Subgroups — #30599517

Order 15616.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461754	C₂	1	6	yes	maximal
2	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519845	—	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	#14519845	—	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	SD₁₆	1	24	yes	—
23	C₂²×D₆₁	#127	1	1	yes	characteristic
24	C₂×#6923	—	8	12	no	—
25	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
26	C₂×Dic₆₁	C₄⋊Q₈	1	3	yes	—
27	C₄×D₆₁	C₂×SD₁₆	1	12	yes	—
28	C₈×C₆₁	#106	1	12	yes	complemented
29	#92418	#106	1	12	yes	complemented
30	#8813	—	61	1	no	maximal, Sylow
31	C₂²×C₆₁	#384	1	1	yes	characteristic
32	C₂×D₆₁	#492	1	6	yes	—
33	#6923	—	16	4	no	—
34	C₄×C₆₁	C₂×#106	1	6	yes	—
35	Dic₆₁	#498	1	6	yes	—
36	#1844	—	61	6	no	—
37	C₂×C₈²	—	61	1	no	—
38	C₂×C₆₁	#1844	1	3	yes	—
39	D₆₁	#1705	1	4	yes	—
40	C₂×#106	—	122	12	no	—
41	C₂×C₄×C₈	—	61	3	no	—
42	#384	—	61	4	no	—
43	C₈²	—	61	3	no	—
44	C₈²	#6923	1	1	yes	characteristic, complemented
45	C₆₁	#8813	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₈	—	61	6	no	—
47	C₂×C₄⋊C₄	—	122	12	no	—
48	C₂×C₄²	—	61	1	no	—
49	#106	—	244	24	no	—
50	C₄×C₈	—	61	6	no	—
51	C₄×C₈	—	61	3	no	—
52	C₄×C₈	C₂×#6923	1	3	yes	—
53	C₂²×C₄	—	61	3	no	—
54	C₂²×C₄	—	244	4	no	—
55	C₂×C₈	—	61	6	no	—
56	C₂×C₈	—	61	24	no	—
57	C₂×C₈	C₄⋊#6923	1	6	yes	—
58	C₄⋊C₄	—	244	24	no	—
59	C₄²	—	61	3	no	—
60	C₄²	C₂²×#6923	1	1	yes	characteristic
61	C₂³	—	61	1	no	—
62	C₂×C₄	—	61	3	no	—
63	C₂×C₄	—	61	12	no	—
64	C₂×C₄	—	488	12	no	—
65	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
66	C₈	—	61	12	no	—
67	C₈	#14519845	1	12	yes	complemented
68	C₂²	—	61	6	no	—
69	C₂²	#16840708	1	1	yes	characteristic
70	C₄	—	61	6	no	—
71	C₄	—	976	4	no	—
72	C₄	C₂×#14519845	1	6	yes	—
73	C₂	—	61	4	no	—
74	C₂	#21461754	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	#8813
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
28. C₈×C₆₁	#106	semidirect
29. #92418	#106	semidirect
44. C₈²	#6923	semidirect
45. C₆₁	#8813	semidirect
67. C₈	#14519845	semidirect

Maximal subgroups

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

subgroup	index
#21461754	2
C₈²×D₆₁	2
#8813	61