Subgroups — C₆₁×#10162 #29513595

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3206	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3141	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1492	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#532	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#492	—	2	1	no	—
12	C₆₁×#489	—	2	2	no	—
13	C₆₁×#487	—	2	2	no	—
14	C₆₁×#487	—	2	1	no	—
15	C₆₁×#487	—	2	1	no	—
16	C₆₁×#487	—	2	2	no	—
17	C₆₁×#486	—	2	1	no	—
18	C₆₁×#461	—	2	2	no	—
19	C₆₁×#448	—	2	2	no	—
20	C₆₁×#431	C₂²	1	1	yes	characteristic
21	C₆₁×#423	C₂²	1	1	yes	characteristic
22	C₆₁×#417	—	2	2	no	—
23	C₂×#103×C₆₁	—	2	2	no	—
24	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#343	—	2	2	no	—
27	C₆₁×#333	—	2	2	no	—
28	C₆₁×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₆₁	—	2	1	no	—
30	C₂²×Q₈×C₆₁	—	2	1	no	—
31	C₂²×Q₈×C₆₁	D₄	1	2	yes	—
32	C₂×SD₁₆×C₆₁	—	2	4	no	—
33	C₂×M₁₆×C₆₁	—	2	1	no	—
34	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
35	C₂²×C₈×C₆₁	—	2	1	no	—
36	C₄⋊Q₈×C₆₁	—	2	1	no	—
37	C₄⋊Q₈×C₆₁	—	2	1	no	—
38	C₄⋊Q₈×C₆₁	—	2	1	no	—
39	C₄⋊Q₈×C₆₁	—	2	1	no	—
40	C₄⋊Q₈×C₆₁	—	2	1	no	—
41	C₄⋊Q₈×C₆₁	—	2	1	no	—
42	C₄⋊Q₈×C₆₁	—	2	1	no	—
43	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
44	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
45	#127×C₆₁	D₄	1	1	yes	characteristic
46	#124×C₆₁	—	2	1	no	—
47	#121×C₆₁	—	2	1	no	—
48	#121×C₆₁	—	2	1	no	—
49	C₄×D₄×C₆₁	—	2	1	no	—
50	C₄×D₄×C₆₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
55	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₁	—	2	1	no	—
57	C₄⋊C₈×C₆₁	—	2	1	no	—
58	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
59	#103×C₆₁	—	2	8	no	—
60	#103×C₆₁	—	2	4	no	—
61	#103×C₆₁	—	2	4	no	—
62	#102×C₆₁	—	2	4	no	—
63	C₂²⋊C₈×C₆₁	—	2	1	no	—
64	#97×C₆₁	—	2	1	no	—
65	C₄×C₈×C₆₁	—	2	1	no	—
66	C₄×C₈×C₆₁	—	2	1	no	—
67	C₄×C₈×C₆₁	—	2	1	no	—
68	C₄×C₈×C₆₁	D₄	1	2	yes	—
69	#41×C₆₁	—	4	2	no	—
70	#41×C₆₁	—	4	2	no	—
71	C₂×Q₈×C₆₁	—	2	1	no	—
72	C₂×Q₈×C₆₁	—	2	2	no	—
73	C₂×Q₈×C₆₁	—	2	2	no	—
74	C₂×Q₈×C₆₁	—	2	2	no	—
75	C₂×Q₈×C₆₁	—	4	4	no	—
76	C₂×Q₈×C₆₁	—	4	1	no	—
77	C₂×Q₈×C₆₁	—	8	2	no	—
78	C₂×D₄×C₆₁	—	2	1	no	—
79	C₂×D₄×C₆₁	—	2	1	no	—
80	C₂×D₄×C₆₁	—	2	1	no	—
81	C₂×D₄×C₆₁	—	4	1	no	—
82	C₂²×C₄×C₆₁	—	2	2	no	—
83	C₂²×C₄×C₆₁	—	4	1	no	—
84	C₂²×C₄×C₆₁	—	4	1	no	—
85	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
88	SD₁₆×C₆₁	—	8	4	no	—
89	M₁₆×C₆₁	—	2	2	no	—
90	M₁₆×C₆₁	—	2	2	no	—
91	M₁₆×C₆₁	—	4	2	no	—
92	C₂×C₈×C₆₁	—	2	2	no	—
93	C₂×C₈×C₆₁	—	2	2	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	4	2	no	—
96	C₂×C₈×C₆₁	—	4	1	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₄⋊C₄×C₆₁	—	2	1	no	—
101	C₄⋊C₄×C₆₁	—	4	1	no	—
102	C₄⋊C₄×C₆₁	—	4	1	no	—
103	C₄⋊C₄×C₆₁	—	4	1	no	—
104	C₄⋊C₄×C₆₁	—	4	1	no	—
105	C₄⋊C₄×C₆₁	—	4	1	no	—
106	C₄⋊C₄×C₆₁	—	4	1	no	—
107	C₂²⋊C₄×C₆₁	—	4	1	no	—
108	C₂²⋊C₄×C₆₁	—	4	1	no	—
109	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₆₁	—	4	1	no	—
114	C₂³×C₆₁	#121	1	1	yes	characteristic
115	Q₈×C₆₁	—	4	4	no	—
116	Q₈×C₆₁	—	4	2	no	—
117	Q₈×C₆₁	—	4	4	no	—
118	Q₈×C₆₁	—	8	2	no	—
119	Q₈×C₆₁	—	8	2	no	—
120	D₄×C₆₁	—	4	2	no	—
121	D₄×C₆₁	—	4	2	no	—
122	D₄×C₆₁	—	4	2	no	—
123	D₄×C₆₁	—	8	2	no	—
124	C₂×C₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	4	1	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	8	2	no	—
130	C₂×C₄×C₆₁	—	8	1	no	—
131	C₂×C₄×C₆₁	—	8	1	no	—
132	C₂×C₄×C₆₁	—	8	2	no	—
133	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
134	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
135	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
136	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
137	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
138	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
139	C₂×C₄×C₆₁	#135	1	2	yes	—
140	C₂×C₄×C₆₁	#137	1	2	yes	—
141	C₈×C₆₁	—	4	2	no	—
142	C₈×C₆₁	—	4	2	no	—
143	C₈×C₆₁	—	4	2	no	—
144	C₈×C₆₁	—	8	1	no	—
145	#10162	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₆₁	—	2	1	no	—
147	C₂²×C₆₁	—	2	1	no	—
148	C₂²×C₆₁	—	8	1	no	—
149	C₂²×C₆₁	—	8	1	no	—
150	C₂²×C₆₁	—	8	1	no	—
151	C₂²×C₆₁	#392	1	1	yes	characteristic
152	C₂²×C₆₁	#464	1	2	yes	—
153	C₄×C₆₁	—	2	1	no	—
154	C₄×C₆₁	—	2	1	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	—	8	1	no	—
157	C₄×C₆₁	—	8	2	no	—
158	C₄×C₆₁	#452	1	2	yes	—
159	C₄×C₆₁	#455	1	2	yes	—
160	C₄×C₆₁	#471	1	2	yes	—
161	C₄×C₆₁	#495	1	1	yes	characteristic
162	C₄×C₆₁	#497	1	1	yes	characteristic
163	#3206	C₂×C₆₁	1	1	yes	characteristic
164	#3141	C₂×C₆₁	1	1	yes	characteristic
165	C₂×#487	C₂×C₆₁	1	1	yes	characteristic, complemented
166	#1997	C₂×C₆₁	1	1	yes	characteristic, complemented
167	#1548	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#1492	C₂×C₆₁	1	1	yes	characteristic
169	#1278	C₂×C₆₁	1	1	yes	characteristic, complemented
170	C₂×C₆₁	—	2	2	no	—
171	C₂×C₆₁	—	16	1	no	—
172	C₂×C₆₁	#2004	1	1	yes	characteristic
173	C₂×C₆₁	#2015	1	1	yes	characteristic
174	C₂×C₆₁	#2019	1	1	yes	characteristic
175	#532	C₂²×C₆₁	1	1	yes	characteristic
176	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
178	#492	—	2	1	no	—
179	#489	—	2	2	no	—
180	#487	—	2	2	no	—
181	#487	—	2	1	no	—
182	#487	—	2	1	no	—
183	#487	—	2	2	no	—
184	#486	—	2	1	no	—
185	#461	—	2	2	no	—
186	#448	—	2	2	no	—
187	#431	C₂²×C₆₁	1	1	yes	characteristic
188	#423	C₂²×C₆₁	1	1	yes	characteristic
189	#417	—	2	2	no	—
190	C₂×#103	—	2	2	no	—
191	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
193	#343	—	2	2	no	—
194	#333	—	2	2	no	—
195	D₄⋊C₈	—	2	2	no	—
196	C₆₁	#10162	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×#41	—	2	1	no	—
198	C₂²×Q₈	—	2	1	no	—
199	C₂²×Q₈	D₄×C₆₁	1	2	yes	—
200	C₂×SD₁₆	—	2	4	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
203	C₂²×C₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
212	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
213	#127	D₄×C₆₁	1	1	yes	characteristic
214	#124	—	2	1	no	—
215	#121	—	2	1	no	—
216	#121	—	2	1	no	—
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
227	#103	—	2	8	no	—
228	#103	—	2	4	no	—
229	#103	—	2	4	no	—
230	#102	—	2	4	no	—
231	C₂²⋊C₈	—	2	1	no	—
232	#97	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	D₄×C₆₁	1	2	yes	—
237	#41	—	4	2	no	—
238	#41	—	4	2	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	8	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
255	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
256	SD₁₆	—	8	4	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
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	C₂³	—	4	1	no	—
282	C₂³	#121×C₆₁	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
302	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
303	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
304	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
305	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
306	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
307	C₂×C₄	#135×C₆₁	1	2	yes	—
308	C₂×C₄	#137×C₆₁	1	2	yes	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	8	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	C₆₁×#392	1	1	yes	characteristic
319	C₂²	C₆₁×#464	1	2	yes	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	C₆₁×#452	1	2	yes	—
326	C₄	C₆₁×#455	1	2	yes	—
327	C₄	C₆₁×#471	1	2	yes	—
328	C₄	C₆₁×#495	1	1	yes	characteristic
329	C₄	C₆₁×#497	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	16	1	no	—
332	C₂	C₆₁×#2004	1	1	yes	characteristic
333	C₂	C₆₁×#2015	1	1	yes	characteristic
334	C₂	C₆₁×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₂×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
3. C₂×C₆₁×#487	C₂	semidirect
4. C₆₁×#1997	C₂	semidirect
5. C₆₁×#1548	C₂	semidirect
7. C₆₁×#1278	C₂	semidirect
145. #10162	C₆₁	direct (N × H)
165. C₂×#487	C₂×C₆₁	semidirect
166. #1997	C₂×C₆₁	semidirect
167. #1548	C₂×C₆₁	semidirect
169. #1278	C₂×C₆₁	semidirect
196. C₆₁	#10162	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×#3206	2
C₆₁×#3141	2
C₂×C₆₁×#487	2
C₆₁×#1997	2
C₆₁×#1548	2
C₆₁×#1492	2
C₆₁×#1278	2
#10162	61

Subgroups — C61×#10162 #29513595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10162 #29513595