Subgroups — C₆₁×#7217 #29510650

Order 15616.

226 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2162	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1834	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#423	—	2	1	no	—
5	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
6	C₆₁×#384	—	2	1	no	—
7	C₆₁×C₄⋊C₁₆	—	2	1	no	—
8	C₆₁×C₂²⋊C₁₆	—	2	1	no	—
9	C₄×C₁₆×C₆₁	C₄	1	2	yes	complemented
10	C₆₁×#340	—	2	1	no	—
11	C₆₁×#340	C₄	1	2	yes	—
12	C₆₁×#335	—	2	1	no	—
13	C₆₁×#335	C₄	1	2	yes	complemented
14	C₂×M₁₆×C₆₁	—	4	1	no	—
15	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
19	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
20	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₁	—	2	2	no	—
22	C₄⋊C₈×C₆₁	—	2	2	no	—
23	C₄⋊C₈×C₆₁	—	4	1	no	—
24	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	—	4	2	no	—
29	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
32	M₁₆×C₆₁	—	8	1	no	—
33	M₁₆×C₆₁	—	8	1	no	—
34	C₂×C₈×C₆₁	—	4	1	no	—
35	C₂×C₈×C₆₁	—	4	1	no	—
36	C₂×C₈×C₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₄⋊C₄×C₆₁	—	4	2	no	—
42	C₄⋊C₄×C₆₁	—	4	2	no	—
43	C₄⋊C₄×C₆₁	—	8	2	no	—
44	C₄⋊C₄×C₆₁	—	8	1	no	—
45	C₄⋊C₄×C₆₁	—	8	1	no	—
46	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
49	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
50	C₁₆×C₆₁	—	4	2	no	—
51	C₂³×C₆₁	#102	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	—	2	1	no	—
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	8	2	no	—
56	C₂×C₄×C₆₁	—	8	2	no	—
57	C₂×C₄×C₆₁	—	8	2	no	—
58	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
64	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
66	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
67	C₂×C₄×C₆₁	Q₃₂	1	1	yes	characteristic
68	C₈×C₆₁	—	2	1	no	—
69	C₈×C₆₁	—	2	1	no	—
70	C₈×C₆₁	—	2	1	no	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	8	2	no	—
73	#7217	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₆₁	—	2	1	no	—
75	C₂²×C₆₁	—	2	1	no	—
76	C₂²×C₆₁	#340	1	1	yes	characteristic
77	C₂²×C₆₁	#357	1	1	yes	characteristic
78	C₂²×C₆₁	#358	1	1	yes	characteristic
79	C₄×C₆₁	—	2	1	no	—
80	C₄×C₆₁	—	2	1	no	—
81	C₄×C₆₁	—	16	2	no	—
82	C₄×C₆₁	H(3,ℤ/4)	1	1	yes	characteristic
83	C₄×C₆₁	#344	1	1	yes	characteristic
84	C₄×C₆₁	#357	1	1	yes	characteristic
85	C₄×C₆₁	#358	1	1	yes	characteristic
86	C₄×C₆₁	#365	1	2	yes	—
87	C₄×C₆₁	#366	1	1	yes	characteristic
88	C₄×C₆₁	#367	1	1	yes	characteristic
89	#2162	C₂×C₆₁	1	1	yes	characteristic
90	#1834	C₂×C₆₁	1	1	yes	characteristic
91	#1567	C₂×C₆₁	1	1	yes	characteristic
92	C₂×C₆₁	—	2	1	no	—
93	C₂×C₆₁	—	2	1	no	—
94	C₂×C₆₁	#1286	1	1	yes	characteristic
95	C₂×C₆₁	#1380	1	1	yes	characteristic
96	C₂×C₆₁	#1383	1	1	yes	characteristic
97	#423	—	2	1	no	—
98	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
99	#384	—	2	1	no	—
100	C₄⋊C₁₆	—	2	1	no	—
101	C₂²⋊C₁₆	—	2	1	no	—
102	C₄×C₁₆	C₄×C₆₁	1	2	yes	complemented
103	#340	—	2	1	no	—
104	#340	C₄×C₆₁	1	2	yes	—
105	#335	—	2	1	no	—
106	#335	C₄×C₆₁	1	2	yes	complemented
107	C₆₁	#7217	1	1	yes	characteristic, direct product factor, Sylow
108	C₂×M₁₆	—	4	1	no	—
109	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
110	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	1	no	—
113	C₂×C₄⋊C₄	—	4	1	no	—
114	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
115	C₂×C₁₆	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
120	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
121	C₄×C₈	Q₈×C₆₁	1	1	yes	characteristic
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
124	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
125	C₂²×C₄	SD₁₆×C₆₁	1	1	yes	characteristic
126	M₁₆	—	8	1	no	—
127	M₁₆	—	8	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	C₄²×C₆₁	1	1	yes	characteristic
132	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
133	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
134	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	8	2	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	1	no	—
140	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
141	C₄²	C₄⋊C₄×C₆₁	1	1	yes	characteristic
142	C₄²	SD₁₆×C₆₁	1	1	yes	characteristic
143	C₄²	Q₁₆×C₆₁	1	1	yes	characteristic
144	C₁₆	—	4	2	no	—
145	C₂³	#102×C₆₁	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	#95×C₆₁	1	1	yes	characteristic
153	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
154	C₂×C₄	#103×C₆₁	1	1	yes	characteristic
155	C₂×C₄	#103×C₆₁	1	1	yes	characteristic
156	C₂×C₄	#106×C₆₁	1	1	yes	characteristic
157	C₂×C₄	#107×C₆₁	1	1	yes	characteristic
158	C₂×C₄	D₁₆×C₆₁	1	1	yes	characteristic
159	C₂×C₄	SD₃₂×C₆₁	1	1	yes	characteristic
160	C₂×C₄	SD₃₂×C₆₁	1	1	yes	characteristic
161	C₂×C₄	Q₃₂×C₆₁	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	8	2	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₆₁×#340	1	1	yes	characteristic
170	C₂²	C₆₁×#357	1	1	yes	characteristic
171	C₂²	C₆₁×#358	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	16	2	no	—
175	C₄	C₆₁×H(3,ℤ/4)	1	1	yes	characteristic
176	C₄	C₆₁×#344	1	1	yes	characteristic
177	C₄	C₆₁×#357	1	1	yes	characteristic
178	C₄	C₆₁×#358	1	1	yes	characteristic
179	C₄	C₆₁×#365	1	2	yes	—
180	C₄	C₆₁×#366	1	1	yes	characteristic
181	C₄	C₆₁×#367	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	C₆₁×#1286	1	1	yes	characteristic
185	C₂	C₆₁×#1380	1	1	yes	characteristic
186	C₂	C₆₁×#1383	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#340
commutator	C₂×C₈	C₄²×C₆₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₁
prime radical	C₂³×C₆₁	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₆×C₆₁	C₄	semidirect
13. C₆₁×#335	C₄	semidirect
73. #7217	C₆₁	direct (N × H)
102. C₄×C₁₆	C₄×C₆₁	semidirect
106. #335	C₄×C₆₁	semidirect
107. C₆₁	#7217	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#2162	2
C₆₁×#1834	2
C₆₁×#1567	2
#7217	61

Subgroups — C61×#7217 #29510650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7217 #29510650