Subgroups — C₂×C₆₁×#2258 #29537406

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#504	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2258	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₆₁×M₆₄	C₂	1	3	yes	maximal, complemented
4	C₂²×C₃₂×C₆₁	C₂	1	3	yes	maximal, complemented
5	C₂×#131×C₆₁	C₄	1	1	yes	characteristic
6	C₆₁×#504	C₄	1	4	yes	—
7	C₆₁×#504	C₂²	1	4	yes	—
8	C₂×M₃₂×C₆₁	C₄	1	3	yes	—
9	C₂²×C₁₆×C₆₁	C₂²	1	3	yes	—
10	C₆₁×M₆₄	C₂²	1	12	yes	complemented
11	C₂×C₃₂×C₆₁	C₂²	1	3	yes	—
12	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₆₁	C₂²	1	12	yes	complemented
14	C₂×#41×C₆₁	C₈	1	1	yes	characteristic
15	#131×C₆₁	C₈	1	4	yes	—
16	#131×C₆₁	C₂×C₄	1	4	yes	—
17	C₂×M₁₆×C₆₁	C₈	1	3	yes	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
19	M₃₂×C₆₁	C₂×C₄	1	12	yes	—
20	C₂×C₁₆×C₆₁	C₂×C₄	1	3	yes	—
21	C₂×C₁₆×C₆₁	C₂×C₄	1	6	yes	—
22	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	C₂³	1	6	yes	—
24	C₃₂×C₆₁	C₂³	1	6	yes	—
25	C₃₂×C₆₁	C₂³	1	2	yes	—
26	#41×C₆₁	C₁₆	1	4	yes	—
27	#41×C₆₁	C₂×C₈	1	4	yes	—
28	C₂×Q₈×C₆₁	C₁₆	1	1	yes	characteristic
29	C₂×D₄×C₆₁	C₁₆	1	3	yes	—
30	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
31	M₁₆×C₆₁	C₂×C₈	1	12	yes	—
32	C₂×C₈×C₆₁	C₂×C₈	1	3	yes	—
33	C₂×C₈×C₆₁	C₂×C₈	1	6	yes	—
34	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂²×C₄	1	6	yes	—
36	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₁₆×C₆₁	C₂²×C₄	1	6	yes	—
38	C₁₆×C₆₁	C₂⁴	1	1	yes	characteristic
39	C₂³×C₆₁	C₂×C₁₆	1	3	yes	—
40	Q₈×C₆₁	C₂×C₁₆	1	4	yes	—
41	D₄×C₆₁	C₂×C₁₆	1	12	yes	—
42	C₂×C₄×C₆₁	C₂×C₁₆	1	3	yes	—
43	C₂×C₄×C₆₁	C₂×C₁₆	1	6	yes	—
44	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
46	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₆₁	C₂²×C₈	1	6	yes	—
48	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
49	C₂×#2258	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₆₁	—	2	6	no	—
51	C₂²×C₆₁	C₂²×C₁₆	1	6	yes	—
52	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
53	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₆₁	C₂²×C₁₆	1	6	yes	—
55	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
56	C₂×#504	C₂×C₆₁	1	1	yes	characteristic
57	#2258	C₂×C₆₁	1	8	yes	direct product factor
58	C₂×M₆₄	C₂×C₆₁	1	3	yes	complemented
59	C₂²×C₃₂	C₂×C₆₁	1	3	yes	complemented
60	C₂×C₆₁	—	2	6	no	—
61	C₂×C₆₁	#2258	1	2	yes	direct product factor
62	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
63	C₂×#131	C₄×C₆₁	1	1	yes	characteristic
64	#504	C₄×C₆₁	1	4	yes	—
65	#504	C₂²×C₆₁	1	4	yes	—
66	C₂×M₃₂	C₄×C₆₁	1	3	yes	—
67	C₂²×C₁₆	C₂²×C₆₁	1	3	yes	—
68	M₆₄	C₂²×C₆₁	1	12	yes	complemented
69	C₂×C₃₂	C₂²×C₆₁	1	3	yes	—
70	C₂×C₃₂	C₂²×C₆₁	1	1	yes	characteristic
71	C₂×C₃₂	C₂²×C₆₁	1	12	yes	complemented
72	C₆₁	C₂×#2258	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×#41	C₈×C₆₁	1	1	yes	characteristic
74	#131	C₈×C₆₁	1	4	yes	—
75	#131	C₂×C₄×C₆₁	1	4	yes	—
76	C₂×M₁₆	C₈×C₆₁	1	3	yes	—
77	C₂²×C₈	C₂×C₄×C₆₁	1	3	yes	—
78	M₃₂	C₂×C₄×C₆₁	1	12	yes	—
79	C₂×C₁₆	C₂×C₄×C₆₁	1	3	yes	—
80	C₂×C₁₆	C₂×C₄×C₆₁	1	6	yes	—
81	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
82	C₂×C₁₆	C₂³×C₆₁	1	6	yes	—
83	C₃₂	C₂³×C₆₁	1	2	yes	—
84	C₃₂	C₂³×C₆₁	1	6	yes	—
85	#41	C₁₆×C₆₁	1	4	yes	—
86	#41	C₂×C₈×C₆₁	1	4	yes	—
87	C₂×Q₈	C₁₆×C₆₁	1	1	yes	characteristic
88	C₂×D₄	C₁₆×C₆₁	1	3	yes	—
89	C₂²×C₄	C₂×C₈×C₆₁	1	3	yes	—
90	M₁₆	C₂×C₈×C₆₁	1	12	yes	—
91	C₂×C₈	C₂×C₈×C₆₁	1	3	yes	—
92	C₂×C₈	C₂×C₈×C₆₁	1	6	yes	—
93	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₄×C₆₁	1	6	yes	—
95	C₁₆	C₂²×C₄×C₆₁	1	1	yes	characteristic
96	C₁₆	C₂²×C₄×C₆₁	1	6	yes	—
97	C₁₆	C₂⁴×C₆₁	1	1	yes	characteristic
98	C₂³	C₂×C₁₆×C₆₁	1	3	yes	—
99	Q₈	C₂×C₁₆×C₆₁	1	4	yes	—
100	D₄	C₂×C₁₆×C₆₁	1	12	yes	—
101	C₂×C₄	C₂×C₁₆×C₆₁	1	3	yes	—
102	C₂×C₄	C₂×C₁₆×C₆₁	1	6	yes	—
103	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈×C₆₁	1	6	yes	—
105	C₈	C₂²×C₈×C₆₁	1	1	yes	characteristic
106	C₈	C₂²×C₈×C₆₁	1	6	yes	—
107	C₈	C₂³×C₄×C₆₁	1	1	yes	characteristic
108	C₂²	—	2	6	no	—
109	C₂²	C₂²×C₁₆×C₆₁	1	6	yes	—
110	C₂²	C₂²×C₁₆×C₆₁	1	1	yes	characteristic
111	C₄	C₂²×C₁₆×C₆₁	1	1	yes	characteristic
112	C₄	C₂²×C₁₆×C₆₁	1	6	yes	—
113	C₄	C₂³×C₈×C₆₁	1	1	yes	characteristic
114	C₂	—	2	6	no	—
115	C₂	C₆₁×#2258	1	2	yes	direct product factor
116	C₂	C₂³×C₁₆×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₆₁	C₂²
commutator	C₂	C₂³×C₁₆×C₆₁
Frattini, cosocle	C₁₆	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×C₁₆
prime radical	C₂×#41×C₆₁	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2258	C₂	direct (N × H)
3. C₂×C₆₁×M₆₄	C₂	semidirect
4. C₂²×C₃₂×C₆₁	C₂	semidirect
10. C₆₁×M₆₄	C₂²	semidirect
13. C₂×C₃₂×C₆₁	C₂²	semidirect
49. C₂×#2258	C₆₁	direct (N × H)
57. #2258	C₂×C₆₁	direct (N × H)
58. C₂×M₆₄	C₂×C₆₁	semidirect
59. C₂²×C₃₂	C₂×C₆₁	semidirect
61. C₂×C₆₁	#2258	direct (N × H)
68. M₆₄	C₂²×C₆₁	semidirect
71. C₂×C₃₂	C₂²×C₆₁	semidirect
72. C₆₁	C₂×#2258	direct (N × H)
115. C₂	C₆₁×#2258	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#504	2
C₆₁×#2258	2
C₂×C₆₁×M₆₄	2
C₂²×C₃₂×C₆₁	2
C₂×#2258	61

Subgroups — C2×C61×#2258 #29537406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#2258 #29537406