Subgroups — C₆₁×#8144 #29511577

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈×C₆₁	C₂	1	2	yes	maximal
3	C₂×C₆₁×#322	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1273	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₆₁	C₄	1	2	yes	—
6	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₆₁	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₆₁	—	2	4	no	—
9	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#336	C₄	1	4	yes	—
14	C₆₁×#322	—	2	4	no	—
15	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₆₁	—	2	4	no	—
18	C₂²×C₈×C₆₁	—	2	4	no	—
19	C₂²×C₈×C₆₁	C₈	1	8	yes	complemented
20	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₁	—	2	2	no	—
24	C₂×C₄²×C₆₁	—	2	2	no	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
28	C₂²⋊C₈×C₆₁	—	2	8	no	—
29	C₄×C₈×C₆₁	—	2	4	no	—
30	C₄×C₈×C₆₁	—	2	4	no	—
31	C₄×C₈×C₆₁	—	2	2	no	—
32	C₄×C₈×C₆₁	—	2	4	no	—
33	C₄×C₈×C₆₁	D₄	1	4	yes	—
34	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	—	2	1	no	—
36	C₂²×C₄×C₆₁	—	2	1	no	—
37	C₂²×C₄×C₆₁	—	2	4	no	—
38	C₂²×C₄×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	—	2	4	no	—
40	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₆₁	M₁₆	1	8	yes	—
44	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	—	2	16	no	—
49	C₂×C₈×C₆₁	—	2	8	no	—
50	C₂×C₈×C₆₁	—	2	8	no	—
51	C₂×C₈×C₆₁	—	2	8	no	—
52	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
53	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
55	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
56	C₄²×C₆₁	—	2	1	no	—
57	C₄²×C₆₁	—	2	1	no	—
58	C₄²×C₆₁	—	2	4	no	—
59	C₄²×C₆₁	C₂×D₄	1	2	yes	—
60	C₄²×C₆₁	#41	1	2	yes	—
61	C₂³×C₆₁	—	2	1	no	—
62	C₂³×C₆₁	—	2	1	no	—
63	C₂³×C₆₁	—	2	2	no	—
64	C₂³×C₆₁	C₄×C₈	1	2	yes	—
65	C₂³×C₆₁	#97	1	4	yes	—
66	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	4	no	—
72	C₂×C₄×C₆₁	—	2	4	no	—
73	C₂×C₄×C₆₁	—	2	16	no	—
74	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
75	C₂×C₄×C₆₁	#97	1	4	yes	—
76	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₆₁	#131	1	4	yes	—
83	C₈×C₆₁	—	2	4	no	—
84	C₈×C₆₁	—	4	8	no	—
85	C₈×C₆₁	C₂²⋊C₈	1	8	yes	complemented
86	#8144	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₆₁	—	2	2	no	—
88	C₂²×C₆₁	—	2	4	no	—
89	C₂²×C₆₁	—	2	2	no	—
90	C₂²×C₆₁	—	2	4	no	—
91	C₂²×C₆₁	#322	1	4	yes	complemented
92	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²×C₆₁	C₂×#97	1	2	yes	—
95	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
96	C₂²×C₆₁	#405	1	2	yes	—
97	C₄×C₆₁	—	2	2	no	—
98	C₄×C₆₁	—	2	4	no	—
99	C₄×C₆₁	C₂×C₂²⋊C₈	1	2	yes	—
100	C₄×C₆₁	#431	1	2	yes	—
101	C₄×C₆₁	#435	1	8	yes	—
102	C₂²×C₄×C₈	C₂×C₆₁	1	1	yes	characteristic
103	C₄×C₂²⋊C₈	C₂×C₆₁	1	2	yes	—
104	C₂×#322	C₂×C₆₁	1	2	yes	complemented
105	#1273	C₂×C₆₁	1	2	yes	complemented
106	C₂×C₆₁	—	2	4	no	—
107	C₂×C₆₁	C₂×#322	1	1	yes	characteristic
108	C₂×C₆₁	#1450	1	1	yes	characteristic
109	C₂×C₆₁	#1451	1	2	yes	—
110	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₆₁	#1752	1	2	yes	—
112	C₂³×C₈	C₄×C₆₁	1	2	yes	—
113	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
114	C₂×C₂²⋊C₈	C₄×C₆₁	1	4	yes	—
115	C₂×C₄×C₈	—	2	4	no	—
116	C₂×C₄×C₈	C₄×C₆₁	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₆₁	1	4	yes	—
119	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
120	#336	C₄×C₆₁	1	4	yes	—
121	#322	—	2	4	no	—
122	C₆₁	#8144	1	1	yes	characteristic, direct product factor, Sylow
123	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
124	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₈×C₆₁	1	8	yes	complemented
128	C₂²×C₈	C₂×C₄×C₆₁	1	8	yes	—
129	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
136	C₂²⋊C₈	—	2	8	no	—
137	C₄×C₈	—	2	4	no	—
138	C₄×C₈	—	2	4	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	2	4	no	—
141	C₄×C₈	D₄×C₆₁	1	4	yes	—
142	C₂⁴	C₄²×C₆₁	1	1	yes	characteristic
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	C₄²×C₆₁	1	1	yes	characteristic
149	C₂²×C₄	C₄²×C₆₁	1	2	yes	—
150	C₂²×C₄	C₂×C₈×C₆₁	1	4	yes	—
151	C₂²×C₄	M₁₆×C₆₁	1	8	yes	—
152	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	16	no	—
157	C₂×C₈	—	2	8	no	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	C₂²⋊C₄×C₆₁	1	4	yes	—
161	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
162	C₂×C₈	C₂×C₈×C₆₁	1	4	yes	—
163	C₂×C₈	M₁₆×C₆₁	1	4	yes	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	4	no	—
167	C₄²	C₂×D₄×C₆₁	1	2	yes	—
168	C₄²	#41×C₆₁	1	2	yes	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₄×C₈×C₆₁	1	2	yes	—
173	C₂³	#97×C₆₁	1	4	yes	—
174	C₂³	C₂×C₄²×C₆₁	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	C₄×C₈×C₆₁	1	2	yes	—
183	C₂×C₄	#97×C₆₁	1	4	yes	—
184	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
185	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
186	C₂×C₄	C₄×D₄×C₆₁	1	4	yes	—
187	C₂×C₄	C₂²×C₈×C₆₁	1	2	yes	—
188	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
189	C₂×C₄	C₂×M₁₆×C₆₁	1	4	yes	—
190	C₂×C₄	#131×C₆₁	1	4	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	4	8	no	—
193	C₈	C₂²⋊C₈×C₆₁	1	8	yes	complemented
194	C₂²	—	2	4	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₆₁×#322	1	4	yes	complemented
199	C₂²	C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
200	C₂²	C₂×C₄×C₈×C₆₁	1	1	yes	characteristic
201	C₂²	C₂×#97×C₆₁	1	2	yes	—
202	C₂²	C₄×M₁₆×C₆₁	1	1	yes	characteristic
203	C₂²	C₆₁×#405	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	C₂×C₂²⋊C₈×C₆₁	1	2	yes	—
207	C₄	C₆₁×#431	1	2	yes	—
208	C₄	C₆₁×#435	1	8	yes	—
209	C₂	—	2	4	no	—
210	C₂	C₂×C₆₁×#322	1	1	yes	characteristic
211	C₂	C₆₁×#1450	1	1	yes	characteristic
212	C₂	C₆₁×#1451	1	2	yes	—
213	C₂	C₄×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
214	C₂	C₆₁×#1752	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₆₁	C₂³
commutator	C₂²	C₂×C₄×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂³×C₆₁	C₂×C₄²
prime radical	C₂⁴×C₆₁	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₆₁×#322	C₂	semidirect
4. C₆₁×#1273	C₂	semidirect
19. C₂²×C₈×C₆₁	C₈	semidirect
85. C₈×C₆₁	C₂²⋊C₈	semidirect
86. #8144	C₆₁	direct (N × H)
91. C₂²×C₆₁	#322	semidirect
104. C₂×#322	C₂×C₆₁	semidirect
105. #1273	C₂×C₆₁	semidirect
122. C₆₁	#8144	direct (N × H)
127. C₂²×C₈	C₈×C₆₁	semidirect
193. C₈	C₂²⋊C₈×C₆₁	semidirect
198. C₂²	C₆₁×#322	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₆₁	2
C₄×C₂²⋊C₈×C₆₁	2
C₂×C₆₁×#322	2
C₆₁×#1273	2
#8144	61

Subgroups — C61×#8144 #29511577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8144 #29511577