Subgroups — C₆₁×#19712 #29523145

Order 15616.

704 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	C₆₁×#3228	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3210	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1723	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1722	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1720	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1717	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#533	C₂²	1	2	yes	—
11	C₆₁×#532	C₂²	1	2	yes	—
12	C₆₁×#500	C₂²	1	1	yes	characteristic
13	C₆₁×#500	C₂²	1	1	yes	characteristic
14	C₆₁×#500	C₂²	1	2	yes	—
15	C₆₁×#498	C₂²	1	2	yes	—
16	C₆₁×#494	C₂²	1	1	yes	characteristic
17	C₆₁×#494	C₂²	1	2	yes	—
18	C₆₁×#493	C₂²	1	1	yes	characteristic
19	C₆₁×#492	C₂²	1	2	yes	—
20	C₆₁×#487	C₂²	1	2	yes	—
21	C₆₁×#487	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	—
26	C₆₁×#482	—	2	4	no	—
27	C₆₁×#478	—	2	4	no	—
28	C₆₁×#467	—	2	2	no	—
29	C₆₁×#466	—	2	2	no	—
30	C₆₁×#464	—	2	4	no	—
31	C₆₁×#452	—	2	4	no	—
32	C₄×Q₁₆×C₆₁	—	2	2	no	—
33	C₄×D₈×C₆₁	—	2	2	no	—
34	C₆₁×#431	C₂²	1	1	yes	characteristic
35	C₆₁×#431	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#322	C₂²	1	2	yes	—
38	C₆₁×#322	C₂²	1	2	yes	—
39	C₂×#41×C₆₁	—	2	2	no	—
40	C₂×Q₁₆×C₆₁	—	2	2	no	—
41	C₂×Q₁₆×C₆₁	—	2	4	no	—
42	C₂×SD₁₆×C₆₁	—	2	4	no	—
43	C₂×D₈×C₆₁	—	2	2	no	—
44	C₂²×C₈×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	C₂³	1	2	yes	—
49	#125×C₆₁	—	2	2	no	—
50	#124×C₆₁	—	2	2	no	—
51	C₂²⋊Q₈×C₆₁	—	2	2	no	—
52	C₂²⋊Q₈×C₆₁	—	2	2	no	—
53	#121×C₆₁	—	2	2	no	—
54	#121×C₆₁	—	2	2	no	—
55	C₄×Q₈×C₆₁	—	2	2	no	—
56	C₄×Q₈×C₆₁	—	2	2	no	—
57	C₄×D₄×C₆₁	—	2	2	no	—
58	C₄×D₄×C₆₁	—	2	2	no	—
59	#117×C₆₁	—	2	2	no	—
60	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
61	#107×C₆₁	—	2	2	no	—
62	#107×C₆₁	—	2	2	no	—
63	#107×C₆₁	—	2	4	no	—
64	#106×C₆₁	—	2	4	no	—
65	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
66	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
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	2	yes	—
74	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
75	C₄×C₈×C₆₁	D₄	1	2	yes	—
76	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂³	1	2	yes	—
79	C₄×C₈×C₆₁	C₂³	1	2	yes	—
80	#41×C₆₁	—	8	4	no	—
81	C₂×Q₈×C₆₁	—	2	2	no	—
82	C₂×Q₈×C₆₁	—	2	2	no	—
83	C₂×Q₈×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	—	4	2	no	—
88	C₂²×C₄×C₆₁	—	4	2	no	—
89	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₆₁	—	4	4	no	—
93	Q₁₆×C₆₁	—	8	4	no	—
94	SD₁₆×C₆₁	—	8	4	no	—
95	D₈×C₆₁	—	4	4	no	—
96	C₂×C₈×C₆₁	—	4	2	no	—
97	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
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	—	2	2	no	—
107	C₄⋊C₄×C₆₁	—	4	2	no	—
108	C₄⋊C₄×C₆₁	—	4	2	no	—
109	C₂²⋊C₄×C₆₁	—	4	2	no	—
110	C₂²⋊C₄×C₆₁	—	4	2	no	—
111	C₂²⋊C₄×C₆₁	—	4	2	no	—
112	C₄²×C₆₁	—	4	2	no	—
113	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₁	—	4	2	no	—
118	C₂³×C₆₁	#127	1	1	yes	characteristic
119	Q₈×C₆₁	—	4	4	no	—
120	Q₈×C₆₁	—	4	4	no	—
121	Q₈×C₆₁	—	4	4	no	—
122	D₄×C₆₁	—	4	4	no	—
123	D₄×C₆₁	—	4	4	no	—
124	D₄×C₆₁	—	4	4	no	—
125	D₄×C₆₁	—	8	4	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	8	4	no	—
133	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
134	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
135	C₂×C₄×C₆₁	#127	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	C₈×C₆₁	—	2	2	no	—
140	C₈×C₆₁	—	2	2	no	—
141	C₈×C₆₁	—	2	4	no	—
142	C₈×C₆₁	—	2	4	no	—
143	#19712	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₆₁	—	2	1	no	—
145	C₂²×C₆₁	—	2	1	no	—
146	C₂²×C₆₁	—	2	1	no	—
147	C₂²×C₆₁	—	8	2	no	—
148	C₂²×C₆₁	—	8	2	no	—
149	C₂²×C₆₁	—	8	2	no	—
150	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
151	C₄×C₆₁	—	2	1	no	—
152	C₄×C₆₁	—	2	1	no	—
153	C₄×C₆₁	—	2	1	no	—
154	C₄×C₆₁	—	8	2	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	—	8	2	no	—
157	C₄×C₆₁	#575	1	2	yes	—
158	C₄×C₆₁	#576	1	2	yes	—
159	C₄×C₆₁	#578	1	2	yes	—
160	#3228	C₂×C₆₁	1	2	yes	complemented
161	#3210	C₂×C₆₁	1	2	yes	complemented
162	#3170	C₂×C₆₁	1	1	yes	characteristic, complemented
163	#3169	C₂×C₆₁	1	1	yes	characteristic
164	#1723	C₂×C₆₁	1	2	yes	complemented
165	#1722	C₂×C₆₁	1	2	yes	complemented
166	#1720	C₂×C₆₁	1	2	yes	complemented
167	#1717	C₂×C₆₁	1	2	yes	complemented
168	#1452	C₂×C₆₁	1	1	yes	characteristic, complemented
169	C₂×C₆₁	—	4	1	no	—
170	C₂×C₆₁	—	16	2	no	—
171	C₂×C₆₁	#3150	1	1	yes	characteristic
172	C₂×C₆₁	#3155	1	1	yes	characteristic
173	C₂×C₆₁	#3156	1	1	yes	characteristic
174	#533	C₂²×C₆₁	1	2	yes	—
175	#532	C₂²×C₆₁	1	2	yes	—
176	#500	C₂²×C₆₁	1	1	yes	characteristic
177	#500	C₂²×C₆₁	1	1	yes	characteristic
178	#500	C₂²×C₆₁	1	2	yes	—
179	#498	C₂²×C₆₁	1	2	yes	—
180	#494	C₂²×C₆₁	1	1	yes	characteristic
181	#494	C₂²×C₆₁	1	2	yes	—
182	#493	C₂²×C₆₁	1	1	yes	characteristic
183	#492	C₂²×C₆₁	1	2	yes	—
184	#487	C₂²×C₆₁	1	2	yes	—
185	#487	C₂²×C₆₁	1	2	yes	—
186	#486	C₂²×C₆₁	1	2	yes	—
187	#486	C₂²×C₆₁	1	2	yes	—
188	#486	C₂²×C₆₁	1	2	yes	—
189	#486	C₂²×C₆₁	1	2	yes	—
190	#482	—	2	4	no	—
191	#478	—	2	4	no	—
192	#467	—	2	2	no	—
193	#466	—	2	2	no	—
194	#464	—	2	4	no	—
195	#452	—	2	4	no	—
196	C₄×Q₁₆	—	2	2	no	—
197	C₄×D₈	—	2	2	no	—
198	#431	C₂²×C₆₁	1	1	yes	characteristic
199	#431	C₂²×C₆₁	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
201	#322	C₂²×C₆₁	1	2	yes	—
202	#322	C₂²×C₆₁	1	2	yes	—
203	C₆₁	#19712	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×#41	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂²×C₈	D₄×C₆₁	1	2	yes	—
210	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
211	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
213	#127	C₂³×C₆₁	1	2	yes	—
214	#125	—	2	2	no	—
215	#124	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	#121	—	2	2	no	—
219	#121	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	#117	—	2	2	no	—
225	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	#107	—	2	2	no	—
228	#107	—	2	4	no	—
229	#106	—	2	4	no	—
230	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
231	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
232	#103	—	2	4	no	—
233	#103	—	2	4	no	—
234	#102	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
239	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
240	C₄×C₈	D₄×C₆₁	1	2	yes	—
241	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₆₁	1	2	yes	—
244	C₄×C₈	C₂³×C₆₁	1	2	yes	—
245	#41	—	8	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
257	Q₁₆	—	4	4	no	—
258	Q₁₆	—	8	4	no	—
259	SD₁₆	—	8	4	no	—
260	D₈	—	4	4	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
263	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
281	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
282	C₂³	—	4	2	no	—
283	C₂³	#127×C₆₁	1	1	yes	characteristic
284	Q₈	—	4	4	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	8	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
299	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
300	C₂×C₄	#127×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	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	—	2	4	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₆₁×#575	1	2	yes	—
322	C₄	C₆₁×#576	1	2	yes	—
323	C₄	C₆₁×#578	1	2	yes	—
324	C₂	—	4	1	no	—
325	C₂	—	16	2	no	—
326	C₂	C₆₁×#3150	1	1	yes	characteristic
327	C₂	C₆₁×#3155	1	1	yes	characteristic
328	C₂	C₆₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3169	C₂
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×#3228	C₂	semidirect
2. C₆₁×#3210	C₂	semidirect
3. C₆₁×#3170	C₂	semidirect
5. C₆₁×#1723	C₂	semidirect
6. C₆₁×#1722	C₂	semidirect
7. C₆₁×#1720	C₂	semidirect
8. C₆₁×#1717	C₂	semidirect
9. C₆₁×#1452	C₂	semidirect
143. #19712	C₆₁	direct (N × H)
160. #3228	C₂×C₆₁	semidirect
161. #3210	C₂×C₆₁	semidirect
162. #3170	C₂×C₆₁	semidirect
164. #1723	C₂×C₆₁	semidirect
165. #1722	C₂×C₆₁	semidirect
166. #1720	C₂×C₆₁	semidirect
167. #1717	C₂×C₆₁	semidirect
168. #1452	C₂×C₆₁	semidirect
203. C₆₁	#19712	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×#3228	2
C₆₁×#3210	2
C₆₁×#3170	2
C₆₁×#3169	2
C₆₁×#1723	2
C₆₁×#1722	2
C₆₁×#1720	2
C₆₁×#1717	2
C₆₁×#1452	2
#19712	61

Subgroups — C61×#19712 #29523145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19712 #29523145