Subgroups — #30595313

Order 15616.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460983	C₂	1	1	yes	maximal, characteristic
2	#21460981	C₂	1	1	yes	maximal, characteristic
3	#21444029	C₂	1	1	yes	maximal, characteristic
4	#16840690	—	2	1	no	—
5	#16840688	—	2	1	no	—
6	C₃₂×D₆₁	C₂²	1	1	yes	characteristic
7	#16839481	C₄	1	1	yes	characteristic, complemented
8	C₆₁×C₆₄	C₄	1	1	yes	characteristic, complemented
9	#14519848	—	4	1	no	—
10	#14519846	—	4	1	no	—
11	C₁₆×D₆₁	D₄	1	1	yes	characteristic
12	#14519683	Q₈	1	1	yes	characteristic
13	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₄⋊#6923	—	8	1	no	—
15	#11758213	—	8	1	no	—
16	C₈×D₆₁	D₈	1	1	yes	characteristic
17	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
18	#11758185	Q₁₆	1	1	yes	characteristic
19	C₂×#6923	—	16	1	no	—
20	C₄×D₆₁	D₁₆	1	1	yes	characteristic
21	#92420	—	16	1	no	—
22	C₈×C₆₁	#107	1	1	yes	characteristic
23	#92418	Q₃₂	1	1	yes	characteristic
24	#7548	—	61	1	no	maximal, Sylow
25	C₂×D₆₁	SD₆₄	1	1	yes	characteristic
26	#6923	—	32	1	no	—
27	C₄×C₆₁	#366	1	1	yes	characteristic
28	Dic₆₁	SD₆₄	1	1	yes	characteristic
29	M₁₂₈	—	61	1	no	—
30	#1425	—	61	1	no	—
31	#1424	—	61	1	no	—
32	C₂×C₆₁	#1424	1	1	yes	characteristic
33	D₆₁	—	2	1	no	—
34	C₂×C₃₂	—	61	1	no	—
35	#368	—	122	1	no	—
36	#366	—	122	1	no	—
37	C₆₄	—	61	1	no	—
38	C₆₄	#6923	1	1	yes	characteristic, complemented
39	C₆₁	#7548	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₁₆	—	61	1	no	—
41	#108	—	244	1	no	—
42	#107	—	244	1	no	—
43	C₃₂	—	61	1	no	—
44	C₃₂	C₂×#6923	1	1	yes	characteristic
45	M₁₆	—	488	1	no	—
46	C₂×C₈	—	61	1	no	—
47	C₄⋊C₄	—	488	1	no	—
48	C₁₆	—	61	1	no	—
49	C₁₆	C₄⋊#6923	1	1	yes	characteristic
50	C₂×C₄	—	61	1	no	—
51	C₂×C₄	—	976	1	no	—
52	C₈	—	61	1	no	—
53	C₈	—	976	1	no	—
54	C₈	#14519846	1	1	yes	characteristic
55	C₂²	—	61	1	no	—
56	C₄	—	61	1	no	—
57	C₄	—	1952	1	no	—
58	C₄	#16840688	1	1	yes	characteristic
59	C₂	—	122	1	no	—
60	C₂	#21460981	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21460981
commutator	C₃₂×C₆₁	C₂×C₄
Frattini	C₃₂	C₂×#6923
Fitting	C₆₁×C₆₄	C₄
socle	C₂×C₆₁	#1424
prime radical	C₂×D₆₁	SD₆₄
cosocle	C₃₂×D₆₁	C₂²
hypercenter	C₆₄	#6923
hypocenter	C₆₁	#7548
LCS	G ▷ C₃₂×C₆₁ ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

Complements

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

N	H ≅ G/N	type
7. #16839481	C₄	semidirect
8. C₆₁×C₆₄	C₄	semidirect
38. C₆₄	#6923	semidirect
39. C₆₁	#7548	semidirect

Maximal subgroups

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

subgroup	index
#21460983	2
#21460981	2
#21444029	2
#7548	61