Subgroups — C₆₁×#7456 #29510889

Order 15616.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#1422	C₂	1	2	yes	maximal
2	C₆₁×#1311	C₂	1	1	yes	maximal, characteristic
3	C₆₁×M₆₄	—	2	2	no	—
4	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
6	C₈²×C₆₁	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₆₁	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₆₁	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
10	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₆₁	—	2	1	no	—
12	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₆₁	—	4	2	no	—
14	C₂×C₈×C₆₁	—	2	2	no	—
15	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
18	C₁₆×C₆₁	—	2	1	no	—
19	C₁₆×C₆₁	—	2	1	no	—
20	C₁₆×C₆₁	C₁₆	1	2	yes	—
21	C₁₆×C₆₁	SD₁₆	1	2	yes	—
22	C₂×C₄×C₆₁	—	2	1	no	—
23	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₆₁	—	4	2	no	—
25	C₈×C₆₁	#106	1	1	yes	characteristic
26	C₈×C₆₁	#108	1	1	yes	characteristic
27	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₆₁	M₃₂	1	1	yes	characteristic
29	#7456	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₆₁	#364	1	1	yes	characteristic
31	C₄×C₆₁	—	2	2	no	—
32	C₄×C₆₁	#334	1	1	yes	characteristic
33	C₄×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
34	#1422	C₂×C₆₁	1	2	yes	—
35	#1311	C₂×C₆₁	1	1	yes	characteristic
36	C₂×C₆₁	—	2	1	no	—
37	C₂×C₆₁	#1367	1	1	yes	characteristic
38	M₆₄	—	2	2	no	—
39	C₄×C₁₆	C₄×C₆₁	1	1	yes	characteristic
40	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
41	C₈²	C₄×C₆₁	1	1	yes	characteristic
42	C₆₁	#7456	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₁₆	C₈×C₆₁	1	1	yes	characteristic
44	C₂×C₁₆	C₈×C₆₁	1	1	yes	characteristic
45	C₂×C₁₆	D₄×C₆₁	1	1	yes	characteristic
46	C₂×C₁₆	Q₈×C₆₁	1	1	yes	characteristic
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
49	C₃₂	—	4	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
52	C₂×C₈	C₂×C₈×C₆₁	1	1	yes	characteristic
53	C₄²	M₁₆×C₆₁	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	2	1	no	—
56	C₁₆	C₁₆×C₆₁	1	2	yes	—
57	C₁₆	SD₁₆×C₆₁	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊C₈×C₆₁	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	#106×C₆₁	1	1	yes	characteristic
62	C₈	#108×C₆₁	1	1	yes	characteristic
63	C₈	C₂×C₁₆×C₆₁	1	1	yes	characteristic
64	C₈	M₃₂×C₆₁	1	1	yes	characteristic
65	C₂²	C₆₁×#364	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	C₆₁×#334	1	1	yes	characteristic
68	C₄	C₆₁×C₄⋊C₁₆	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	C₆₁×#1367	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₁	#106
commutator	C₈	C₂×C₁₆×C₆₁
Frattini, cosocle	C₄×C₁₆	C₂²×C₆₁
socle	C₂×C₆₁	#1367
prime radical	C₂²×C₆₁	#364
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₁ ◁ C₂×C₁₆×C₆₁ ◁ C₄×C₁₆×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
29. #7456	C₆₁	direct (N × H)
42. C₆₁	#7456	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×#1422	2
C₆₁×#1311	2
#7456	61

Subgroups — C61×#7456 #29510889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7456 #29510889