Subgroups — C₆₁×#9859 #29513292

Order 15616.

390 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1837	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1493	C₂	1	2	yes	maximal
4	C₆₁×#1493	C₂	1	2	yes	maximal
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#533	C₂²	1	2	yes	—
7	C₆₁×#500	C₄	1	1	yes	characteristic
8	C₆₁×#500	C₄	1	1	yes	characteristic
9	C₆₁×#499	—	2	2	no	—
10	C₆₁×#498	C₄	1	2	yes	—
11	C₆₁×#427	—	2	2	no	—
12	C₆₁×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#341	—	2	2	no	—
17	C₆₁×Q₈⋊C₈	—	2	4	no	—
18	C₆₁×Q₈⋊C₈	—	2	4	no	—
19	C₆₁×#322	—	2	2	no	—
20	C₂×M₁₆×C₆₁	—	2	2	no	—
21	C₂²×C₈×C₆₁	—	2	1	no	—
22	C₂²×C₈×C₆₁	—	2	1	no	—
23	C₂²×C₈×C₆₁	D₄	1	2	yes	—
24	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
26	#125×C₆₁	—	2	2	no	—
27	C₂²⋊Q₈×C₆₁	—	2	2	no	—
28	C₂²⋊Q₈×C₆₁	—	2	2	no	—
29	C₄×Q₈×C₆₁	—	2	2	no	—
30	C₄×Q₈×C₆₁	—	2	2	no	—
31	#117×C₆₁	—	2	2	no	—
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#107×C₆₁	—	2	2	no	—
34	#107×C₆₁	—	2	2	no	—
35	#106×C₆₁	—	2	4	no	—
36	C₄⋊C₈×C₆₁	—	2	2	no	—
37	C₄⋊C₈×C₆₁	—	2	2	no	—
38	C₄×C₈×C₆₁	—	2	1	no	—
39	C₄×C₈×C₆₁	—	2	1	no	—
40	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	D₄	1	2	yes	—
43	C₄×C₈×C₆₁	D₄	1	2	yes	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₂×Q₈×C₆₁	—	2	2	no	—
46	C₂×Q₈×C₆₁	—	2	2	no	—
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
50	M₁₆×C₆₁	—	4	4	no	—
51	C₂×C₈×C₆₁	—	2	4	no	—
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	—	2	4	no	—
54	C₂×C₈×C₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	—	4	2	no	—
56	C₂×C₈×C₆₁	—	4	2	no	—
57	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₆₁	—	2	2	no	—
60	C₄⋊C₄×C₆₁	—	2	2	no	—
61	C₄⋊C₄×C₆₁	—	2	2	no	—
62	C₄⋊C₄×C₆₁	—	2	2	no	—
63	C₄⋊C₄×C₆₁	—	4	2	no	—
64	C₄⋊C₄×C₆₁	—	4	2	no	—
65	C₂²⋊C₄×C₆₁	—	4	2	no	—
66	C₄²×C₆₁	—	4	2	no	—
67	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₆₁	#124	1	1	yes	characteristic
72	Q₈×C₆₁	—	4	4	no	—
73	Q₈×C₆₁	—	4	4	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	4	2	no	—
77	C₂×C₄×C₆₁	—	4	2	no	—
78	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#137	1	4	yes	—
85	C₈×C₆₁	—	2	4	no	—
86	C₈×C₆₁	—	2	2	no	—
87	C₈×C₆₁	—	2	2	no	—
88	C₈×C₆₁	—	4	2	no	—
89	C₈×C₆₁	—	8	2	no	—
90	#9859	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₆₁	—	2	1	no	—
92	C₂²×C₆₁	—	2	1	no	—
93	C₂²×C₆₁	#390	1	1	yes	characteristic
94	C₂²×C₆₁	#491	1	2	yes	—
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	8	2	no	—
98	C₄×C₆₁	—	8	2	no	—
99	C₄×C₆₁	#419	1	2	yes	—
100	C₄×C₆₁	#443	1	2	yes	—
101	C₄×C₆₁	#444	1	2	yes	—
102	C₄×C₆₁	#497	1	2	yes	—
103	#3160	C₂×C₆₁	1	1	yes	characteristic
104	#1837	C₂×C₆₁	1	1	yes	characteristic
105	#1493	C₂×C₆₁	1	2	yes	—
106	#1493	C₂×C₆₁	1	2	yes	—
107	C₂×#322	C₂×C₆₁	1	1	yes	characteristic
108	C₂×C₆₁	—	2	2	no	—
109	C₂×C₆₁	#1960	1	1	yes	characteristic
110	C₂×C₆₁	#1974	1	1	yes	characteristic
111	C₂×C₆₁	#1975	1	1	yes	characteristic
112	#533	C₂²×C₆₁	1	2	yes	—
113	#500	C₄×C₆₁	1	1	yes	characteristic
114	#500	C₄×C₆₁	1	1	yes	characteristic
115	#499	—	2	2	no	—
116	#498	C₄×C₆₁	1	2	yes	—
117	#427	—	2	2	no	—
118	#423	C₂²×C₆₁	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
122	#341	—	2	2	no	—
123	Q₈⋊C₈	—	2	4	no	—
124	Q₈⋊C₈	—	2	4	no	—
125	#322	—	2	2	no	—
126	C₆₁	#9859	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×M₁₆	—	2	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	D₄×C₆₁	1	2	yes	—
131	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
132	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
133	#125	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	—	2	2	no	—
138	#117	—	2	2	no	—
139	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
140	#107	—	2	2	no	—
141	#107	—	2	2	no	—
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
148	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
149	C₄×C₈	D₄×C₆₁	1	2	yes	—
150	C₄×C₈	D₄×C₆₁	1	2	yes	—
151	C₄×C₈	D₄×C₆₁	1	2	yes	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
155	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
156	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
157	M₁₆	—	4	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
165	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
175	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
177	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
178	C₂³	#124×C₆₁	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	4	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
186	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
187	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
188	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
189	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
190	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
191	C₂×C₄	#137×C₆₁	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	8	2	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₆₁×#390	1	1	yes	characteristic
200	C₂²	C₆₁×#491	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	2	no	—
204	C₄	—	8	2	no	—
205	C₄	C₆₁×#419	1	2	yes	—
206	C₄	C₆₁×#443	1	2	yes	—
207	C₄	C₆₁×#444	1	2	yes	—
208	C₄	C₆₁×#497	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	C₆₁×#1960	1	1	yes	characteristic
211	C₂	C₆₁×#1974	1	1	yes	characteristic
212	C₂	C₆₁×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#127
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
90. #9859	C₆₁	direct (N × H)
126. C₆₁	#9859	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×#3160	2
C₆₁×#1837	2
C₆₁×#1493	2
C₂×C₆₁×#322	2
#9859	61

Subgroups — C61×#9859 #29513292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9859 #29513292