Subgroups — C₆₁×#23801 #29527234

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₆₁	C₂	1	4	yes	maximal
2	C₄×C₄⋊Q₈×C₆₁	C₂	1	2	yes	maximal
3	C₆₁×#2095	C₂	1	4	yes	maximal
4	C₆₁×#2016	C₂	1	4	yes	maximal
5	C₆₁×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₁	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₆₁	C₂²	1	4	yes	—
9	C₄×Q₁₆×C₆₁	—	2	16	no	—
10	C₂×#107×C₆₁	C₂²	1	4	yes	—
11	C₂×#103×C₆₁	C₂²	1	8	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#391	C₂²	1	4	yes	—
14	C₆₁×#389	—	2	4	no	—
15	C₆₁×#384	C₂²	1	2	yes	—
16	C₄×C₄⋊C₄×C₆₁	—	2	4	no	—
17	C₄³×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#336	C₂²	1	4	yes	—
19	C₂²×Q₈×C₆₁	C₂³	1	4	yes	—
20	C₂×Q₁₆×C₆₁	—	2	16	no	—
21	C₂×Q₁₆×C₆₁	—	2	8	no	—
22	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
23	C₄⋊Q₈×C₆₁	—	2	8	no	—
24	C₄×Q₈×C₆₁	—	2	16	no	—
25	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
29	C₂×C₄²×C₆₁	—	2	4	no	—
30	C₂×C₄²×C₆₁	D₄	1	2	yes	—
31	C₂×C₄²×C₆₁	D₄	1	2	yes	—
32	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	#107×C₆₁	—	2	8	no	—
35	#103×C₆₁	—	2	16	no	—
36	C₄×C₈×C₆₁	D₄	1	8	yes	—
37	#95×C₆₁	—	2	8	no	—
38	C₂×Q₈×C₆₁	—	2	4	no	—
39	C₂×Q₈×C₆₁	—	2	8	no	—
40	C₂×Q₈×C₆₁	—	2	8	no	—
41	C₂×Q₈×C₆₁	#41	1	8	yes	—
42	C₂²×C₄×C₆₁	—	2	4	no	—
43	C₂²×C₄×C₆₁	—	2	4	no	—
44	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
48	Q₁₆×C₆₁	—	2	32	no	—
49	C₂×C₈×C₆₁	—	2	8	no	—
50	C₂×C₈×C₆₁	C₂×D₄	1	8	yes	—
51	C₄⋊C₄×C₆₁	—	2	16	no	—
52	C₄⋊C₄×C₆₁	—	2	8	no	—
53	C₄⋊C₄×C₆₁	—	4	8	no	—
54	C₄⋊C₄×C₆₁	—	4	8	no	—
55	C₄²×C₆₁	—	2	4	no	—
56	C₄²×C₆₁	—	4	2	no	—
57	C₄²×C₆₁	—	4	8	no	—
58	C₄²×C₆₁	Q₁₆	1	8	yes	complemented
59	C₄²×C₆₁	C₂×D₄	1	4	yes	—
60	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
61	Q₈×C₆₁	—	2	16	no	—
62	Q₈×C₆₁	—	4	8	no	—
63	C₂×C₄×C₆₁	—	2	4	no	—
64	C₂×C₄×C₆₁	—	2	4	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	8	no	—
67	C₂×C₄×C₆₁	—	4	4	no	—
68	C₂×C₄×C₆₁	—	4	4	no	—
69	C₂×C₄×C₆₁	—	4	4	no	—
70	C₂×C₄×C₆₁	—	4	8	no	—
71	C₂×C₄×C₆₁	#127	1	2	yes	—
72	C₂×C₄×C₆₁	#127	1	2	yes	—
73	C₂×C₄×C₆₁	C₂×Q₁₆	1	8	yes	—
74	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
75	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
77	C₈×C₆₁	—	2	8	no	—
78	#23801	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₆₁	#525	1	2	yes	—
80	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
81	C₂²×C₆₁	C₂²×Q₁₆	1	2	yes	—
82	C₂²×C₆₁	#575	1	2	yes	—
83	C₄×C₆₁	—	2	8	no	—
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	—	4	8	no	—
86	C₄×C₆₁	—	8	4	no	—
87	C₄×C₆₁	#494	1	4	yes	—
88	C₄×C₆₁	#525	1	4	yes	—
89	C₂×C₄×Q₁₆	C₂×C₆₁	1	4	yes	—
90	C₄×C₄⋊Q₈	C₂×C₆₁	1	2	yes	—
91	#2095	C₂×C₆₁	1	4	yes	—
92	#2016	C₂×C₆₁	1	4	yes	—
93	#1744	C₂×C₆₁	1	1	yes	characteristic
94	C₂×C₆₁	#2397	1	1	yes	characteristic
95	C₂×C₆₁	#3104	1	4	yes	—
96	C₂×C₆₁	C₂×#494	1	1	yes	characteristic
97	C₂×C₆₁	#3150	1	1	yes	characteristic
98	C₂²×Q₁₆	C₂²×C₆₁	1	4	yes	—
99	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
100	C₂×C₄×Q₈	C₂²×C₆₁	1	4	yes	—
101	C₄×Q₁₆	—	2	16	no	—
102	C₂×#107	C₂²×C₆₁	1	4	yes	—
103	C₂×#103	C₂²×C₆₁	1	8	yes	—
104	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
105	#391	C₂²×C₆₁	1	4	yes	—
106	#389	—	2	4	no	—
107	#384	C₂²×C₆₁	1	2	yes	—
108	C₄×C₄⋊C₄	—	2	4	no	—
109	C₄³	C₂²×C₆₁	1	1	yes	characteristic
110	#336	C₂²×C₆₁	1	4	yes	—
111	C₆₁	#23801	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×Q₈	C₂³×C₆₁	1	4	yes	—
113	C₂×Q₁₆	—	2	16	no	—
114	C₂×Q₁₆	—	2	8	no	—
115	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
116	C₄⋊Q₈	—	2	8	no	—
117	C₄×Q₈	—	2	16	no	—
118	C₂×C₄⋊C₄	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
122	C₂×C₄²	—	2	4	no	—
123	C₂×C₄²	D₄×C₆₁	1	2	yes	—
124	C₂×C₄²	D₄×C₆₁	1	2	yes	—
125	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
126	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
127	#107	—	2	8	no	—
128	#103	—	2	16	no	—
129	C₄×C₈	D₄×C₆₁	1	8	yes	—
130	#95	—	2	8	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	#41×C₆₁	1	8	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
139	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
140	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
141	Q₁₆	—	2	32	no	—
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	C₂×D₄×C₆₁	1	8	yes	—
144	C₄⋊C₄	—	2	16	no	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	4	8	no	—
147	C₄⋊C₄	—	4	8	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	8	no	—
151	C₄²	Q₁₆×C₆₁	1	8	yes	complemented
152	C₄²	C₂×D₄×C₆₁	1	4	yes	—
153	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
154	Q₈	—	2	16	no	—
155	Q₈	—	4	8	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	#127×C₆₁	1	2	yes	—
165	C₂×C₄	#127×C₆₁	1	2	yes	—
166	C₂×C₄	C₂×Q₁₆×C₆₁	1	8	yes	—
167	C₂×C₄	C₂×Q₁₆×C₆₁	1	4	yes	—
168	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
169	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
170	C₈	—	2	8	no	—
171	C₂²	C₆₁×#525	1	2	yes	—
172	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
173	C₂²	C₂²×Q₁₆×C₆₁	1	2	yes	—
174	C₂²	C₆₁×#575	1	2	yes	—
175	C₄	—	2	8	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	8	no	—
178	C₄	—	8	4	no	—
179	C₄	C₆₁×#494	1	4	yes	—
180	C₄	C₆₁×#525	1	4	yes	—
181	C₂	C₆₁×#2397	1	1	yes	characteristic
182	C₂	C₆₁×#3104	1	4	yes	—
183	C₂	C₂×C₆₁×#494	1	1	yes	characteristic
184	C₂	C₆₁×#3150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle, prime radical	C₂³×C₆₁	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
58. C₄²×C₆₁	Q₁₆	semidirect
78. #23801	C₆₁	direct (N × H)
111. C₆₁	#23801	direct (N × H)
151. C₄²	Q₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₆₁	2
C₄×C₄⋊Q₈×C₆₁	2
C₆₁×#2095	2
C₆₁×#2016	2
C₆₁×#1744	2
#23801	61

Subgroups — C61×#23801 #29527234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23801 #29527234