Subgroups — C₆₁×#19652 #29523085

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3208	C₂	1	4	yes	maximal, complemented
2	C₆₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#493	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1715	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1706	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₆₁	—	2	2	no	—
8	C₆₁×#533	C₂²	1	2	yes	—
9	C₂×#127×C₆₁	C₂²	1	2	yes	—
10	C₆₁×#500	C₂²	1	2	yes	complemented
11	C₆₁×#499	—	2	2	no	—
12	C₆₁×#498	C₂²	1	2	yes	complemented
13	C₆₁×#493	—	2	2	no	—
14	C₆₁×#493	C₂²	1	2	yes	—
15	C₆₁×#493	C₂²	1	2	yes	—
16	C₆₁×#492	C₂²	1	8	yes	complemented
17	C₆₁×#486	C₂²	1	8	yes	complemented
18	C₆₁×#460	—	2	8	no	—
19	C₆₁×#450	—	2	8	no	—
20	C₆₁×#431	C₂²	1	2	yes	—
21	C₆₁×#427	—	2	2	no	—
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#322	C₂²	1	2	yes	complemented
24	C₈²×C₆₁	C₂²	1	2	yes	complemented
25	C₂²×D₄×C₆₁	—	2	2	no	—
26	C₂²×D₄×C₆₁	—	2	4	no	—
27	C₂×SD₁₆×C₆₁	—	2	16	no	—
28	C₂×D₈×C₆₁	—	2	4	no	—
29	C₂×D₈×C₆₁	—	2	4	no	—
30	C₂×D₈×C₆₁	—	4	8	no	—
31	C₂²×C₈×C₆₁	D₄	1	2	yes	—
32	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
33	#127×C₆₁	—	2	4	no	—
34	#127×C₆₁	—	2	2	no	—
35	#127×C₆₁	C₂³	1	4	yes	—
36	#125×C₆₁	—	2	2	no	—
37	C₂²⋊Q₈×C₆₁	—	2	4	no	—
38	C₄×Q₈×C₆₁	—	2	4	no	—
39	#117×C₆₁	—	2	2	no	—
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	#107×C₆₁	—	2	4	no	—
42	#106×C₆₁	—	2	4	no	—
43	C₄⋊C₈×C₆₁	D₄	1	4	yes	—
44	#102×C₆₁	—	2	16	no	—
45	C₂²⋊C₈×C₆₁	D₄	1	4	yes	—
46	C₄×C₈×C₆₁	D₄	1	2	yes	—
47	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₆₁	C₂³	1	4	yes	—
49	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	C₂⁴×C₆₁	—	4	2	no	—
51	C₂×Q₈×C₆₁	—	2	4	no	—
52	C₂×D₄×C₆₁	—	2	8	no	—
53	C₂×D₄×C₆₁	—	2	4	no	—
54	C₂×D₄×C₆₁	—	4	4	no	—
55	C₂×D₄×C₆₁	—	4	2	no	—
56	C₂×D₄×C₆₁	—	4	8	no	—
57	C₂×D₄×C₆₁	—	8	8	no	—
58	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	SD₁₆×C₆₁	—	8	16	no	—
61	D₈×C₆₁	—	4	8	no	—
62	D₈×C₆₁	—	4	8	no	—
63	C₂×C₈×C₆₁	SD₁₆	1	8	yes	complemented
64	C₂×C₈×C₆₁	C₂×D₄	1	8	yes	—
65	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
67	C₄⋊C₄×C₆₁	—	2	4	no	—
68	C₄⋊C₄×C₆₁	—	2	4	no	—
69	C₄⋊C₄×C₆₁	—	4	4	no	—
70	C₂²⋊C₄×C₆₁	—	4	2	no	—
71	C₄²×C₆₁	—	4	2	no	—
72	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₆₁	C₂×D₄	1	2	yes	—
74	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
75	C₂³×C₆₁	—	4	4	no	—
76	C₂³×C₆₁	—	8	8	no	—
77	C₂³×C₆₁	—	8	4	no	—
78	C₂³×C₆₁	#127	1	1	yes	characteristic
79	Q₈×C₆₁	—	4	8	no	—
80	D₄×C₆₁	—	4	8	no	—
81	D₄×C₆₁	—	4	16	no	—
82	D₄×C₆₁	—	8	8	no	—
83	D₄×C₆₁	—	8	4	no	—
84	C₂×C₄×C₆₁	—	2	4	no	—
85	C₂×C₄×C₆₁	—	4	4	no	—
86	C₂×C₄×C₆₁	#127	1	2	yes	—
87	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
89	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
91	C₈×C₆₁	—	2	8	no	—
92	C₈×C₆₁	C₂×SD₁₆	1	4	yes	—
93	C₈×C₆₁	C₂×SD₁₆	1	4	yes	—
94	#19652	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₆₁	—	2	2	no	—
96	C₂²×C₆₁	—	8	4	no	—
97	C₂²×C₆₁	—	8	8	no	—
98	C₂²×C₆₁	—	16	8	no	—
99	C₂²×C₆₁	#492	1	2	yes	—
100	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
101	C₄×C₆₁	—	2	2	no	—
102	C₄×C₆₁	—	8	4	no	—
103	C₄×C₆₁	#492	1	2	yes	—
104	C₄×C₆₁	C₂²×SD₁₆	1	2	yes	—
105	C₄×C₆₁	#576	1	4	yes	—
106	#3208	C₂×C₆₁	1	4	yes	complemented
107	#3160	C₂×C₆₁	1	1	yes	characteristic, complemented
108	C₂×#493	C₂×C₆₁	1	1	yes	characteristic
109	#1715	C₂×C₆₁	1	4	yes	complemented
110	#1706	C₂×C₆₁	1	4	yes	complemented
111	#1450	C₂×C₆₁	1	1	yes	characteristic, complemented
112	C₂×C₆₁	—	2	2	no	—
113	C₂×C₆₁	—	16	4	no	—
114	C₂×C₆₁	C₂×#492	1	1	yes	characteristic
115	C₂×C₆₁	#3155	1	2	yes	—
116	C₂²×D₈	—	2	2	no	—
117	#533	C₂²×C₆₁	1	2	yes	—
118	C₂×#127	C₂²×C₆₁	1	2	yes	—
119	#500	C₂²×C₆₁	1	2	yes	complemented
120	#499	—	2	2	no	—
121	#498	C₂²×C₆₁	1	2	yes	complemented
122	#493	—	2	2	no	—
123	#493	C₂²×C₆₁	1	2	yes	—
124	#493	C₂²×C₆₁	1	2	yes	—
125	#492	C₂²×C₆₁	1	8	yes	complemented
126	#486	C₂²×C₆₁	1	8	yes	complemented
127	#460	—	2	8	no	—
128	#450	—	2	8	no	—
129	#431	C₂²×C₆₁	1	2	yes	—
130	#427	—	2	2	no	—
131	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
132	#322	C₂²×C₆₁	1	2	yes	complemented
133	C₈²	C₂²×C₆₁	1	2	yes	complemented
134	C₆₁	#19652	1	1	yes	characteristic, direct product factor, Sylow
135	C₂²×D₄	—	2	2	no	—
136	C₂²×D₄	—	2	4	no	—
137	C₂×SD₁₆	—	2	16	no	—
138	C₂×D₈	—	2	4	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	4	8	no	—
141	C₂²×C₈	D₄×C₆₁	1	2	yes	—
142	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
143	#127	—	2	4	no	—
144	#127	—	2	2	no	—
145	#127	C₂³×C₆₁	1	4	yes	—
146	#125	—	2	2	no	—
147	C₂²⋊Q₈	—	2	4	no	—
148	C₄×Q₈	—	2	4	no	—
149	#117	—	2	2	no	—
150	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
151	#107	—	2	4	no	—
152	#106	—	2	4	no	—
153	C₄⋊C₈	D₄×C₆₁	1	4	yes	—
154	#102	—	2	16	no	—
155	C₂²⋊C₈	D₄×C₆₁	1	4	yes	—
156	C₄×C₈	D₄×C₆₁	1	2	yes	—
157	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₆₁	1	4	yes	—
159	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
160	C₂⁴	—	4	2	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×D₄	—	2	8	no	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	8	no	—
167	C₂×D₄	—	8	8	no	—
168	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
170	SD₁₆	—	8	16	no	—
171	D₈	—	4	8	no	—
172	D₈	—	4	8	no	—
173	C₂×C₈	SD₁₆×C₆₁	1	8	yes	complemented
174	C₂×C₈	C₂×D₄×C₆₁	1	8	yes	—
175	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
176	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
183	C₄²	C₂×D₄×C₆₁	1	2	yes	—
184	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
185	C₂³	—	4	4	no	—
186	C₂³	—	8	8	no	—
187	C₂³	—	8	4	no	—
188	C₂³	#127×C₆₁	1	1	yes	characteristic
189	Q₈	—	4	8	no	—
190	D₄	—	4	8	no	—
191	D₄	—	4	16	no	—
192	D₄	—	8	8	no	—
193	D₄	—	8	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	#127×C₆₁	1	2	yes	—
197	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
198	C₂×C₄	C₂×SD₁₆×C₆₁	1	4	yes	—
199	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
200	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
201	C₈	—	2	8	no	—
202	C₈	C₂×SD₁₆×C₆₁	1	4	yes	—
203	C₈	C₂×SD₁₆×C₆₁	1	4	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	—	8	4	no	—
206	C₂²	—	8	8	no	—
207	C₂²	—	16	8	no	—
208	C₂²	C₆₁×#492	1	2	yes	—
209	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	8	4	no	—
212	C₄	C₆₁×#492	1	2	yes	—
213	C₄	C₂²×SD₁₆×C₆₁	1	2	yes	—
214	C₄	C₆₁×#576	1	4	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	16	4	no	—
217	C₂	C₂×C₆₁×#492	1	1	yes	characteristic
218	C₂	C₆₁×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₂×C₆₁×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3208	C₂	semidirect
2. C₆₁×#3160	C₂	semidirect
4. C₆₁×#1715	C₂	semidirect
5. C₆₁×#1706	C₂	semidirect
6. C₆₁×#1450	C₂	semidirect
10. C₆₁×#500	C₂²	semidirect
12. C₆₁×#498	C₂²	semidirect
16. C₆₁×#492	C₂²	semidirect
17. C₆₁×#486	C₂²	semidirect
23. C₆₁×#322	C₂²	semidirect
24. C₈²×C₆₁	C₂²	semidirect
63. C₂×C₈×C₆₁	SD₁₆	semidirect
94. #19652	C₆₁	direct (N × H)
106. #3208	C₂×C₆₁	semidirect
107. #3160	C₂×C₆₁	semidirect
109. #1715	C₂×C₆₁	semidirect
110. #1706	C₂×C₆₁	semidirect
111. #1450	C₂×C₆₁	semidirect
119. #500	C₂²×C₆₁	semidirect
121. #498	C₂²×C₆₁	semidirect
125. #492	C₂²×C₆₁	semidirect
126. #486	C₂²×C₆₁	semidirect
132. #322	C₂²×C₆₁	semidirect
133. C₈²	C₂²×C₆₁	semidirect
134. C₆₁	#19652	direct (N × H)
173. C₂×C₈	SD₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3208	2
C₆₁×#3160	2
C₂×C₆₁×#493	2
C₆₁×#1715	2
C₆₁×#1706	2
C₆₁×#1450	2
#19652	61

Subgroups — C61×#19652 #29523085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19652 #29523085