Subgroups — #30599522

Order 15616.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461755	C₂	1	6	yes	maximal
2	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519846	—	2	12	no	—
4	#16840708	C₂²	1	4	yes	—
5	C₄×C₈×D₆₁	C₂²	1	3	yes	—
6	C₈×#92418	C₄	1	3	yes	complemented
7	C₈²×C₆₁	C₄	1	1	yes	characteristic, complemented
8	C₂×C₄⋊#6923	—	2	12	no	—
9	#14519846	—	4	24	no	—
10	C₂×C₈×D₆₁	D₄	1	6	yes	—
11	C₄²×D₆₁	C₂³	1	1	yes	characteristic
12	C₄×#92418	C₂×C₄	1	3	yes	—
13	C₈×Dic₆₁	Q₈	1	6	yes	—
14	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
15	C₂²×#6923	—	4	4	no	—
16	C₂×C₄×D₆₁	C₂×D₄	1	3	yes	—
17	C₄⋊#6923	—	4	24	no	—
18	C₂×C₈×C₆₁	C₄⋊C₄	1	6	yes	—
19	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
20	C₄×Dic₆₁	C₂×Q₈	1	3	yes	—
21	C₂×#92418	C₄⋊C₄	1	6	yes	—
22	C₈×D₆₁	D₈	1	12	yes	—
23	C₈×D₆₁	Q₁₆	1	12	yes	—
24	C₂²×D₆₁	#127	1	1	yes	characteristic
25	C₂×#6923	—	8	12	no	—
26	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
27	C₂×Dic₆₁	C₄⋊Q₈	1	3	yes	—
28	C₄×D₆₁	C₂×D₈	1	6	yes	—
29	C₄×D₆₁	C₂×Q₁₆	1	6	yes	—
30	C₈×C₆₁	#107	1	12	yes	complemented
31	#92418	#107	1	12	yes	complemented
32	#8818	—	61	1	no	maximal, Sylow
33	C₂²×C₆₁	#384	1	1	yes	characteristic
34	C₂×D₆₁	#493	1	3	yes	—
35	C₂×D₆₁	#494	1	3	yes	—
36	#6923	—	16	4	no	—
37	C₄×C₆₁	C₂×#107	1	6	yes	—
38	Dic₆₁	#500	1	6	yes	—
39	#1845	—	61	6	no	—
40	C₂×C₈²	—	61	1	no	—
41	C₂×C₆₁	#1845	1	3	yes	—
42	D₆₁	#1712	1	1	yes	characteristic
43	D₆₁	#1713	1	3	yes	—
44	C₂×#107	—	122	12	no	—
45	C₂×C₄×C₈	—	61	3	no	—
46	#384	—	61	4	no	—
47	C₈²	—	61	3	no	—
48	C₈²	#6923	1	1	yes	characteristic, complemented
49	C₆₁	#8818	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₈	—	61	6	no	—
51	C₂×C₄⋊C₄	—	122	12	no	—
52	C₂×C₄²	—	61	1	no	—
53	#107	—	244	24	no	—
54	C₄×C₈	—	61	6	no	—
55	C₄×C₈	—	61	3	no	—
56	C₄×C₈	C₂×#6923	1	3	yes	—
57	C₂²×C₄	—	61	3	no	—
58	C₂²×C₄	—	244	4	no	—
59	C₂×C₈	—	61	6	no	—
60	C₂×C₈	—	61	12	no	—
61	C₂×C₈	—	61	12	no	—
62	C₂×C₈	C₄⋊#6923	1	6	yes	—
63	C₄⋊C₄	—	244	24	no	—
64	C₄²	—	61	3	no	—
65	C₄²	C₂²×#6923	1	1	yes	characteristic
66	C₂³	—	61	1	no	—
67	C₂×C₄	—	61	3	no	—
68	C₂×C₄	—	61	6	no	—
69	C₂×C₄	—	61	6	no	—
70	C₂×C₄	—	488	12	no	—
71	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
72	C₈	—	61	12	no	—
73	C₈	#14519846	1	12	yes	complemented
74	C₂²	—	61	3	no	—
75	C₂²	—	61	3	no	—
76	C₂²	#16840708	1	1	yes	characteristic
77	C₄	—	61	6	no	—
78	C₄	—	976	4	no	—
79	C₄	C₂×#14519846	1	6	yes	—
80	C₂	—	61	1	no	—
81	C₂	—	61	3	no	—
82	C₂	#21461755	1	3	yes	—

	subgroup	G/H
center	C₂²	#16840708
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	#384
prime radical	C₂²×D₆₁	#127
cosocle	C₄²×D₆₁	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#8818
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
6. C₈×#92418	C₄	semidirect
7. C₈²×C₆₁	C₄	semidirect
30. C₈×C₆₁	#107	semidirect
31. #92418	#107	semidirect
48. C₈²	#6923	semidirect
49. C₆₁	#8818	semidirect
73. C₈	#14519846	semidirect

Maximal subgroups

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

subgroup	index
#21461755	2
C₈²×D₆₁	2
#8818	61