Subgroups — C₆₁×#8780 #29512213

Order 15616.

310 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#1726	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1564	C₂	1	2	yes	maximal
3	C₆₁×#1281	C₂	1	2	yes	maximal
4	C₆₁×#1274	C₂	1	2	yes	maximal
5	C₆₁×#431	C₂²	1	4	yes	—
6	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
7	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
8	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#376	C₂²	1	1	yes	characteristic
13	C₆₁×#335	—	2	2	no	—
14	C₆₁×#334	—	2	2	no	—
15	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	—	2	4	no	—
23	C₂²⋊C₈×C₆₁	—	2	4	no	—
24	#97×C₆₁	—	2	2	no	—
25	#97×C₆₁	—	2	2	no	—
26	#97×C₆₁	—	2	2	no	—
27	C₄×C₈×C₆₁	—	2	2	no	—
28	C₄×C₈×C₆₁	—	2	4	no	—
29	C₄×C₈×C₆₁	C₈	1	4	yes	complemented
30	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
34	#95×C₆₁	—	2	2	no	—
35	C₂²×C₄×C₆₁	—	2	1	no	—
36	C₂²×C₄×C₆₁	—	2	1	no	—
37	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	#41	1	2	yes	—
39	C₂×C₈×C₆₁	—	2	1	no	—
40	C₂×C₈×C₆₁	—	2	1	no	—
41	C₂×C₈×C₆₁	—	2	1	no	—
42	C₂×C₈×C₆₁	—	2	1	no	—
43	C₂×C₈×C₆₁	—	2	1	no	—
44	C₂×C₈×C₆₁	—	2	1	no	—
45	C₂×C₈×C₆₁	—	4	4	no	—
46	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
53	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
54	C₄²×C₆₁	—	2	1	no	—
55	C₄²×C₆₁	—	2	1	no	—
56	C₄²×C₆₁	C₂×C₈	1	2	yes	—
57	C₄²×C₆₁	M₁₆	1	2	yes	—
58	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₆₁	#41	1	2	yes	—
61	C₂³×C₆₁	#117	1	1	yes	characteristic
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	4	2	no	—
65	C₂×C₄×C₆₁	—	4	2	no	—
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#124	1	2	yes	—
70	C₂×C₄×C₆₁	#125	1	2	yes	—
71	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
75	C₈×C₆₁	—	2	1	no	—
76	C₈×C₆₁	—	2	1	no	—
77	C₈×C₆₁	—	2	2	no	—
78	C₈×C₆₁	—	2	2	no	—
79	C₈×C₆₁	—	8	4	no	—
80	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
81	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
82	#8780	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₆₁	—	2	2	no	—
84	C₂²×C₆₁	#382	1	1	yes	characteristic
85	C₂²×C₆₁	#431	1	1	yes	characteristic
86	C₂²×C₆₁	#432	1	1	yes	characteristic
87	C₄×C₆₁	—	2	2	no	—
88	C₄×C₆₁	—	2	2	no	—
89	C₄×C₆₁	—	4	2	no	—
90	C₄×C₆₁	—	4	2	no	—
91	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₄×C₆₁	#423	1	1	yes	characteristic
93	C₄×C₆₁	#431	1	1	yes	characteristic
94	C₄×C₆₁	#432	1	1	yes	characteristic
95	#1726	C₂×C₆₁	1	1	yes	characteristic
96	#1564	C₂×C₆₁	1	2	yes	—
97	#1281	C₂×C₆₁	1	2	yes	—
98	#1274	C₂×C₆₁	1	2	yes	—
99	C₂×C₆₁	—	2	1	no	—
100	C₂×C₆₁	—	2	1	no	—
101	C₂×C₆₁	#1831	1	1	yes	characteristic
102	C₂×C₆₁	#1833	1	2	yes	—
103	#431	C₂²×C₆₁	1	4	yes	—
104	C₂×#97	C₄×C₆₁	1	1	yes	characteristic
105	C₂×#97	C₄×C₆₁	1	1	yes	characteristic
106	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
107	C₂×C₄×C₈	C₄×C₆₁	1	1	yes	characteristic
108	C₂×C₄×C₈	C₄×C₆₁	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
110	#376	C₂²×C₆₁	1	1	yes	characteristic
111	#335	—	2	2	no	—
112	#334	—	2	2	no	—
113	C₆₁	#8780	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
118	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
119	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
121	C₄⋊C₈	—	2	4	no	—
122	C₂²⋊C₈	—	2	4	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	4	no	—
128	C₄×C₈	C₈×C₆₁	1	4	yes	complemented
129	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
130	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
131	C₄×C₈	Q₈×C₆₁	1	1	yes	characteristic
132	C₄×C₈	Q₈×C₆₁	1	1	yes	characteristic
133	#95	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
137	C₂²×C₄	#41×C₆₁	1	2	yes	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	4	4	no	—
145	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
146	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
147	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
148	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
149	C₂×C₈	C₂×C₈×C₆₁	1	2	yes	—
150	C₂×C₈	C₂×C₈×C₆₁	1	2	yes	—
151	C₂×C₈	M₁₆×C₆₁	1	2	yes	—
152	C₂×C₈	M₁₆×C₆₁	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₂×C₈×C₆₁	1	2	yes	—
156	C₄²	M₁₆×C₆₁	1	2	yes	—
157	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
158	C₄²	C₂×Q₈×C₆₁	1	1	yes	characteristic
159	C₄²	#41×C₆₁	1	2	yes	—
160	C₂³	#117×C₆₁	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
167	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
168	C₂×C₄	#124×C₆₁	1	2	yes	—
169	C₂×C₄	#125×C₆₁	1	2	yes	—
170	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
171	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
172	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
173	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	8	4	no	—
179	C₈	C₄⋊C₈×C₆₁	1	2	yes	—
180	C₈	C₄⋊C₈×C₆₁	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	C₆₁×#382	1	1	yes	characteristic
183	C₂²	C₆₁×#431	1	1	yes	characteristic
184	C₂²	C₆₁×#432	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	C₂×C₄⋊C₈×C₆₁	1	1	yes	characteristic
190	C₄	C₆₁×#423	1	1	yes	characteristic
191	C₄	C₆₁×#431	1	1	yes	characteristic
192	C₄	C₆₁×#432	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₆₁×#1831	1	1	yes	characteristic
196	C₂	C₆₁×#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₁	#117
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#382
prime radical	C₂³×C₆₁	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₆₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
29. C₄×C₈×C₆₁	C₈	semidirect
82. #8780	C₆₁	direct (N × H)
113. C₆₁	#8780	direct (N × H)
128. C₄×C₈	C₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#1726	2
C₆₁×#1564	2
C₆₁×#1281	2
C₆₁×#1274	2
#8780	61

Subgroups — C61×#8780 #29512213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8780 #29512213