Subgroups — C₆₁×#13142 #29516575

Order 15616.

356 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#361	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2140	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1972	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#427	C₂²	1	1	yes	characteristic
13	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#102×C₆₁	—	2	1	no	—
15	C₆₁×#405	—	2	1	no	—
16	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#390	—	2	1	no	—
18	C₆₁×#365	—	2	2	no	—
19	C₆₁×#361	—	2	2	no	—
20	C₆₁×#357	C₄	1	4	yes	complemented
21	C₆₁×#339	—	2	1	no	—
22	C₂²×D₄×C₆₁	—	2	1	no	—
23	C₂×D₈×C₆₁	—	2	1	no	—
24	C₂×D₈×C₆₁	—	2	1	no	—
25	C₂×D₈×C₆₁	—	2	1	no	—
26	C₂×D₈×C₆₁	—	2	1	no	—
27	C₂×D₈×C₆₁	—	2	1	no	—
28	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₆₁	—	2	1	no	—
30	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	#117×C₆₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₆₁	—	4	2	no	—
35	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
36	M₃₂×C₆₁	—	2	1	no	—
37	M₃₂×C₆₁	—	2	1	no	—
38	M₃₂×C₆₁	—	2	1	no	—
39	M₃₂×C₆₁	—	2	1	no	—
40	C₂×C₁₆×C₆₁	—	2	1	no	—
41	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
42	#108×C₆₁	—	2	1	no	—
43	#108×C₆₁	D₄	1	2	yes	—
44	#107×C₆₁	C₂×C₄	1	2	yes	—
45	#106×C₆₁	—	2	1	no	—
46	#106×C₆₁	—	2	1	no	—
47	#106×C₆₁	—	2	1	no	—
48	#102×C₆₁	—	4	2	no	—
49	#97×C₆₁	—	2	1	no	—
50	#97×C₆₁	—	2	1	no	—
51	#97×C₆₁	—	4	1	no	—
52	C₄×C₈×C₆₁	—	4	1	no	—
53	C₂⁴×C₆₁	—	4	1	no	—
54	C₂×D₄×C₆₁	—	2	2	no	—
55	C₂×D₄×C₆₁	—	4	1	no	—
56	C₂×D₄×C₆₁	—	4	1	no	—
57	C₂×D₄×C₆₁	—	4	1	no	—
58	C₂×D₄×C₆₁	—	4	1	no	—
59	C₂×D₄×C₆₁	—	4	1	no	—
60	C₂²×C₄×C₆₁	—	2	1	no	—
61	C₂²×C₄×C₆₁	—	4	1	no	—
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	D₈×C₆₁	—	2	2	no	—
64	D₈×C₆₁	—	2	2	no	—
65	D₈×C₆₁	—	4	1	no	—
66	D₈×C₆₁	—	4	1	no	—
67	M₁₆×C₆₁	—	2	2	no	—
68	M₁₆×C₆₁	—	4	1	no	—
69	C₂×C₈×C₆₁	—	2	1	no	—
70	C₂×C₈×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	—	4	1	no	—
72	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
76	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
77	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
78	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₆₁	—	2	2	no	—
80	C₄⋊C₄×C₆₁	—	4	1	no	—
81	C₄⋊C₄×C₆₁	—	4	1	no	—
82	C₂²⋊C₄×C₆₁	—	4	1	no	—
83	C₂²⋊C₄×C₆₁	—	8	4	no	—
84	C₄²×C₆₁	—	2	1	no	—
85	C₄²×C₆₁	—	2	1	no	—
86	C₄²×C₆₁	—	4	1	no	—
87	C₁₆×C₆₁	—	2	2	no	—
88	C₁₆×C₆₁	—	4	1	no	—
89	C₂³×C₆₁	—	4	2	no	—
90	C₂³×C₆₁	—	8	2	no	—
91	C₂³×C₆₁	—	8	2	no	—
92	C₂³×C₆₁	—	8	1	no	—
93	C₂³×C₆₁	—	8	1	no	—
94	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
95	D₄×C₆₁	—	4	2	no	—
96	D₄×C₆₁	—	4	2	no	—
97	D₄×C₆₁	—	8	1	no	—
98	D₄×C₆₁	—	8	1	no	—
99	C₂×C₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	—	4	1	no	—
102	C₂×C₄×C₆₁	—	8	2	no	—
103	C₂×C₄×C₆₁	—	8	1	no	—
104	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
109	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₈×C₆₁	—	2	2	no	—
111	C₈×C₆₁	—	4	2	no	—
112	C₈×C₆₁	#117	1	1	yes	characteristic
113	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
114	C₈×C₆₁	#124	1	1	yes	characteristic
115	C₈×C₆₁	#126	1	1	yes	characteristic
116	#13142	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₆₁	—	2	2	no	—
118	C₂²×C₆₁	—	8	4	no	—
119	C₂²×C₆₁	—	8	2	no	—
120	C₂²×C₆₁	—	16	2	no	—
121	C₂²×C₆₁	—	16	1	no	—
122	C₂²×C₆₁	—	16	1	no	—
123	C₂²×C₆₁	#442	1	1	yes	characteristic
124	C₂²×C₆₁	#460	1	1	yes	characteristic
125	C₂²×C₆₁	#482	1	1	yes	characteristic
126	C₄×C₆₁	—	8	1	no	—
127	C₄×C₆₁	—	8	2	no	—
128	C₄×C₆₁	—	16	1	no	—
129	C₄×C₆₁	#388	1	1	yes	characteristic
130	C₄×C₆₁	C₄×SD₁₆	1	1	yes	characteristic
131	C₄×C₆₁	#463	1	1	yes	characteristic
132	C₄×C₆₁	#481	1	1	yes	characteristic
133	C₂×#365	C₂×C₆₁	1	1	yes	characteristic, complemented
134	C₂×#361	C₂×C₆₁	1	1	yes	characteristic
135	#2140	C₂×C₆₁	1	1	yes	characteristic
136	#1972	C₂×C₆₁	1	1	yes	characteristic
137	#1773	C₂×C₆₁	1	1	yes	characteristic, complemented
138	#1386	C₂×C₆₁	1	1	yes	characteristic, complemented
139	#1375	C₂×C₆₁	1	1	yes	characteristic, complemented
140	C₂×C₆₁	—	2	1	no	—
141	C₂×C₆₁	—	2	1	no	—
142	C₂×C₆₁	—	16	2	no	—
143	C₂×C₆₁	#1926	1	1	yes	characteristic
144	C₂×C₆₁	AGL(1,ℤ/16)	1	2	yes	—
145	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
146	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
148	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
149	#427	C₂²×C₆₁	1	1	yes	characteristic
150	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
151	C₂×#102	—	2	1	no	—
152	#405	—	2	1	no	—
153	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
154	#390	—	2	1	no	—
155	#365	—	2	2	no	—
156	#361	—	2	2	no	—
157	#357	C₄×C₆₁	1	4	yes	complemented
158	#339	—	2	1	no	—
159	C₆₁	#13142	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×D₄	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	C₂×C₄×C₆₁	1	2	yes	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
170	#117	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₂²⋊C₄	—	4	2	no	—
173	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
180	#108	—	2	1	no	—
181	#108	D₄×C₆₁	1	2	yes	—
182	#107	C₂×C₄×C₆₁	1	2	yes	—
183	#106	—	2	1	no	—
184	#106	—	2	1	no	—
185	#106	—	2	1	no	—
186	#102	—	4	2	no	—
187	#97	—	2	1	no	—
188	#97	—	2	1	no	—
189	#97	—	4	1	no	—
190	C₄×C₈	—	4	1	no	—
191	C₂⁴	—	4	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
201	D₈	—	2	2	no	—
202	D₈	—	2	2	no	—
203	D₈	—	4	1	no	—
204	D₈	—	4	1	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	4	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	4	1	no	—
210	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
211	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
213	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
214	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
215	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
216	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₂²⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	8	4	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	4	1	no	—
225	C₁₆	—	2	2	no	—
226	C₁₆	—	4	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	1	no	—
231	C₂³	—	8	1	no	—
232	C₂³	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	1	no	—
236	D₄	—	8	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
243	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
244	C₂×C₄	#123×C₆₁	1	1	yes	characteristic
245	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
246	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
247	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	#117×C₆₁	1	1	yes	characteristic
251	C₈	C₄×D₄×C₆₁	1	1	yes	characteristic
252	C₈	#124×C₆₁	1	1	yes	characteristic
253	C₈	#126×C₆₁	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	8	4	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	16	2	no	—
258	C₂²	—	16	1	no	—
259	C₂²	—	16	1	no	—
260	C₂²	C₆₁×#442	1	1	yes	characteristic
261	C₂²	C₆₁×#460	1	1	yes	characteristic
262	C₂²	C₆₁×#482	1	1	yes	characteristic
263	C₄	—	8	1	no	—
264	C₄	—	8	2	no	—
265	C₄	—	16	1	no	—
266	C₄	C₆₁×#388	1	1	yes	characteristic
267	C₄	C₄×SD₁₆×C₆₁	1	1	yes	characteristic
268	C₄	C₆₁×#463	1	1	yes	characteristic
269	C₄	C₆₁×#481	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	16	2	no	—
273	C₂	C₆₁×#1926	1	1	yes	characteristic
274	C₂	C₆₁×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	#442
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂²×D₈×C₆₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#97×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#365	C₂	semidirect
5. C₆₁×#1773	C₂	semidirect
6. C₆₁×#1386	C₂	semidirect
7. C₆₁×#1375	C₂	semidirect
20. C₆₁×#357	C₄	semidirect
116. #13142	C₆₁	direct (N × H)
133. C₂×#365	C₂×C₆₁	semidirect
137. #1773	C₂×C₆₁	semidirect
138. #1386	C₂×C₆₁	semidirect
139. #1375	C₂×C₆₁	semidirect
157. #357	C₄×C₆₁	semidirect
159. C₆₁	#13142	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×#365	2
C₂×C₆₁×#361	2
C₆₁×#2140	2
C₆₁×#1972	2
C₆₁×#1773	2
C₆₁×#1386	2
C₆₁×#1375	2
#13142	61

Subgroups — C61×#13142 #29516575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13142 #29516575