Subgroups — C₆₁×#7476 #29510909

Order 15616.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#368	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
4	C₂×#108×C₆₁	—	2	2	no	—
5	C₆₁×M₆₄	—	2	2	no	—
6	C₂×C₃₂×C₆₁	C₄	1	2	yes	—
7	C₆₁×#368	—	2	2	no	—
8	C₆₁×#368	C₄	1	4	yes	—
9	C₂×M₁₆×C₆₁	—	4	2	no	—
10	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
13	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
14	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
15	#108×C₆₁	—	2	4	no	—
16	#108×C₆₁	—	4	2	no	—
17	C₃₂×C₆₁	—	2	2	no	—
18	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
19	M₁₆×C₆₁	—	4	4	no	—
20	M₁₆×C₆₁	—	8	2	no	—
21	C₂×C₈×C₆₁	—	8	2	no	—
22	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
25	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
26	C₁₆×C₆₁	C₄²	1	1	yes	characteristic
27	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
29	C₂³×C₆₁	Q₃₂	1	1	yes	characteristic
30	C₂×C₄×C₆₁	#102	1	2	yes	—
31	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
32	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
34	C₈×C₆₁	—	8	4	no	—
35	C₈×C₆₁	#95	1	1	yes	characteristic
36	C₈×C₆₁	#103	1	2	yes	—
37	C₈×C₆₁	#106	1	1	yes	characteristic
38	#7476	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₆₁	—	4	1	no	—
40	C₂²×C₆₁	#358	1	2	yes	—
41	C₂²×C₆₁	#366	1	1	yes	characteristic
42	C₄×C₆₁	#340	1	1	yes	characteristic
43	C₄×C₆₁	#357	1	2	yes	—
44	C₄×C₆₁	#367	1	1	yes	characteristic
45	C₂×M₆₄	C₂×C₆₁	1	1	yes	characteristic
46	C₂×#368	C₂×C₆₁	1	2	yes	—
47	C₂×C₆₁	—	2	2	no	—
48	C₂×C₆₁	—	2	1	no	—
49	C₂×C₆₁	#1380	1	1	yes	characteristic
50	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
51	C₂×#108	—	2	2	no	—
52	M₆₄	—	2	2	no	—
53	C₂×C₃₂	C₄×C₆₁	1	2	yes	—
54	#368	—	2	2	no	—
55	#368	C₄×C₆₁	1	4	yes	—
56	C₆₁	#7476	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×M₁₆	—	4	2	no	—
58	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
59	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
60	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
61	C₂×C₁₆	D₄×C₆₁	1	2	yes	—
62	C₂×C₁₆	Q₈×C₆₁	1	1	yes	characteristic
63	#108	—	2	4	no	—
64	#108	—	4	2	no	—
65	C₃₂	—	2	2	no	—
66	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
67	M₁₆	—	4	4	no	—
68	M₁₆	—	8	2	no	—
69	C₂×C₈	—	8	2	no	—
70	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
71	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
72	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
73	C₂×C₈	Q₁₆×C₆₁	1	1	yes	characteristic
74	C₁₆	C₄²×C₆₁	1	1	yes	characteristic
75	C₁₆	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
76	C₁₆	C₄⋊C₄×C₆₁	1	2	yes	—
77	C₂³	Q₃₂×C₆₁	1	1	yes	characteristic
78	C₂×C₄	#102×C₆₁	1	2	yes	—
79	C₂×C₄	#107×C₆₁	1	1	yes	characteristic
80	C₂×C₄	D₁₆×C₆₁	1	1	yes	characteristic
81	C₂×C₄	SD₃₂×C₆₁	1	2	yes	—
82	C₈	—	8	4	no	—
83	C₈	#95×C₆₁	1	1	yes	characteristic
84	C₈	#103×C₆₁	1	2	yes	—
85	C₈	#106×C₆₁	1	1	yes	characteristic
86	C₂²	—	4	1	no	—
87	C₂²	C₆₁×#358	1	2	yes	—
88	C₂²	C₆₁×#366	1	1	yes	characteristic
89	C₄	C₆₁×#340	1	1	yes	characteristic
90	C₄	C₆₁×#357	1	2	yes	—
91	C₄	C₆₁×#367	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	—	2	1	no	—
94	C₂	C₆₁×#1380	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₁	#367
commutator	C₁₆	C₄²×C₆₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₆₁
socle	C₂×C₆₁	#1380
prime radical	C₂³×C₆₁	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ C₂²×C₁₆×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
38. #7476	C₆₁	direct (N × H)
56. C₆₁	#7476	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×M₆₄	2
C₂×C₆₁×#368	2
#7476	61

Subgroups — C61×#7476 #29510909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7476 #29510909