Subgroups — C₆₁×#13164 #29516597

Order 15616.

284 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2153	C₂	1	2	yes	maximal
2	C₂×C₆₁×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1772	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
7	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#108×C₆₁	C₄	1	1	yes	characteristic
10	C₆₁×#427	C₄	1	1	yes	characteristic
11	C₆₁×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#364	—	2	4	no	—
14	C₆₁×#364	—	2	1	no	—
15	C₆₁×#364	—	2	1	no	—
16	C₆₁×#346	—	2	1	no	—
17	C₄×C₁₆×C₆₁	—	2	1	no	—
18	C₆₁×#339	C₄	1	2	yes	—
19	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
22	#117×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	M₃₂×C₆₁	—	2	1	no	—
25	M₃₂×C₆₁	—	2	1	no	—
26	M₃₂×C₆₁	—	2	2	no	—
27	M₃₂×C₆₁	—	2	2	no	—
28	M₃₂×C₆₁	D₄	1	2	yes	—
29	M₃₂×C₆₁	Q₈	1	2	yes	—
30	C₂×C₁₆×C₆₁	—	2	1	no	—
31	C₂×C₁₆×C₆₁	—	2	1	no	—
32	C₂×C₁₆×C₆₁	—	2	2	no	—
33	#108×C₆₁	C₈	1	2	yes	—
34	#108×C₆₁	C₈	1	2	yes	—
35	#107×C₆₁	—	2	1	no	—
36	#106×C₆₁	—	2	1	no	—
37	#97×C₆₁	—	2	2	no	—
38	C₄×C₈×C₆₁	—	2	2	no	—
39	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	—	2	1	no	—
44	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	M₁₆×C₆₁	—	2	2	no	—
46	M₁₆×C₆₁	C₂×C₈	1	4	yes	—
47	C₂×C₈×C₆₁	—	2	2	no	—
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	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
59	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₆₁	—	2	2	no	—
61	C₂²⋊C₄×C₆₁	—	2	2	no	—
62	C₄²×C₆₁	—	2	1	no	—
63	C₄²×C₆₁	—	2	2	no	—
64	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
66	C₁₆×C₆₁	—	2	2	no	—
67	C₁₆×C₆₁	—	2	2	no	—
68	C₁₆×C₆₁	—	4	2	no	—
69	C₂³×C₆₁	#131	1	1	yes	characteristic
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	1	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	—	4	2	no	—
74	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
80	C₈×C₆₁	—	2	4	no	—
81	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
82	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
85	C₈×C₆₁	#121	1	1	yes	characteristic
86	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
87	C₈×C₆₁	#125	1	1	yes	characteristic
88	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
89	#13164	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₆₁	—	4	1	no	—
91	C₂²×C₆₁	#424	1	1	yes	characteristic
92	C₂²×C₆₁	#436	1	1	yes	characteristic
93	C₂²×C₆₁	C₈×Q₈	1	1	yes	characteristic
94	C₄×C₆₁	—	2	2	no	—
95	C₄×C₆₁	—	4	1	no	—
96	C₄×C₆₁	—	8	2	no	—
97	C₄×C₆₁	#389	1	1	yes	characteristic
98	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
100	C₄×C₆₁	#446	1	1	yes	characteristic
101	#2153	C₂×C₆₁	1	2	yes	—
102	C₂×#364	C₂×C₆₁	1	1	yes	characteristic
103	C₄×M₃₂	C₂×C₆₁	1	1	yes	characteristic
104	#1772	C₂×C₆₁	1	1	yes	characteristic
105	#1378	C₂×C₆₁	1	2	yes	—
106	C₂×C₆₁	—	2	1	no	—
107	C₂×C₆₁	—	2	1	no	—
108	C₂×C₆₁	—	2	1	no	—
109	C₂×C₆₁	#1939	1	1	yes	characteristic
110	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
111	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
112	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
113	C₂×#108	C₄×C₆₁	1	1	yes	characteristic
114	#427	C₄×C₆₁	1	1	yes	characteristic
115	#405	C₂²×C₆₁	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
117	#364	—	2	4	no	—
118	#364	—	2	1	no	—
119	#364	—	2	1	no	—
120	#346	—	2	1	no	—
121	C₄×C₁₆	—	2	1	no	—
122	#339	C₄×C₆₁	1	2	yes	—
123	C₆₁	#13164	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
125	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
127	#117	C₂×C₄×C₆₁	1	2	yes	—
128	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
129	M₃₂	—	2	1	no	—
130	M₃₂	—	2	1	no	—
131	M₃₂	—	2	2	no	—
132	M₃₂	—	2	2	no	—
133	M₃₂	D₄×C₆₁	1	2	yes	—
134	M₃₂	Q₈×C₆₁	1	2	yes	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	—	2	2	no	—
138	#108	C₈×C₆₁	1	2	yes	—
139	#108	C₈×C₆₁	1	2	yes	—
140	#107	—	2	1	no	—
141	#106	—	2	1	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
145	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₆₁	1	1	yes	characteristic
147	C₄×C₈	Q₈×C₆₁	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	C₂×C₈×C₆₁	1	4	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
154	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
155	C₂×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
156	C₂×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
157	C₂×C₈	M₁₆×C₆₁	1	1	yes	characteristic
158	C₂×C₈	M₁₆×C₆₁	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
161	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
162	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
163	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
164	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₄⋊C₄×C₆₁	1	1	yes	characteristic
170	C₄²	C₄⋊C₄×C₆₁	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	2	no	—
173	C₁₆	—	4	2	no	—
174	C₂³	#131×C₆₁	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₆₁	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
183	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
184	C₂×C₄	#131×C₆₁	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	C₄⋊C₈×C₆₁	1	1	yes	characteristic
187	C₈	C₄⋊C₈×C₆₁	1	1	yes	characteristic
188	C₈	C₄⋊C₈×C₆₁	1	1	yes	characteristic
189	C₈	C₄⋊C₈×C₆₁	1	1	yes	characteristic
190	C₈	#121×C₆₁	1	1	yes	characteristic
191	C₈	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
192	C₈	#125×C₆₁	1	1	yes	characteristic
193	C₈	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	C₆₁×#424	1	1	yes	characteristic
196	C₂²	C₆₁×#436	1	1	yes	characteristic
197	C₂²	C₈×Q₈×C₆₁	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	C₆₁×#389	1	1	yes	characteristic
202	C₄	C₂×C₄⋊C₈×C₆₁	1	1	yes	characteristic
203	C₄	C₈×D₄×C₆₁	1	1	yes	characteristic
204	C₄	C₆₁×#446	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	C₆₁×#1939	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₁	#121
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1939
prime radical	C₂³×C₆₁	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
89. #13164	C₆₁	direct (N × H)
123. C₆₁	#13164	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×#2153	2
C₂×C₆₁×#364	2
C₄×M₃₂×C₆₁	2
C₆₁×#1772	2
C₆₁×#1378	2
#13164	61

Subgroups — C61×#13164 #29516597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13164 #29516597