Subgroups — C₆₁×#21952 #29525385

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#431	C₂	1	2	yes	maximal
2	C₆₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1939	C₂	1	8	yes	maximal, complemented
4	C₆₁×#1798	C₂	1	4	yes	maximal
5	C₂×#117×C₆₁	C₂²	1	2	yes	—
6	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
7	C₆₁×#431	—	2	8	no	—
8	C₂×C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
9	C₂×C₄⋊C₈×C₆₁	C₂²	1	8	yes	—
10	C₂×C₂²⋊C₈×C₆₁	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	4	yes	—
12	C₆₁×#384	C₄	1	2	yes	—
13	C₆₁×#382	C₄	1	2	yes	—
14	C₆₁×#380	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
16	C₆₁×#336	C₂²	1	8	yes	—
17	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	C₂³	1	8	yes	—
20	#117×C₆₁	—	4	4	no	—
21	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
22	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
23	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	4	yes	—
27	C₂×C₄²×C₆₁	Q₈	1	4	yes	—
28	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
29	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
30	C₄⋊C₈×C₆₁	—	2	8	no	—
31	C₄⋊C₈×C₆₁	—	4	8	no	—
32	C₂²⋊C₈×C₆₁	—	2	8	no	—
33	C₄×C₈×C₆₁	—	2	8	no	—
34	#95×C₆₁	C₂×C₄	1	8	yes	—
35	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	—	2	8	no	—
37	C₂²×C₄×C₆₁	—	2	2	no	—
38	C₂²×C₄×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₆₁	—	2	8	no	—
48	C₂×C₈×C₆₁	—	2	8	no	—
49	C₂×C₈×C₆₁	—	2	8	no	—
50	C₄⋊C₄×C₆₁	—	2	16	no	—
51	C₄⋊C₄×C₆₁	—	4	4	no	—
52	C₂²⋊C₄×C₆₁	—	4	4	no	—
53	C₄²×C₆₁	—	2	8	no	—
54	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
55	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
56	C₄²×C₆₁	C₂×D₄	1	4	yes	—
57	C₄²×C₆₁	C₂×Q₈	1	4	yes	—
58	C₂³×C₆₁	—	2	1	no	—
59	C₂³×C₆₁	—	2	1	no	—
60	C₂³×C₆₁	—	2	4	no	—
61	C₂³×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
62	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	1	no	—
64	C₂×C₄×C₆₁	—	2	1	no	—
65	C₂×C₄×C₆₁	—	2	4	no	—
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	—	2	4	no	—
68	C₂×C₄×C₆₁	—	2	16	no	—
69	C₂×C₄×C₆₁	—	2	8	no	—
70	C₂×C₄×C₆₁	—	2	16	no	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₄×C₆₁	#131	1	8	yes	—
75	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
78	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
79	C₈×C₆₁	—	4	8	no	—
80	#21952	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₆₁	—	2	2	no	—
82	C₂²×C₆₁	—	2	4	no	—
83	C₂²×C₆₁	—	4	2	no	—
84	C₂²×C₆₁	—	4	2	no	—
85	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₆₁	#518	1	1	yes	characteristic
87	C₂²×C₆₁	#519	1	1	yes	characteristic
88	C₂²×C₆₁	C₂×#131	1	4	yes	—
89	C₄×C₆₁	—	2	2	no	—
90	C₄×C₆₁	—	2	4	no	—
91	C₄×C₆₁	—	4	8	no	—
92	C₄×C₆₁	#424	1	8	yes	—
93	C₄×C₆₁	#537	1	2	yes	—
94	C₄×C₆₁	#544	1	2	yes	—
95	C₂×#431	C₂×C₆₁	1	2	yes	—
96	#2285	C₂×C₆₁	1	1	yes	characteristic
97	#1939	C₂×C₆₁	1	8	yes	complemented
98	#1798	C₂×C₆₁	1	4	yes	—
99	C₂×C₆₁	—	4	2	no	—
100	C₂×C₆₁	#2316	1	1	yes	characteristic
101	C₂×C₆₁	C₂×#424	1	2	yes	—
102	C₂×C₆₁	#2979	1	2	yes	—
103	C₂×C₆₁	#2986	1	2	yes	—
104	C₂×#117	C₂²×C₆₁	1	2	yes	—
105	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
106	#431	—	2	8	no	—
107	C₂×C₄⋊C₈	C₂²×C₆₁	1	4	yes	—
108	C₂×C₄⋊C₈	C₂²×C₆₁	1	8	yes	—
109	C₂×C₂²⋊C₈	C₂²×C₆₁	1	4	yes	—
110	C₂×C₄×C₈	C₂²×C₆₁	1	4	yes	—
111	#384	C₄×C₆₁	1	2	yes	—
112	#382	C₄×C₆₁	1	2	yes	—
113	#380	C₄×C₆₁	1	4	yes	—
114	C₄×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
115	#336	C₂²×C₆₁	1	8	yes	—
116	C₆₁	#21952	1	1	yes	characteristic, direct product factor, Sylow
117	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
118	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₆₁	1	8	yes	—
120	#117	—	4	4	no	—
121	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	8	yes	—
122	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
123	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
124	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
126	C₂×C₄²	D₄×C₆₁	1	4	yes	—
127	C₂×C₄²	Q₈×C₆₁	1	4	yes	—
128	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
129	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
130	C₄⋊C₈	—	2	8	no	—
131	C₄⋊C₈	—	4	8	no	—
132	C₂²⋊C₈	—	2	8	no	—
133	C₄×C₈	—	2	8	no	—
134	#95	C₂×C₄×C₆₁	1	8	yes	—
135	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	C₄⋊C₄×C₆₁	1	8	yes	—
140	C₂²×C₄	C₂²×C₄×C₆₁	1	8	yes	—
141	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
145	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
146	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	2	8	no	—
150	C₄⋊C₄	—	2	16	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	8	no	—
154	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
155	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
156	C₄²	C₂×D₄×C₆₁	1	4	yes	—
157	C₄²	C₂×Q₈×C₆₁	1	4	yes	—
158	C₂³	—	2	4	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	1	no	—
161	C₂³	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
162	C₂³	C₂³×C₄×C₆₁	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
172	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
173	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	4	yes	—
174	C₂×C₄	#131×C₆₁	1	8	yes	—
175	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
176	C₂×C₄	C₂²×Q₈×C₆₁	1	1	yes	characteristic
177	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
178	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
179	C₈	—	4	8	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂²×C₄⋊C₄×C₆₁	1	1	yes	characteristic
185	C₂²	C₆₁×#518	1	1	yes	characteristic
186	C₂²	C₆₁×#519	1	1	yes	characteristic
187	C₂²	C₂×#131×C₆₁	1	4	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	4	8	no	—
191	C₄	C₆₁×#424	1	8	yes	—
192	C₄	C₆₁×#537	1	2	yes	—
193	C₄	C₆₁×#544	1	2	yes	—
194	C₂	—	4	2	no	—
195	C₂	C₆₁×#2316	1	1	yes	characteristic
196	C₂	C₂×C₆₁×#424	1	2	yes	—
197	C₂	C₆₁×#2979	1	2	yes	—
198	C₂	C₆₁×#2986	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1939	C₂	semidirect
80. #21952	C₆₁	direct (N × H)
97. #1939	C₂×C₆₁	semidirect
116. C₆₁	#21952	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×#431	2
C₆₁×#2285	2
C₆₁×#1939	2
C₆₁×#1798	2
#21952	61

Subgroups — C61×#21952 #29525385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#21952 #29525385