Subgroups — C₆₁×#8164 #29511597

Order 15616.

518 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×#100×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1460	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1451	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1283	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₆₁	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₆₁	—	2	2	no	—
9	C₈×D₄×C₆₁	—	2	2	no	—
10	C₆₁×#431	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×#100×C₆₁	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈×C₆₁	C₄	1	2	yes	—
14	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#343	C₄	1	2	yes	—
17	C₆₁×#323	C₄	1	4	yes	—
18	C₆₁×#322	—	2	2	no	—
19	C₆₁×#322	—	2	2	no	—
20	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₆₁	—	2	2	no	—
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	4	4	no	—
26	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×D₄×C₆₁	—	2	2	no	—
30	C₄×D₄×C₆₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₁	—	2	2	no	—
35	C₄⋊C₈×C₆₁	—	2	2	no	—
36	#100×C₆₁	—	2	4	no	—
37	C₂²⋊C₈×C₆₁	—	2	4	no	—
38	C₂²⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
39	#97×C₆₁	—	2	2	no	—
40	C₄×C₈×C₆₁	—	2	2	no	—
41	C₄×C₈×C₆₁	—	2	2	no	—
42	C₄×C₈×C₆₁	D₄	1	2	yes	—
43	C₄×C₈×C₆₁	D₄	1	2	yes	—
44	C₂⁴×C₆₁	C₄²	1	2	yes	—
45	C₂×D₄×C₆₁	—	2	2	no	—
46	C₂×D₄×C₆₁	—	2	2	no	—
47	C₂×D₄×C₆₁	M₁₆	1	4	yes	—
48	C₂²×C₄×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	—	2	2	no	—
50	C₂²×C₄×C₆₁	—	4	2	no	—
51	C₂²×C₄×C₆₁	—	4	2	no	—
52	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₆₁	—	2	4	no	—
57	M₁₆×C₆₁	—	2	4	no	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	—	2	4	no	—
62	C₂×C₈×C₆₁	—	4	1	no	—
63	C₂×C₈×C₆₁	—	4	4	no	—
64	C₂×C₈×C₆₁	—	4	4	no	—
65	C₂×C₈×C₆₁	—	4	2	no	—
66	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₆₁	—	2	2	no	—
70	C₂²⋊C₄×C₆₁	—	2	4	no	—
71	C₄²×C₆₁	C₂×D₄	1	2	yes	—
72	C₄²×C₆₁	#41	1	2	yes	—
73	C₂³×C₆₁	—	2	2	no	—
74	C₂³×C₆₁	—	2	2	no	—
75	C₂³×C₆₁	—	4	2	no	—
76	C₂³×C₆₁	—	4	2	no	—
77	C₂³×C₆₁	#97	1	4	yes	—
78	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
79	D₄×C₆₁	—	2	8	no	—
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	2	2	no	—
83	C₂×C₄×C₆₁	—	4	1	no	—
84	C₂×C₄×C₆₁	—	4	4	no	—
85	C₂×C₄×C₆₁	—	4	2	no	—
86	C₂×C₄×C₆₁	—	4	2	no	—
87	C₂×C₄×C₆₁	—	4	2	no	—
88	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₆₁	#131	1	2	yes	—
94	C₈×C₆₁	—	2	1	no	—
95	C₈×C₆₁	—	2	2	no	—
96	C₈×C₆₁	—	4	4	no	—
97	C₈×C₆₁	—	4	2	no	—
98	C₈×C₆₁	—	4	4	no	—
99	C₈×C₆₁	#100	1	2	yes	complemented
100	#8164	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₆₁	—	2	1	no	—
102	C₂²×C₆₁	—	2	1	no	—
103	C₂²×C₆₁	—	2	4	no	—
104	C₂²×C₆₁	—	4	4	no	—
105	C₂²×C₆₁	—	4	4	no	—
106	C₂²×C₆₁	—	4	2	no	—
107	C₂²×C₆₁	—	4	2	no	—
108	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
110	C₂²×C₆₁	#405	1	1	yes	characteristic
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	—	2	2	no	—
113	C₄×C₆₁	—	4	2	no	—
114	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
115	C₄×C₆₁	#413	1	1	yes	characteristic
116	C₄×C₆₁	#435	1	4	yes	—
117	C₂×C₈×D₄	C₂×C₆₁	1	1	yes	characteristic
118	C₄×#100	C₂×C₆₁	1	1	yes	characteristic
119	#1460	C₂×C₆₁	1	1	yes	characteristic
120	#1451	C₂×C₆₁	1	2	yes	complemented
121	#1283	C₂×C₆₁	1	2	yes	complemented
122	C₂×C₆₁	—	2	1	no	—
123	C₂×C₆₁	—	2	1	no	—
124	C₂×C₆₁	—	4	4	no	—
125	C₂×C₆₁	#1752	1	1	yes	characteristic
126	C₂×C₆₁	C₄×#100	1	1	yes	characteristic
127	C₂×C₆₁	#1757	1	1	yes	characteristic
128	C₂³×C₈	C₄×C₆₁	1	2	yes	—
129	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
130	C₈×D₄	—	2	2	no	—
131	C₈×D₄	—	2	2	no	—
132	#431	C₂²×C₆₁	1	2	yes	—
133	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
134	C₂×#100	C₄×C₆₁	1	2	yes	—
135	C₂×C₂²⋊C₈	C₄×C₆₁	1	2	yes	—
136	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
138	#343	C₄×C₆₁	1	2	yes	—
139	#323	C₄×C₆₁	1	4	yes	—
140	#322	—	2	2	no	—
141	#322	—	2	2	no	—
142	C₆₁	#8164	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
144	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
145	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	4	4	no	—
149	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
156	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	#100	—	2	4	no	—
160	C₂²⋊C₈	—	2	4	no	—
161	C₂²⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
162	#97	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	D₄×C₆₁	1	2	yes	—
166	C₄×C₈	D₄×C₆₁	1	2	yes	—
167	C₂⁴	C₄²×C₆₁	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	M₁₆×C₆₁	1	4	yes	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	4	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
190	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	C₂×D₄×C₆₁	1	2	yes	—
195	C₄²	#41×C₆₁	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	#97×C₆₁	1	4	yes	—
201	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
202	D₄	—	2	8	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
212	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
213	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
214	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
215	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
216	C₂×C₄	#131×C₆₁	1	2	yes	—
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	4	4	no	—
220	C₈	—	4	2	no	—
221	C₈	—	4	4	no	—
222	C₈	#100×C₆₁	1	2	yes	complemented
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
231	C₂²	C₂×#97×C₆₁	1	1	yes	characteristic
232	C₂²	C₆₁×#405	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	C₂×#100×C₆₁	1	1	yes	characteristic
237	C₄	C₆₁×#413	1	1	yes	characteristic
238	C₄	C₆₁×#435	1	4	yes	—
239	C₂	—	2	1	no	—
240	C₂	—	2	1	no	—
241	C₂	—	4	4	no	—
242	C₂	C₆₁×#1752	1	1	yes	characteristic
243	C₂	C₄×#100×C₆₁	1	1	yes	characteristic
244	C₂	C₆₁×#1757	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1451	C₂	semidirect
5. C₆₁×#1283	C₂	semidirect
99. C₈×C₆₁	#100	semidirect
100. #8164	C₆₁	direct (N × H)
120. #1451	C₂×C₆₁	semidirect
121. #1283	C₂×C₆₁	semidirect
142. C₆₁	#8164	direct (N × H)
222. C₈	#100×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₆₁	2
C₄×#100×C₆₁	2
C₆₁×#1460	2
C₆₁×#1451	2
C₆₁×#1283	2
#8164	61

Subgroups — C61×#8164 #29511597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8164 #29511597