Subgroups — C₆₁×#9719 #29513152

Order 15616.

506 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1980	C₂	1	2	yes	maximal, complemented
3	C₄×#107×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1502	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#532	C₂²	1	2	yes	—
7	C₆₁×#500	C₄	1	1	yes	characteristic, complemented
8	C₆₁×#493	C₄	1	1	yes	characteristic, complemented
9	C₆₁×#486	C₄	1	2	yes	complemented
10	C₆₁×#466	—	2	1	no	—
11	C₆₁×#466	—	2	1	no	—
12	C₄×D₈×C₆₁	—	2	1	no	—
13	C₄×D₈×C₆₁	—	2	1	no	—
14	C₆₁×#431	C₂²	1	1	yes	characteristic
15	C₂×#107×C₆₁	—	2	1	no	—
16	C₂×#104×C₆₁	—	2	4	no	—
17	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
20	C₆₁×#340	—	2	1	no	—
21	C₆₁×#327	—	2	4	no	—
22	C₆₁×#322	—	2	1	no	—
23	C₈²×C₆₁	—	2	1	no	—
24	C₂×#41×C₆₁	—	2	2	no	—
25	C₂×D₈×C₆₁	—	2	1	no	—
26	C₂×D₈×C₆₁	—	2	1	no	—
27	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
28	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
31	#127×C₆₁	C₂×C₄	1	2	yes	—
32	#124×C₆₁	—	2	2	no	—
33	#121×C₆₁	—	2	2	no	—
34	#121×C₆₁	—	2	2	no	—
35	C₄×D₄×C₆₁	—	2	2	no	—
36	C₄×D₄×C₆₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₂×C₄²×C₆₁	—	2	2	no	—
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	#107×C₆₁	—	2	4	no	—
41	#107×C₆₁	—	2	1	no	—
42	#107×C₆₁	—	2	1	no	—
43	C₄⋊C₈×C₆₁	—	2	1	no	—
44	#104×C₆₁	—	4	8	no	—
45	#102×C₆₁	—	2	2	no	—
46	#102×C₆₁	—	2	2	no	—
47	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
48	#97×C₆₁	—	2	1	no	—
49	C₄×C₈×C₆₁	—	2	1	no	—
50	C₄×C₈×C₆₁	—	2	1	no	—
51	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
55	#95×C₆₁	—	2	2	no	—
56	#41×C₆₁	—	4	4	no	—
57	#41×C₆₁	—	4	4	no	—
58	C₂×Q₈×C₆₁	—	2	2	no	—
59	C₂×D₄×C₆₁	—	2	2	no	—
60	C₂×D₄×C₆₁	—	2	2	no	—
61	C₂×D₄×C₆₁	—	2	2	no	—
62	C₂×D₄×C₆₁	—	4	2	no	—
63	C₂²×C₄×C₆₁	—	4	2	no	—
64	C₂²×C₄×C₆₁	—	4	2	no	—
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
68	D₈×C₆₁	—	4	2	no	—
69	D₈×C₆₁	—	4	2	no	—
70	M₁₆×C₆₁	—	2	2	no	—
71	M₁₆×C₆₁	D₈	1	4	yes	complemented
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	2	2	no	—
75	C₂×C₈×C₆₁	—	4	1	no	—
76	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₆₁	—	2	4	no	—
81	C₄⋊C₄×C₆₁	—	2	2	no	—
82	C₄⋊C₄×C₆₁	—	2	2	no	—
83	C₄⋊C₄×C₆₁	—	4	2	no	—
84	C₂²⋊C₄×C₆₁	—	4	2	no	—
85	C₂²⋊C₄×C₆₁	—	4	2	no	—
86	C₄²×C₆₁	—	4	4	no	—
87	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₁	#41	1	1	yes	characteristic
90	C₄²×C₆₁	#41	1	1	yes	characteristic
91	C₂³×C₆₁	—	4	2	no	—
92	C₂³×C₆₁	#127	1	1	yes	characteristic
93	Q₈×C₆₁	—	4	4	no	—
94	D₄×C₆₁	—	4	2	no	—
95	D₄×C₆₁	—	4	2	no	—
96	D₄×C₆₁	—	4	4	no	—
97	D₄×C₆₁	—	8	2	no	—
98	D₄×C₆₁	—	8	2	no	—
99	D₄×C₆₁	—	8	2	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	4	1	no	—
103	C₂×C₄×C₆₁	—	4	4	no	—
104	C₂×C₄×C₆₁	—	4	2	no	—
105	C₂×C₄×C₆₁	—	8	2	no	—
106	C₂×C₄×C₆₁	—	8	2	no	—
107	C₂×C₄×C₆₁	—	8	4	no	—
108	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₆₁	#135	1	2	yes	—
116	C₈×C₆₁	—	2	1	no	—
117	C₈×C₆₁	—	2	1	no	—
118	C₈×C₆₁	—	2	1	no	—
119	C₈×C₆₁	—	2	4	no	—
120	C₈×C₆₁	—	4	2	no	—
121	C₈×C₆₁	#104	1	2	yes	complemented
122	#9719	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₆₁	—	2	1	no	—
124	C₂²×C₆₁	—	2	1	no	—
125	C₂²×C₆₁	—	8	2	no	—
126	C₂²×C₆₁	—	8	2	no	—
127	C₂²×C₆₁	—	8	2	no	—
128	C₂²×C₆₁	#390	1	1	yes	characteristic
129	C₂²×C₆₁	#493	1	1	yes	characteristic
130	C₂²×C₆₁	#495	1	1	yes	characteristic
131	C₄×C₆₁	—	2	1	no	—
132	C₄×C₆₁	—	2	1	no	—
133	C₄×C₆₁	—	2	1	no	—
134	C₄×C₆₁	—	8	4	no	—
135	C₄×C₆₁	—	8	2	no	—
136	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
137	C₄×C₆₁	#421	1	1	yes	characteristic
138	C₄×C₆₁	C₄×D₈	1	2	yes	—
139	C₄×C₆₁	#466	1	2	yes	—
140	#3169	C₂×C₆₁	1	1	yes	characteristic
141	#1980	C₂×C₆₁	1	2	yes	complemented
142	C₄×#107	C₂×C₆₁	1	1	yes	characteristic, complemented
143	#1502	C₂×C₆₁	1	2	yes	complemented
144	C₈×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	—	2	1	no	—
147	C₂×C₆₁	—	16	2	no	—
148	C₂×C₆₁	#1970	1	1	yes	characteristic
149	C₂×C₆₁	#1976	1	1	yes	characteristic
150	C₂×C₆₁	#1980	1	1	yes	characteristic
151	#532	C₂²×C₆₁	1	2	yes	—
152	#500	C₄×C₆₁	1	1	yes	characteristic, complemented
153	#493	C₄×C₆₁	1	1	yes	characteristic, complemented
154	#486	C₄×C₆₁	1	2	yes	complemented
155	#466	—	2	1	no	—
156	#466	—	2	1	no	—
157	C₄×D₈	—	2	1	no	—
158	C₄×D₈	—	2	1	no	—
159	#431	C₂²×C₆₁	1	1	yes	characteristic
160	C₂×#107	—	2	1	no	—
161	C₂×#104	—	2	4	no	—
162	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
164	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
165	#340	—	2	1	no	—
166	#327	—	2	4	no	—
167	#322	—	2	1	no	—
168	C₈²	—	2	1	no	—
169	C₆₁	#9719	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×#41	—	2	2	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
174	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
176	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
177	#127	C₂×C₄×C₆₁	1	2	yes	—
178	#124	—	2	2	no	—
179	#121	—	2	2	no	—
180	#121	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
186	#107	—	2	4	no	—
187	#107	—	2	1	no	—
188	#107	—	2	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	#104	—	4	8	no	—
191	#102	—	2	2	no	—
192	#102	—	2	2	no	—
193	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
198	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
199	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
200	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
201	#95	—	2	2	no	—
202	#41	—	4	4	no	—
203	#41	—	4	4	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
214	D₈	—	4	2	no	—
215	D₈	—	4	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	D₈×C₆₁	1	4	yes	complemented
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	4	4	no	—
233	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
235	C₄²	#41×C₆₁	1	1	yes	characteristic
236	C₄²	#41×C₆₁	1	1	yes	characteristic
237	C₂³	—	4	2	no	—
238	C₂³	#127×C₆₁	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
257	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
258	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
259	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
260	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
261	C₂×C₄	#135×C₆₁	1	2	yes	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	4	no	—
266	C₈	—	4	2	no	—
267	C₈	#104×C₆₁	1	2	yes	complemented
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	C₆₁×#390	1	1	yes	characteristic
274	C₂²	C₆₁×#493	1	1	yes	characteristic
275	C₂²	C₆₁×#495	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	8	4	no	—
280	C₄	—	8	2	no	—
281	C₄	C₂×#104×C₆₁	1	1	yes	characteristic
282	C₄	C₆₁×#421	1	1	yes	characteristic
283	C₄	C₄×D₈×C₆₁	1	2	yes	—
284	C₄	C₆₁×#466	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	16	2	no	—
288	C₂	C₆₁×#1970	1	1	yes	characteristic
289	C₂	C₆₁×#1976	1	1	yes	characteristic
290	C₂	C₆₁×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#121
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#390
prime radical	C₆₁×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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
2. C₆₁×#1980	C₂	semidirect
3. C₄×#107×C₆₁	C₂	semidirect
4. C₆₁×#1502	C₂	semidirect
5. C₈×M₁₆×C₆₁	C₂	semidirect
7. C₆₁×#500	C₄	semidirect
8. C₆₁×#493	C₄	semidirect
9. C₆₁×#486	C₄	semidirect
71. M₁₆×C₆₁	D₈	semidirect
121. C₈×C₆₁	#104	semidirect
122. #9719	C₆₁	direct (N × H)
141. #1980	C₂×C₆₁	semidirect
142. C₄×#107	C₂×C₆₁	semidirect
143. #1502	C₂×C₆₁	semidirect
144. C₈×M₁₆	C₂×C₆₁	semidirect
152. #500	C₄×C₆₁	semidirect
153. #493	C₄×C₆₁	semidirect
154. #486	C₄×C₆₁	semidirect
169. C₆₁	#9719	direct (N × H)
217. M₁₆	D₈×C₆₁	semidirect
267. C₈	#104×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3169	2
C₆₁×#1980	2
C₄×#107×C₆₁	2
C₆₁×#1502	2
C₈×M₁₆×C₆₁	2
#9719	61

Subgroups — C61×#9719 #29513152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9719 #29513152