Subgroups — C₆₁×#33493 #29536926

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2169	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₆×C₆₁	C₂	1	4	yes	maximal, complemented
4	C₆₁×#2162	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2113	C₂	1	4	yes	maximal, complemented
6	C₂×#131×C₆₁	C₂²	1	2	yes	—
7	C₆₁×#532	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₆₁	C₂²	1	4	yes	—
9	C₂²×C₁₆×C₆₁	C₂²	1	4	yes	—
10	C₆₁×#436	C₄	1	4	yes	—
11	C₈×D₄×C₆₁	C₂²	1	4	yes	—
12	C₆₁×#423	C₄	1	1	yes	characteristic
13	C₆₁×#408	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₆₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
18	C₆₁×C₂²⋊C₁₆	C₂²	1	8	yes	—
19	C₄×C₁₆×C₆₁	C₂²	1	4	yes	—
20	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
21	#131×C₆₁	—	4	2	no	—
22	#131×C₆₁	—	4	2	no	—
23	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₆₁	C₈	1	1	yes	characteristic
29	#127×C₆₁	C₈	1	1	yes	characteristic
30	#124×C₆₁	C₈	1	2	yes	—
31	#121×C₆₁	C₈	1	4	yes	—
32	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₆₁	—	4	4	no	—
35	C₂×C₁₆×C₆₁	—	4	4	no	—
36	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
37	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
38	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
39	C₂²⋊C₈×C₆₁	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₁	C₂³	1	2	yes	—
42	#41×C₆₁	—	4	2	no	—
43	#41×C₆₁	—	4	2	no	—
44	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₆₁	C₂×C₈	1	4	yes	—
46	C₂×D₄×C₆₁	C₂×C₈	1	4	yes	—
47	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	M₁₆×C₆₁	—	4	2	no	—
52	M₁₆×C₆₁	—	4	4	no	—
53	C₂×C₈×C₆₁	—	2	4	no	—
54	C₂×C₈×C₆₁	—	2	4	no	—
55	C₂×C₈×C₆₁	—	2	1	no	—
56	C₂×C₈×C₆₁	—	2	1	no	—
57	C₂×C₈×C₆₁	—	4	1	no	—
58	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
63	C₂²⋊C₄×C₆₁	C₂×C₈	1	8	yes	—
64	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
67	C₁₆×C₆₁	—	2	4	no	—
68	C₁₆×C₆₁	—	2	4	no	—
69	C₂³×C₆₁	C₂²×C₈	1	4	yes	—
70	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
71	Q₈×C₆₁	—	4	2	no	—
72	D₄×C₆₁	—	2	8	no	—
73	D₄×C₆₁	—	4	2	no	—
74	D₄×C₆₁	—	4	4	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	4	no	—
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	—	2	4	no	—
79	C₂×C₄×C₆₁	—	2	4	no	—
80	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
85	C₈×C₆₁	—	2	4	no	—
86	C₈×C₆₁	—	2	1	no	—
87	C₈×C₆₁	—	2	2	no	—
88	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
89	C₈×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
90	#33493	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₆₁	—	2	2	no	—
92	C₂²×C₆₁	—	2	8	no	—
93	C₂²×C₆₁	—	2	1	no	—
94	C₂²×C₆₁	—	2	4	no	—
95	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₆₁	—	2	2	no	—
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	—	2	4	no	—
99	C₄×C₆₁	#504	1	4	yes	—
100	C₄×C₆₁	#518	1	1	yes	characteristic
101	C₄×C₆₁	#568	1	1	yes	characteristic
102	#2931	C₂×C₆₁	1	1	yes	characteristic
103	#2169	C₂×C₆₁	1	4	yes	complemented
104	D₄×C₁₆	C₂×C₆₁	1	4	yes	complemented
105	#2162	C₂×C₆₁	1	2	yes	complemented
106	#2113	C₂×C₆₁	1	4	yes	complemented
107	C₂×C₆₁	—	4	1	no	—
108	C₂×C₆₁	—	4	4	no	—
109	C₂×C₆₁	#2972	1	1	yes	characteristic
110	C₂×C₆₁	C₂×#504	1	2	yes	—
111	C₂×#131	C₂²×C₆₁	1	2	yes	—
112	#532	C₄×C₆₁	1	1	yes	characteristic
113	C₂×M₃₂	C₂²×C₆₁	1	4	yes	—
114	C₂²×C₁₆	C₂²×C₆₁	1	4	yes	—
115	#436	C₄×C₆₁	1	4	yes	—
116	C₈×D₄	C₂²×C₆₁	1	4	yes	—
117	#423	C₄×C₆₁	1	1	yes	characteristic
118	#408	C₄×C₆₁	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
120	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
121	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
122	C₂²⋊C₁₆	C₂²×C₆₁	1	4	yes	—
123	C₂²⋊C₁₆	C₂²×C₆₁	1	8	yes	—
124	C₄×C₁₆	C₂²×C₆₁	1	4	yes	—
125	C₆₁	#33493	1	1	yes	characteristic, direct product factor, Sylow
126	C₂×#41	C₂×C₄×C₆₁	1	2	yes	—
127	#131	—	4	2	no	—
128	#131	—	4	2	no	—
129	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
130	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
131	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
134	C₄⋊Q₈	C₈×C₆₁	1	1	yes	characteristic
135	#127	C₈×C₆₁	1	1	yes	characteristic
136	#124	C₈×C₆₁	1	2	yes	—
137	#121	C₈×C₆₁	1	4	yes	—
138	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
139	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
140	M₃₂	—	4	4	no	—
141	C₂×C₁₆	—	4	4	no	—
142	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
143	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
144	C₄⋊C₈	C₂×C₄×C₆₁	1	4	yes	—
145	C₂²⋊C₈	C₂×C₄×C₆₁	1	8	yes	—
146	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
147	C₄×C₈	C₂³×C₆₁	1	2	yes	—
148	#41	—	4	2	no	—
149	#41	—	4	2	no	—
150	C₂×Q₈	C₂×C₈×C₆₁	1	2	yes	—
151	C₂×D₄	C₂×C₈×C₆₁	1	4	yes	—
152	C₂×D₄	C₂×C₈×C₆₁	1	4	yes	—
153	C₂×D₄	C₂×C₈×C₆₁	1	2	yes	—
154	C₂²×C₄	C₂×C₈×C₆₁	1	2	yes	—
155	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
157	M₁₆	—	4	2	no	—
158	M₁₆	—	4	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
166	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
167	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
168	C₄⋊C₄	C₂×C₈×C₆₁	1	4	yes	—
169	C₂²⋊C₄	C₂×C₈×C₆₁	1	8	yes	—
170	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
171	C₄²	C₂×C₈×C₆₁	1	1	yes	characteristic
172	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	2	4	no	—
175	C₂³	C₂²×C₈×C₆₁	1	4	yes	—
176	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
177	Q₈	—	4	2	no	—
178	D₄	—	2	8	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₈×C₆₁	1	2	yes	—
188	C₂×C₄	C₂²×C₈×C₆₁	1	4	yes	—
189	C₂×C₄	C₂²×C₈×C₆₁	1	2	yes	—
190	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	2	4	no	—
193	C₈	—	2	1	no	—
194	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
195	C₈	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	4	no	—
200	C₂²	C₂³×C₈×C₆₁	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	4	no	—
204	C₄	C₆₁×#504	1	4	yes	—
205	C₄	C₆₁×#518	1	1	yes	characteristic
206	C₄	C₆₁×#568	1	1	yes	characteristic
207	C₂	—	4	1	no	—
208	C₂	—	4	4	no	—
209	C₂	C₆₁×#2972	1	1	yes	characteristic
210	C₂	C₂×C₆₁×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂³×C₈
prime radical	C₆₁×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2169	C₂	semidirect
3. D₄×C₁₆×C₆₁	C₂	semidirect
4. C₆₁×#2162	C₂	semidirect
5. C₆₁×#2113	C₂	semidirect
90. #33493	C₆₁	direct (N × H)
103. #2169	C₂×C₆₁	semidirect
104. D₄×C₁₆	C₂×C₆₁	semidirect
105. #2162	C₂×C₆₁	semidirect
106. #2113	C₂×C₆₁	semidirect
125. C₆₁	#33493	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×#2931	2
C₆₁×#2169	2
D₄×C₁₆×C₆₁	2
C₆₁×#2162	2
C₆₁×#2113	2
#33493	61

Subgroups — C61×#33493 #29536926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33493 #29536926