Subgroups — C₆₁×#9669 #29513102

Order 15616.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1479	C₂	1	2	yes	maximal
4	C₆₁×#1461	C₂	1	2	yes	maximal
5	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
8	C₄×Q₁₆×C₆₁	—	2	1	no	—
9	C₄×Q₁₆×C₆₁	—	2	1	no	—
10	C₄×Q₁₆×C₆₁	—	2	1	no	—
11	C₄×Q₁₆×C₆₁	—	2	1	no	—
12	C₆₁×#431	C₂²	1	2	yes	—
13	C₆₁×#431	C₂²	1	1	yes	characteristic
14	C₆₁×#431	C₂²	1	1	yes	characteristic
15	C₂×#107×C₆₁	C₄	1	1	yes	characteristic
16	C₂×#103×C₆₁	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#335	—	2	1	no	—
19	C₆₁×#334	—	2	1	no	—
20	C₆₁×Q₈⋊C₈	—	2	4	no	—
21	C₆₁×#324	—	2	4	no	—
22	C₆₁×#322	—	2	1	no	—
23	C₈²×C₆₁	—	2	1	no	—
24	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₆₁	—	2	1	no	—
26	C₂×Q₁₆×C₆₁	—	2	1	no	—
27	C₂×Q₁₆×C₆₁	—	2	2	no	—
28	C₂×Q₁₆×C₆₁	C₈	1	4	yes	complemented
29	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₆₁	—	2	2	no	—
32	C₄×Q₈×C₆₁	—	2	2	no	—
33	C₄×Q₈×C₆₁	—	2	2	no	—
34	C₄×Q₈×C₆₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₁	—	2	2	no	—
38	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	#107×C₆₁	—	2	1	no	—
40	#107×C₆₁	—	2	1	no	—
41	C₄⋊C₈×C₆₁	—	2	2	no	—
42	C₄⋊C₈×C₆₁	—	2	1	no	—
43	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
44	#103×C₆₁	—	2	2	no	—
45	#103×C₆₁	—	2	2	no	—
46	C₂²⋊C₈×C₆₁	—	2	1	no	—
47	C₂²⋊C₈×C₆₁	—	2	2	no	—
48	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
49	C₄×C₈×C₆₁	—	2	1	no	—
50	C₄×C₈×C₆₁	—	2	1	no	—
51	C₄×C₈×C₆₁	—	2	2	no	—
52	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₆₁	—	2	2	no	—
57	C₂×Q₈×C₆₁	—	2	2	no	—
58	C₂×Q₈×C₆₁	—	2	2	no	—
59	C₂×Q₈×C₆₁	—	2	2	no	—
60	C₂×Q₈×C₆₁	—	2	2	no	—
61	C₂×Q₈×C₆₁	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₆₁	—	2	2	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	Q₁₆×C₆₁	—	2	4	no	—
68	Q₁₆×C₆₁	—	2	4	no	—
69	C₂×C₈×C₆₁	—	2	1	no	—
70	C₂×C₈×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	—	2	2	no	—
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	4	2	no	—
74	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
80	C₄⋊C₄×C₆₁	—	2	2	no	—
81	C₄⋊C₄×C₆₁	—	2	2	no	—
82	C₄⋊C₄×C₆₁	—	4	2	no	—
83	C₄⋊C₄×C₆₁	—	4	2	no	—
84	C₄²×C₆₁	—	4	2	no	—
85	C₄²×C₆₁	—	4	2	no	—
86	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₁	#41	1	1	yes	characteristic
90	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
91	Q₈×C₆₁	—	2	4	no	—
92	Q₈×C₆₁	—	2	4	no	—
93	Q₈×C₆₁	—	4	2	no	—
94	Q₈×C₆₁	—	4	2	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	—	2	4	no	—
97	C₂×C₄×C₆₁	—	4	1	no	—
98	C₂×C₄×C₆₁	—	4	2	no	—
99	C₂×C₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	—	8	2	no	—
102	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
107	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
112	C₈×C₆₁	—	2	1	no	—
113	C₈×C₆₁	—	2	1	no	—
114	C₈×C₆₁	—	4	2	no	—
115	C₈×C₆₁	—	4	2	no	—
116	C₈×C₆₁	—	8	2	no	—
117	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
118	C₈×C₆₁	C₂²⋊C₈	1	2	yes	—
119	#9669	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₆₁	—	2	1	no	—
121	C₂²×C₆₁	—	2	1	no	—
122	C₂²×C₆₁	#390	1	1	yes	characteristic
123	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₆₁	#436	1	1	yes	characteristic
125	C₄×C₆₁	—	2	1	no	—
126	C₄×C₆₁	—	2	1	no	—
127	C₄×C₆₁	—	2	1	no	—
128	C₄×C₆₁	—	4	4	no	—
129	C₄×C₆₁	—	8	2	no	—
130	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₆₁	#408	1	1	yes	characteristic
132	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
133	C₄×C₆₁	#436	1	1	yes	characteristic
134	C₄×C₆₁	#473	1	2	yes	—
135	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic
136	#1564	C₂×C₆₁	1	1	yes	characteristic
137	#1479	C₂×C₆₁	1	2	yes	—
138	#1461	C₂×C₆₁	1	2	yes	—
139	#1450	C₂×C₆₁	1	1	yes	characteristic
140	C₂×C₆₁	—	2	1	no	—
141	C₂×C₆₁	—	2	1	no	—
142	C₂×C₆₁	#1954	1	1	yes	characteristic
143	C₂×C₆₁	#1976	1	1	yes	characteristic
144	C₂×C₆₁	#1979	1	1	yes	characteristic
145	C₂²×Q₁₆	C₄×C₆₁	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
147	C₄×Q₁₆	—	2	1	no	—
148	C₄×Q₁₆	—	2	1	no	—
149	C₄×Q₁₆	—	2	1	no	—
150	C₄×Q₁₆	—	2	1	no	—
151	#431	C₂²×C₆₁	1	2	yes	—
152	#431	C₂²×C₆₁	1	1	yes	characteristic
153	#431	C₂²×C₆₁	1	1	yes	characteristic
154	C₂×#107	C₄×C₆₁	1	1	yes	characteristic
155	C₂×#103	C₄×C₆₁	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
157	#335	—	2	1	no	—
158	#334	—	2	1	no	—
159	Q₈⋊C₈	—	2	4	no	—
160	#324	—	2	4	no	—
161	#322	—	2	1	no	—
162	C₈²	—	2	1	no	—
163	C₆₁	#9669	1	1	yes	characteristic, direct product factor, Sylow
164	C₂²×Q₈	C₂×C₄×C₆₁	1	2	yes	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	C₈×C₆₁	1	4	yes	complemented
169	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	—	2	2	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
179	#107	—	2	1	no	—
180	#107	—	2	1	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
184	#103	—	2	2	no	—
185	#103	—	2	2	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
193	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
194	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
195	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	C₂×C₈×C₆₁	1	4	yes	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
206	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
207	Q₁₆	—	2	4	no	—
208	Q₁₆	—	2	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
215	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
216	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
218	C₂×C₈	C₂×C₈×C₆₁	1	2	yes	—
219	C₂×C₈	M₁₆×C₆₁	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
229	C₄²	#41×C₆₁	1	1	yes	characteristic
230	C₂³	C₄×D₄×C₆₁	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
244	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
245	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
246	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
247	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
249	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
250	C₂×C₄	#131×C₆₁	1	1	yes	characteristic
251	C₂×C₄	#131×C₆₁	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	8	2	no	—
257	C₈	C₂²⋊C₈×C₆₁	1	2	yes	—
258	C₈	C₂²⋊C₈×C₆₁	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	C₆₁×#390	1	1	yes	characteristic
262	C₂²	C₈×D₄×C₆₁	1	1	yes	characteristic
263	C₂²	C₆₁×#436	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
270	C₄	C₆₁×#408	1	1	yes	characteristic
271	C₄	C₈×D₄×C₆₁	1	1	yes	characteristic
272	C₄	C₆₁×#436	1	1	yes	characteristic
273	C₄	C₆₁×#473	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	C₆₁×#1954	1	1	yes	characteristic
277	C₂	C₆₁×#1976	1	1	yes	characteristic
278	C₂	C₆₁×#1979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#121
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
28. C₂×Q₁₆×C₆₁	C₈	semidirect
119. #9669	C₆₁	direct (N × H)
163. C₆₁	#9669	direct (N × H)
168. C₂×Q₁₆	C₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₆₁	2
C₆₁×#1564	2
C₆₁×#1479	2
C₆₁×#1461	2
C₆₁×#1450	2
#9669	61

Subgroups — C61×#9669 #29513102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9669 #29513102