	—
115	C₄²×C₆₁	—	4	1	no	—
116	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₁	C₂×D₄	1	2	yes	—
118	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
119	C₂³×C₆₁	—	4	2	no	—
120	C₂³×C₆₁	—	4	2	no	—
121	C₂³×C₆₁	—	8	2	no	—
122	C₂³×C₆₁	—	8	4	no	—
123	C₂³×C₆₁	#127	1	1	yes	characteristic
124	Q₈×C₆₁	—	4	4	no	—
125	Q₈×C₆₁	—	4	4	no	—
126	D₄×C₆₁	—	4	4	no	—
127	D₄×C₆₁	—	4	4	no	—
128	D₄×C₆₁	—	4	4	no	—
129	D₄×C₆₁	—	4	4	no	—
130	D₄×C₆₁	—	4	4	no	—
131	D₄×C₆₁	—	4	4	no	—
132	D₄×C₆₁	—	8	2	no	—
133	D₄×C₆₁	—	8	4	no	—
134	D₄×C₆₁	—	8	4	no	—
135	C₂×C₄×C₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	4	no	—
137	C₂×C₄×C₆₁	—	4	2	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	—	8	4	no	—
140	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
141	C₂×C₄×C₆₁	#127	1	2	yes	—
142	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
144	C₈×C₆₁	—	2	2	no	—
145	C₈×C₆₁	—	2	2	no	—
146	C₈×C₆₁	—	2	4	no	—
147	C₈×C₆₁	—	2	4	no	—
148	#19729	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₆₁	—	2	1	no	—
150	C₂²×C₆₁	—	2	2	no	—
151	C₂²×C₆₁	—	8	2	no	—
152	C₂²×C₆₁	—	8	2	no	—
153	C₂²×C₆₁	—	8	2	no	—
154	C₂²×C₆₁	—	8	2	no	—
155	C₂²×C₆₁	—	8	2	no	—
156	C₂²×C₆₁	—	8	2	no	—
157	C₂²×C₆₁	—	16	4	no	—
158	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
159	C₄×C₆₁	—	2	1	no	—
160	C₄×C₆₁	—	2	2	no	—
161	C₄×C₆₁	—	8	2	no	—
162	C₄×C₆₁	—	8	2	no	—
163	C₄×C₆₁	#575	1	2	yes	—
164	C₄×C₆₁	#576	1	4	yes	—
165	#3210	C₂×C₆₁	1	2	yes	complemented
166	#3205	C₂×C₆₁	1	2	yes	complemented
167	#3184	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#3138	C₂×C₆₁	1	1	yes	characteristic, complemented
169	#1721	C₂×C₆₁	1	2	yes	complemented
170	#1720	C₂×C₆₁	1	2	yes	complemented
171	#1717	C₂×C₆₁	1	2	yes	complemented
172	#1709	C₂×C₆₁	1	2	yes	complemented
173	#1453	C₂×C₆₁	1	1	yes	characteristic, complemented
174	C₂×C₆₁	—	4	1	no	—
175	C₂×C₆₁	—	16	2	no	—
176	C₂×C₆₁	—	16	2	no	—
177	C₂×C₆₁	#3153	1	1	yes	characteristic
178	C₂×C₆₁	#3155	1	2	yes	—
179	#533	C₂²×C₆₁	1	1	yes	characteristic
180	#532	C₂²×C₆₁	1	2	yes	—
181	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
182	#498	C₂²×C₆₁	1	1	yes	characteristic, complemented
183	#494	C₂²×C₆₁	1	2	yes	complemented
184	#493	C₂²×C₆₁	1	2	yes	—
185	#493	C₂²×C₆₁	1	2	yes	—
186	#493	C₂²×C₆₁	1	1	yes	characteristic, complemented
187	#492	C₂²×C₆₁	1	2	yes	complemented
188	#490	—	2	2	no	—
189	#487	C₂²×C₆₁	1	2	yes	complemented
190	#486	C₂²×C₆₁	1	2	yes	—
191	#486	C₂²×C₆₁	1	2	yes	—
192	#486	C₂²×C₆₁	1	2	yes	—
193	#486	C₂²×C₆₁	1	2	yes	—
194	#486	C₂²×C₆₁	1	2	yes	complemented
195	#486	C₂²×C₆₁	1	2	yes	complemented
196	#469	—	2	2	no	—
197	#464	—	2	4	no	—
198	#459	—	2	4	no	—
199	#452	—	2	4	no	—
200	#447	—	2	4	no	—
201	#442	—	2	2	no	—
202	#431	C₂²×C₆₁	1	2	yes	—
203	#418	—	2	2	no	—
204	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
205	#322	C₂²×C₆₁	1	1	yes	characteristic, complemented
206	#322	C₂²×C₆₁	1	2	yes	complemented
207	C₈²	C₂²×C₆₁	1	1	yes	characteristic, complemented
208	C₆₁	#19729	1	1	yes	characteristic, direct product factor, Sylow
209	C₂×#41	—	2	2	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂²×D₄	—	2	2	no	—
212	C₂×Q₁₆	—	2	4	no	—
213	C₂×SD₁₆	—	2	4	no	—
214	C₂×D₈	—	2	2	no	—
215	C₂×D₈	—	2	4	no	—
216	C₂×D₈	—	2	4	no	—
217	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
218	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
219	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
220	#127	—	2	2	no	—
221	#127	—	2	1	no	—
222	#127	C₂³×C₆₁	1	2	yes	—
223	#127	C₂³×C₆₁	1	2	yes	—
224	#125	—	2	1	no	—
225	#124	—	2	2	no	—
226	C₂²⋊Q₈	—	2	2	no	—
227	#121	—	2	2	no	—
228	#121	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	2	2	no	—
232	#117	—	2	1	no	—
233	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	C₄⋊C₈	D₄×C₆₁	1	4	yes	—
236	#103	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	#102	—	2	4	no	—
242	#102	—	2	4	no	—
243	C₂²⋊C₈	D₄×C₆₁	1	4	yes	—
244	#97	D₄×C₆₁	1	2	yes	—
245	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₆₁	1	2	yes	—
247	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₆₁	1	2	yes	—
249	C₂⁴	—	4	1	no	—
250	#41	—	8	4	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	8	4	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
267	Q₁₆	—	8	4	no	—
268	SD₁₆	—	8	4	no	—
269	D₈	—	4	4	no	—
270	D₈	—	8	4	no	—
271	D₈	—	8	4	no	—
272	M₁₆	—	4	2	no	—
273	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
274	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
275	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
276	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₆₁	1	2	yes	—
288	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
289	C₂³	—	4	2	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	4	no	—
293	C₂³	#127×C₆₁	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	4	no	—
301	D₄	—	4	4	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
311	C₂×C₄	#127×C₆₁	1	2	yes	—
312	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	2	4	no	—
317	C₈	—	2	4	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	16	4	no	—
327	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₆₁×#575	1	2	yes	—
333	C₄	C₆₁×#576	1	4	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	16	2	no	—
336	C₂	—	16	2	no	—
337	C₂	C₆₁×#3153	1	1	yes	characteristic
338	C₂	C₆₁×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
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×#3210	C₂	semidirect
2. C₆₁×#3205	C₂	semidirect
3. C₆₁×#3184	C₂	semidirect
4. C₆₁×#3138	C₂	semidirect
5. C₆₁×#1721	C₂	semidirect
6. C₆₁×#1720	C₂	semidirect
7. C₆₁×#1717	C₂	semidirect
8. C₆₁×#1709	C₂	semidirect
9. C₆₁×#1453	C₂	semidirect
13. C₆₁×#498	C₂²	semidirect
14. C₆₁×#494	C₂²	semidirect
17. C₆₁×#493	C₂²	semidirect
18. C₆₁×#492	C₂²	semidirect
20. C₆₁×#487	C₂²	semidirect
25. C₆₁×#486	C₂²	semidirect
26. C₆₁×#486	C₂²	semidirect
36. C₆₁×#322	C₂²	semidirect
37. C₆₁×#322	C₂²	semidirect
38. C₈²×C₆₁	C₂²	semidirect
148. #19729	C₆₁	direct (N × H)
165. #3210	C₂×C₆₁	semidirect
166. #3205	C₂×C₆₁	semidirect
167. #3184	C₂×C₆₁	semidirect
168. #3138	C₂×C₆₁	semidirect
169. #1721	C₂×C₆₁	semidirect
170. #1720	C₂×C₆₁	semidirect
171. #1717	C₂×C₆₁	semidirect
172. #1709	C₂×C₆₁	semidirect
173. #1453	C₂×C₆₁	semidirect
182. #498	C₂²×C₆₁	semidirect
183. #494	C₂²×C₆₁	semidirect
186. #493	C₂²×C₆₁	semidirect
187. #492	C₂²×C₆₁	semidirect
189. #487	C₂²×C₆₁	semidirect
194. #486	C₂²×C₆₁	semidirect
195. #486	C₂²×C₆₁	semidirect
205. #322	C₂²×C₆₁	semidirect
206. #322	C₂²×C₆₁	semidirect
207. C₈²	C₂²×C₆₁	semidirect
208. C₆₁	#19729	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×#3210	2
C₆₁×#3205	2
C₆₁×#3184	2
C₆₁×#3138	2
C₆₁×#1721	2
C₆₁×#1720	2
C₆₁×#1717	2
C₆₁×#1709	2
C₆₁×#1453	2
#19729	61

Subgroups — C61×#19729 #29523162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19729 #29523162