Subgroups — C₂×C₆₁×#2246 #29537142

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₆₁	C₂	1	4	yes	maximal, complemented
2	C₂×C₆₁×#493	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2246	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₆₁	—	2	2	no	—
6	C₂²×D₈×C₆₁	C₂²	1	4	yes	—
7	C₂×#127×C₆₁	—	2	2	no	—
8	C₂×D₁₆×C₆₁	—	2	16	no	—
9	C₂×D₁₆×C₆₁	C₂²	1	16	yes	complemented
10	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#493	—	2	4	no	—
12	C₆₁×#493	C₂²	1	8	yes	complemented
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₁	C₂²	1	4	yes	complemented
15	C₂²×D₄×C₆₁	—	2	4	no	—
16	C₂²×D₄×C₆₁	—	4	4	no	—
17	C₂×D₈×C₆₁	—	2	16	no	—
18	C₂×D₈×C₆₁	—	2	8	no	—
19	C₂×D₈×C₆₁	—	2	4	no	—
20	C₂×D₈×C₆₁	—	4	8	no	—
21	C₂×D₈×C₆₁	C₂³	1	8	yes	—
22	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
24	#127×C₆₁	—	2	8	no	—
25	#127×C₆₁	—	4	4	no	—
26	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
27	D₁₆×C₆₁	—	2	32	no	—
28	C₂×C₁₆×C₆₁	D₄	1	8	yes	complemented
29	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
30	C₄×C₈×C₆₁	D₄	1	2	yes	—
31	C₄×C₈×C₆₁	C₂³	1	2	yes	—
32	C₂⁴×C₆₁	—	4	4	no	—
33	C₂×D₄×C₆₁	—	2	8	no	—
34	C₂×D₄×C₆₁	—	4	8	no	—
35	C₂×D₄×C₆₁	—	4	8	no	—
36	C₂×D₄×C₆₁	—	4	16	no	—
37	C₂×D₄×C₆₁	—	4	4	no	—
38	C₂×D₄×C₆₁	—	8	16	no	—
39	C₂²×C₄×C₆₁	D₈	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	D₈×C₆₁	—	2	16	no	—
42	D₈×C₆₁	—	4	8	no	—
43	D₈×C₆₁	—	4	16	no	—
44	C₂×C₈×C₆₁	D₈	1	4	yes	—
45	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
47	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₄²×C₆₁	D₈	1	2	yes	—
50	C₄²×C₆₁	C₂×D₄	1	2	yes	—
51	C₁₆×C₆₁	C₂×D₄	1	8	yes	complemented
52	C₂³×C₆₁	—	4	8	no	—
53	C₂³×C₆₁	—	8	8	no	—
54	C₂³×C₆₁	—	8	16	no	—
55	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
56	D₄×C₆₁	—	4	8	no	—
57	D₄×C₆₁	—	4	8	no	—
58	D₄×C₆₁	—	8	8	no	—
59	D₄×C₆₁	—	8	32	no	—
60	C₂×C₄×C₆₁	D₁₆	1	8	yes	complemented
61	C₂×C₄×C₆₁	#127	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
65	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
66	C₈×C₆₁	#127	1	2	yes	—
67	C₈×C₆₁	C₂×D₈	1	4	yes	—
68	C₈×C₆₁	C₂²×D₄	1	2	yes	—
69	C₂×#2246	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₆₁	—	8	8	no	—
71	C₂²×C₆₁	—	8	16	no	—
72	C₂²×C₆₁	—	16	16	no	—
73	C₂²×C₆₁	#493	1	2	yes	—
74	C₂²×C₆₁	C₂×D₁₆	1	4	yes	—
75	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
76	C₄×C₆₁	#493	1	2	yes	—
77	C₄×C₆₁	C₂×D₁₆	1	8	yes	complemented
78	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
79	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
80	C₂²×D₁₆	C₂×C₆₁	1	4	yes	complemented
81	C₂×#493	C₂×C₆₁	1	2	yes	complemented
82	#2246	C₂×C₆₁	1	8	yes	direct product factor
83	C₂×C₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
84	C₂×C₆₁	—	16	8	no	—
85	C₂×C₆₁	#2246	1	4	yes	direct product factor
86	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
87	C₂×C₆₁	C₂²×D₁₆	1	2	yes	—
88	C₂²×D₈	—	2	2	no	—
89	C₂²×D₈	C₂²×C₆₁	1	4	yes	—
90	C₂×#127	—	2	2	no	—
91	C₂×D₁₆	—	2	16	no	—
92	C₂×D₁₆	C₂²×C₆₁	1	16	yes	complemented
93	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
94	#493	—	2	4	no	—
95	#493	C₂²×C₆₁	1	8	yes	complemented
96	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
97	C₄×C₁₆	C₂²×C₆₁	1	4	yes	complemented
98	C₆₁	C₂×#2246	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×D₄	—	2	4	no	—
100	C₂²×D₄	—	4	4	no	—
101	C₂×D₈	—	2	16	no	—
102	C₂×D₈	—	2	8	no	—
103	C₂×D₈	—	2	4	no	—
104	C₂×D₈	—	4	8	no	—
105	C₂×D₈	C₂³×C₆₁	1	8	yes	—
106	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
108	#127	—	2	8	no	—
109	#127	—	4	4	no	—
110	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
111	D₁₆	—	2	32	no	—
112	C₂×C₁₆	D₄×C₆₁	1	8	yes	complemented
113	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
114	C₄×C₈	D₄×C₆₁	1	2	yes	—
115	C₄×C₈	C₂³×C₆₁	1	2	yes	—
116	C₂⁴	—	4	4	no	—
117	C₂×D₄	—	2	8	no	—
118	C₂×D₄	—	4	8	no	—
119	C₂×D₄	—	4	8	no	—
120	C₂×D₄	—	4	16	no	—
121	C₂×D₄	—	4	4	no	—
122	C₂×D₄	—	8	16	no	—
123	C₂²×C₄	D₈×C₆₁	1	2	yes	—
124	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
125	D₈	—	2	16	no	—
126	D₈	—	4	8	no	—
127	D₈	—	4	16	no	—
128	C₂×C₈	D₈×C₆₁	1	4	yes	—
129	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
130	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
131	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
132	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
133	C₄²	D₈×C₆₁	1	2	yes	—
134	C₄²	C₂×D₄×C₆₁	1	2	yes	—
135	C₁₆	C₂×D₄×C₆₁	1	8	yes	complemented
136	C₂³	—	4	8	no	—
137	C₂³	—	8	8	no	—
138	C₂³	—	8	16	no	—
139	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
140	D₄	—	4	8	no	—
141	D₄	—	4	8	no	—
142	D₄	—	8	8	no	—
143	D₄	—	8	32	no	—
144	C₂×C₄	D₁₆×C₆₁	1	8	yes	complemented
145	C₂×C₄	#127×C₆₁	1	2	yes	—
146	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
147	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
148	C₂×C₄	C₂×D₈×C₆₁	1	4	yes	—
149	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
150	C₈	#127×C₆₁	1	2	yes	—
151	C₈	C₂×D₈×C₆₁	1	4	yes	—
152	C₈	C₂²×D₄×C₆₁	1	2	yes	—
153	C₂²	—	8	8	no	—
154	C₂²	—	8	16	no	—
155	C₂²	—	16	16	no	—
156	C₂²	C₆₁×#493	1	2	yes	—
157	C₂²	C₂×D₁₆×C₆₁	1	4	yes	—
158	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
159	C₄	C₆₁×#493	1	2	yes	—
160	C₄	C₂×D₁₆×C₆₁	1	8	yes	complemented
161	C₄	C₂×#127×C₆₁	1	1	yes	characteristic
162	C₄	C₂²×D₈×C₆₁	1	1	yes	characteristic
163	C₂	—	16	8	no	—
164	C₂	C₆₁×#2246	1	4	yes	direct product factor
165	C₂	C₂×C₆₁×#493	1	1	yes	characteristic
166	C₂	C₂²×D₁₆×C₆₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₆×C₆₁	C₂	semidirect
2. C₂×C₆₁×#493	C₂	semidirect
3. C₆₁×#2246	C₂	direct (N × H)
4. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
9. C₂×D₁₆×C₆₁	C₂²	semidirect
12. C₆₁×#493	C₂²	semidirect
14. C₄×C₁₆×C₆₁	C₂²	semidirect
28. C₂×C₁₆×C₆₁	D₄	semidirect
51. C₁₆×C₆₁	C₂×D₄	semidirect
60. C₂×C₄×C₆₁	D₁₆	semidirect
69. C₂×#2246	C₆₁	direct (N × H)
77. C₄×C₆₁	C₂×D₁₆	semidirect
80. C₂²×D₁₆	C₂×C₆₁	semidirect
81. C₂×#493	C₂×C₆₁	semidirect
82. #2246	C₂×C₆₁	direct (N × H)
83. C₂×C₄×C₁₆	C₂×C₆₁	semidirect
85. C₂×C₆₁	#2246	direct (N × H)
92. C₂×D₁₆	C₂²×C₆₁	semidirect
95. #493	C₂²×C₆₁	semidirect
97. C₄×C₁₆	C₂²×C₆₁	semidirect
98. C₆₁	C₂×#2246	direct (N × H)
112. C₂×C₁₆	D₄×C₆₁	semidirect
135. C₁₆	C₂×D₄×C₆₁	semidirect
144. C₂×C₄	D₁₆×C₆₁	semidirect
160. C₄	C₂×D₁₆×C₆₁	semidirect
164. C₂	C₆₁×#2246	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₆₁	2
C₂×C₆₁×#493	2
C₆₁×#2246	2
C₂×C₄×C₁₆×C₆₁	2
C₂×#2246	61

Subgroups — C2×C61×#2246 #29537142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#2246 #29537142