Subgroups — C₆₁×#11972 #29515405

Order 15616.

330 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2162	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1365	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
6	C₆₁×#500	—	2	1	no	—
7	C₆₁×#500	—	2	1	no	—
8	C₆₁×#500	C₄	1	2	yes	—
9	C₆₁×#500	C₂²	1	2	yes	—
10	C₂×#107×C₆₁	—	2	1	no	—
11	C₂×#107×C₆₁	C₄	1	2	yes	—
12	C₆₁×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₆₁×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₆₁	—	2	1	no	—
17	C₆₁×#335	—	2	1	no	—
18	C₆₁×#335	C₂²	1	2	yes	—
19	C₂²×Q₈×C₆₁	—	4	1	no	—
20	C₂×M₁₆×C₆₁	—	4	1	no	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₆₁	—	2	2	no	—
24	C₄⋊Q₈×C₆₁	—	2	2	no	—
25	C₄⋊Q₈×C₆₁	—	4	1	no	—
26	C₄⋊Q₈×C₆₁	—	4	1	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
29	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₆₁	—	2	2	no	—
31	#107×C₆₁	—	2	2	no	—
32	#107×C₆₁	—	2	2	no	—
33	#107×C₆₁	—	2	2	no	—
34	#107×C₆₁	C₂×C₄	1	4	yes	—
35	C₄⋊C₈×C₆₁	—	2	2	no	—
36	C₄⋊C₈×C₆₁	—	4	1	no	—
37	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₆₁	—	4	2	no	—
42	C₂×Q₈×C₆₁	—	4	2	no	—
43	C₂×Q₈×C₆₁	—	8	4	no	—
44	C₂²×C₄×C₆₁	—	4	2	no	—
45	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
48	M₁₆×C₆₁	—	8	2	no	—
49	C₂×C₈×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	4	2	no	—
52	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₆₁	—	2	4	no	—
57	C₄⋊C₄×C₆₁	—	4	2	no	—
58	C₄⋊C₄×C₆₁	—	4	2	no	—
59	C₄⋊C₄×C₆₁	—	4	2	no	—
60	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₆₁	D₈	1	1	yes	characteristic
62	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
63	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₁₆×C₆₁	—	4	2	no	—
65	C₂³×C₆₁	#102	1	1	yes	characteristic
66	Q₈×C₆₁	—	8	4	no	—
67	Q₈×C₆₁	—	8	4	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	4	4	no	—
72	C₂×C₄×C₆₁	—	8	4	no	—
73	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
76	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
77	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₈×C₆₁	—	2	1	no	—
82	C₈×C₆₁	—	2	1	no	—
83	C₈×C₆₁	—	8	2	no	—
84	C₈×C₆₁	#101	1	2	yes	—
85	C₈×C₆₁	#137	1	2	yes	—
86	#11972	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₆₁	—	2	1	no	—
88	C₂²×C₆₁	—	2	1	no	—
89	C₂²×C₆₁	#358	1	2	yes	—
90	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
91	C₄×C₆₁	—	2	1	no	—
92	C₄×C₆₁	—	2	1	no	—
93	C₄×C₆₁	—	8	4	no	—
94	C₄×C₆₁	#358	1	2	yes	—
95	C₄×C₆₁	#361	1	2	yes	—
96	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
97	C₄×C₆₁	#419	1	1	yes	characteristic
98	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
99	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
100	C₂×#500	C₂×C₆₁	1	1	yes	characteristic
101	#2162	C₂×C₆₁	1	1	yes	characteristic
102	#1567	C₂×C₆₁	1	1	yes	characteristic
103	#1365	C₂×C₆₁	1	4	yes	complemented
104	C₂×C₆₁	—	2	2	no	—
105	C₂×C₆₁	#1546	1	1	yes	characteristic
106	C₂×C₆₁	C₂×#358	1	1	yes	characteristic
107	C₂×C₆₁	C₂×#361	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	—	2	1	no	—
109	#500	—	2	1	no	—
110	#500	—	2	1	no	—
111	#500	C₄×C₆₁	1	2	yes	—
112	#500	C₂²×C₆₁	1	2	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	C₄×C₆₁	1	2	yes	—
115	#423	—	2	1	no	—
116	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
117	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
118	C₂²⋊C₁₆	—	2	1	no	—
119	C₄×C₁₆	—	2	1	no	—
120	#335	—	2	1	no	—
121	#335	C₂²×C₆₁	1	2	yes	—
122	C₆₁	#11972	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×Q₈	—	4	1	no	—
124	C₂×M₁₆	—	4	1	no	—
125	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
126	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	4	1	no	—
130	C₄⋊Q₈	—	4	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	C₂×C₄×C₆₁	1	4	yes	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	4	1	no	—
141	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
142	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
143	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
144	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	8	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
150	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
151	C₂²×C₄	SD₁₆×C₆₁	1	1	yes	characteristic
152	M₁₆	—	8	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
158	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄²	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
165	C₄²	D₈×C₆₁	1	1	yes	characteristic
166	C₄²	SD₁₆×C₆₁	1	1	yes	characteristic
167	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
168	C₁₆	—	4	2	no	—
169	C₂³	#102×C₆₁	1	1	yes	characteristic
170	Q₈	—	8	4	no	—
171	Q₈	—	8	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
178	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
179	C₂×C₄	#102×C₆₁	1	1	yes	characteristic
180	C₂×C₄	SD₃₂×C₆₁	1	2	yes	—
181	C₂×C₄	Q₃₂×C₆₁	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
183	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	8	2	no	—
188	C₈	#101×C₆₁	1	2	yes	—
189	C₈	#137×C₆₁	1	2	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	C₆₁×#358	1	2	yes	—
193	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	4	no	—
197	C₄	C₆₁×#358	1	2	yes	—
198	C₄	C₆₁×#361	1	2	yes	—
199	C₄	C₂×#101×C₆₁	1	1	yes	characteristic
200	C₄	C₆₁×#419	1	1	yes	characteristic
201	C₄	C₂×SD₃₂×C₆₁	1	1	yes	characteristic
202	C₄	C₂×Q₃₂×C₆₁	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₆₁×#1546	1	1	yes	characteristic
205	C₂	C₂×C₆₁×#358	1	1	yes	characteristic
206	C₂	C₂×C₆₁×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	#102
LCS	G ▷ C₂×C₈ ▷ C₂×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
4. C₆₁×#1365	C₂	semidirect
86. #11972	C₆₁	direct (N × H)
103. #1365	C₂×C₆₁	semidirect
122. C₆₁	#11972	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×#500	2
C₆₁×#2162	2
C₆₁×#1567	2
C₆₁×#1365	2
#11972	61

Subgroups — C61×#11972 #29515405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11972 #29515405