Subgroups — C₆₁×#8209 #29511642

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#446	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1770	C₂	1	1	yes	maximal, characteristic
3	C₄×#103×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1279	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#446	—	2	2	no	—
10	C₆₁×#446	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#103×C₆₁	C₄	1	2	yes	—
15	C₂×#97×C₆₁	—	2	1	no	—
16	C₂×#97×C₆₁	—	2	1	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₆₁×#340	C₄	1	2	yes	—
22	C₆₁×#336	—	2	1	no	—
23	C₆₁×Q₈⋊C₈	C₄	1	4	yes	complemented
24	C₆₁×#322	—	2	2	no	—
25	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	—	2	1	no	—
27	C₂²×C₈×C₆₁	—	2	1	no	—
28	C₂²×C₈×C₆₁	—	2	1	no	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₁	—	2	1	no	—
33	C₄×Q₈×C₆₁	—	2	1	no	—
34	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	—	2	1	no	—
40	C₂×C₄²×C₆₁	—	2	1	no	—
41	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	—	2	4	no	—
43	C₄⋊C₈×C₆₁	—	2	1	no	—
44	C₄⋊C₈×C₆₁	—	2	1	no	—
45	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
47	#103×C₆₁	—	2	4	no	—
48	#97×C₆₁	—	2	4	no	—
49	#97×C₆₁	—	4	2	no	—
50	C₄×C₈×C₆₁	—	2	1	no	—
51	C₄×C₈×C₆₁	—	2	1	no	—
52	C₄×C₈×C₆₁	—	2	1	no	—
53	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₆₁	D₄	1	2	yes	—
56	#95×C₆₁	—	2	1	no	—
57	C₂×Q₈×C₆₁	—	2	1	no	—
58	C₂×Q₈×C₆₁	—	2	2	no	—
59	C₂×Q₈×C₆₁	—	2	2	no	—
60	C₂×Q₈×C₆₁	C₄²	1	2	yes	—
61	C₂²×C₄×C₆₁	—	2	1	no	—
62	C₂²×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	—	2	2	no	—
64	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
67	C₂×C₈×C₆₁	—	2	2	no	—
68	C₂×C₈×C₆₁	—	2	2	no	—
69	C₂×C₈×C₆₁	—	2	1	no	—
70	C₂×C₈×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	—	2	1	no	—
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	4	2	no	—
77	C₂×C₈×C₆₁	—	4	1	no	—
78	C₂×C₈×C₆₁	—	4	1	no	—
79	C₂×C₈×C₆₁	—	4	1	no	—
80	C₂×C₈×C₆₁	—	4	1	no	—
81	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₄⋊C₄×C₆₁	—	2	2	no	—
86	C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₄⋊C₄×C₆₁	—	2	2	no	—
89	C₄⋊C₄×C₆₁	C₄²	1	2	yes	—
90	C₄²×C₆₁	—	2	2	no	—
91	C₄²×C₆₁	—	2	2	no	—
92	C₄²×C₆₁	—	4	1	no	—
93	C₄²×C₆₁	—	4	1	no	—
94	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₁	#41	1	1	yes	characteristic
98	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₆₁	—	2	2	no	—
100	Q₈×C₆₁	—	2	2	no	—
101	Q₈×C₆₁	—	4	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	4	2	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	—	4	2	no	—
108	C₂×C₄×C₆₁	—	4	2	no	—
109	C₂×C₄×C₆₁	—	4	2	no	—
110	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₆₁	#131	1	2	yes	—
118	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
121	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
122	C₈×C₆₁	—	2	2	no	—
123	C₈×C₆₁	—	2	2	no	—
124	C₈×C₆₁	—	2	4	no	—
125	C₈×C₆₁	—	4	2	no	—
126	C₈×C₆₁	—	4	2	no	—
127	C₈×C₆₁	—	4	2	no	—
128	#8209	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₆₁	#436	1	2	yes	—
131	C₂²×C₆₁	#440	1	1	yes	characteristic
132	C₂²×C₆₁	#441	1	1	yes	characteristic
133	C₂²×C₆₁	#444	1	2	yes	—
134	C₄×C₆₁	—	2	1	no	—
135	C₄×C₆₁	—	2	1	no	—
136	C₄×C₆₁	—	4	2	no	—
137	C₄×C₆₁	—	4	2	no	—
138	C₄×C₆₁	—	8	2	no	—
139	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
140	C₄×C₆₁	#405	1	1	yes	characteristic
141	C₄×C₆₁	#417	1	1	yes	characteristic
142	C₄×C₆₁	#419	1	1	yes	characteristic
143	C₄×C₆₁	#435	1	2	yes	—
144	C₄×C₆₁	#440	1	1	yes	characteristic
145	C₄×C₆₁	#441	1	1	yes	characteristic
146	C₂×#446	C₂×C₆₁	1	1	yes	characteristic
147	#1770	C₂×C₆₁	1	1	yes	characteristic
148	C₄×#103	C₂×C₆₁	1	1	yes	characteristic
149	C₂×Q₈⋊C₈	C₂×C₆₁	1	1	yes	characteristic
150	C₂×#322	C₂×C₆₁	1	1	yes	characteristic
151	#1279	C₂×C₆₁	1	1	yes	characteristic
152	#1277	C₂×C₆₁	1	1	yes	characteristic
153	C₂×C₆₁	#1592	1	2	yes	—
154	C₂×C₆₁	#1593	1	2	yes	—
155	C₂×C₆₁	#1753	1	1	yes	characteristic
156	C₂×C₆₁	#1763	1	1	yes	characteristic
157	C₂×C₆₁	#1765	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
159	#446	—	2	2	no	—
160	#446	C₄×C₆₁	1	4	yes	complemented
161	C₂×C₄⋊C₈	—	2	1	no	—
162	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
164	C₂×#103	C₄×C₆₁	1	2	yes	—
165	C₂×#97	—	2	1	no	—
166	C₂×#97	—	2	1	no	—
167	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
170	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
171	#340	C₄×C₆₁	1	2	yes	—
172	#336	—	2	1	no	—
173	Q₈⋊C₈	C₄×C₆₁	1	4	yes	complemented
174	#322	—	2	2	no	—
175	C₆₁	#8209	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
181	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
182	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
186	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
197	C₄⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
198	#103	—	2	4	no	—
199	#97	—	2	4	no	—
200	#97	—	4	2	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
205	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
206	C₄×C₈	D₄×C₆₁	1	2	yes	—
207	#95	—	2	1	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	C₄²×C₆₁	1	2	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
217	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
233	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
234	C₂×C₈	M₁₆×C₆₁	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	C₄²×C₆₁	1	2	yes	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
246	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₄²	#41×C₆₁	1	1	yes	characteristic
249	C₂³	C₄×D₄×C₆₁	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	—	2	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
263	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
265	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
266	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
267	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
268	C₂×C₄	#131×C₆₁	1	2	yes	—
269	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
270	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
271	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
272	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	—	4	2	no	—
277	C₈	—	4	2	no	—
278	C₈	—	4	2	no	—
279	C₂²	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
280	C₂²	C₆₁×#436	1	2	yes	—
281	C₂²	C₆₁×#440	1	1	yes	characteristic
282	C₂²	C₆₁×#441	1	1	yes	characteristic
283	C₂²	C₆₁×#444	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	8	2	no	—
289	C₄	C₄×M₁₆×C₆₁	1	1	yes	characteristic
290	C₄	C₆₁×#405	1	1	yes	characteristic
291	C₄	C₆₁×#417	1	1	yes	characteristic
292	C₄	C₆₁×#419	1	1	yes	characteristic
293	C₄	C₆₁×#435	1	2	yes	—
294	C₄	C₆₁×#440	1	1	yes	characteristic
295	C₄	C₆₁×#441	1	1	yes	characteristic
296	C₂	C₆₁×#1592	1	2	yes	—
297	C₂	C₆₁×#1593	1	2	yes	—
298	C₂	C₆₁×#1753	1	1	yes	characteristic
299	C₂	C₆₁×#1763	1	1	yes	characteristic
300	C₂	C₆₁×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₆₁×#446	C₄	semidirect
23. C₆₁×Q₈⋊C₈	C₄	semidirect
128. #8209	C₆₁	direct (N × H)
160. #446	C₄×C₆₁	semidirect
173. Q₈⋊C₈	C₄×C₆₁	semidirect
175. C₆₁	#8209	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#446	2
C₆₁×#1770	2
C₄×#103×C₆₁	2
C₂×C₆₁×Q₈⋊C₈	2
C₂×C₆₁×#322	2
C₆₁×#1279	2
C₆₁×#1277	2
#8209	61

Subgroups — C61×#8209 #29511642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8209 #29511642