Subgroups — C₆₁×#7475 #29510908

Order 15616.

170 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	C₆₁×#2158	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
4	C₆₁×#427	—	2	2	no	—
5	C₆₁×M₆₄	—	2	2	no	—
6	C₂×C₃₂×C₆₁	C₄	1	2	yes	complemented
7	C₆₁×#367	C₄	1	4	yes	complemented
8	C₆₁×#366	—	2	2	no	—
9	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
10	#117×C₆₁	—	4	2	no	—
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	#107×C₆₁	—	2	4	no	—
16	#106×C₆₁	—	4	2	no	—
17	C₃₂×C₆₁	—	2	2	no	—
18	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
19	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
20	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
22	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
23	C₄⋊C₄×C₆₁	—	4	4	no	—
24	C₂²⋊C₄×C₆₁	—	8	2	no	—
25	C₄²×C₆₁	—	8	2	no	—
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	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₆₁	—	8	4	no	—
31	C₂×C₄×C₆₁	#102	1	2	yes	—
32	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
33	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
34	C₂×C₄×C₆₁	Q₃₂	1	1	yes	characteristic
35	C₈×C₆₁	#95	1	1	yes	characteristic
36	C₈×C₆₁	#103	1	2	yes	—
37	C₈×C₆₁	#106	1	1	yes	characteristic
38	#7475	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₆₁	—	4	1	no	—
40	C₂²×C₆₁	#357	1	2	yes	—
41	C₂²×C₆₁	#366	1	1	yes	characteristic
42	C₄×C₆₁	—	16	4	no	—
43	C₄×C₆₁	#340	1	1	yes	characteristic
44	C₄×C₆₁	#358	1	2	yes	—
45	C₄×C₆₁	#367	1	1	yes	characteristic
46	C₂×M₆₄	C₂×C₆₁	1	1	yes	characteristic
47	#2158	C₂×C₆₁	1	2	yes	—
48	C₂×C₆₁	—	2	2	no	—
49	C₂×C₆₁	—	2	1	no	—
50	C₂×C₆₁	#1380	1	1	yes	characteristic
51	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
52	#427	—	2	2	no	—
53	M₆₄	—	2	2	no	—
54	C₂×C₃₂	C₄×C₆₁	1	2	yes	complemented
55	#367	C₄×C₆₁	1	4	yes	complemented
56	#366	—	2	2	no	—
57	C₆₁	#7475	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
59	#117	—	4	2	no	—
60	C₂×C₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
61	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
62	C₂×C₁₆	D₄×C₆₁	1	2	yes	—
63	C₂×C₁₆	Q₈×C₆₁	1	1	yes	characteristic
64	#107	—	2	4	no	—
65	#106	—	4	2	no	—
66	C₃₂	—	2	2	no	—
67	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
69	C₂×C₈	C₄⋊C₄×C₆₁	1	1	yes	characteristic
70	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
71	C₂×C₈	Q₁₆×C₆₁	1	1	yes	characteristic
72	C₄⋊C₄	—	4	4	no	—
73	C₂²⋊C₄	—	8	2	no	—
74	C₄²	—	8	2	no	—
75	C₁₆	C₄²×C₆₁	1	1	yes	characteristic
76	C₁₆	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
77	C₁₆	C₄⋊C₄×C₆₁	1	2	yes	—
78	C₂³	D₁₆×C₆₁	1	1	yes	characteristic
79	C₂×C₄	—	8	4	no	—
80	C₂×C₄	#102×C₆₁	1	2	yes	—
81	C₂×C₄	#107×C₆₁	1	1	yes	characteristic
82	C₂×C₄	SD₃₂×C₆₁	1	2	yes	—
83	C₂×C₄	Q₃₂×C₆₁	1	1	yes	characteristic
84	C₈	#95×C₆₁	1	1	yes	characteristic
85	C₈	#103×C₆₁	1	2	yes	—
86	C₈	#106×C₆₁	1	1	yes	characteristic
87	C₂²	—	4	1	no	—
88	C₂²	C₆₁×#357	1	2	yes	—
89	C₂²	C₆₁×#366	1	1	yes	characteristic
90	C₄	—	16	4	no	—
91	C₄	C₆₁×#340	1	1	yes	characteristic
92	C₄	C₆₁×#358	1	2	yes	—
93	C₄	C₆₁×#367	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	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	D₁₆
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
6. C₂×C₃₂×C₆₁	C₄	semidirect
7. C₆₁×#367	C₄	semidirect
38. #7475	C₆₁	direct (N × H)
54. C₂×C₃₂	C₄×C₆₁	semidirect
55. #367	C₄×C₆₁	semidirect
57. C₆₁	#7475	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
C₆₁×#2158	2
#7475	61

Subgroups — C61×#7475 #29510908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7475 #29510908