Subgroups — #30595309

Order 15616.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460980	C₂	1	2	yes	maximal
2	C₆₄×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#16840688	—	2	2	no	—
4	C₃₂×D₆₁	C₂²	1	1	yes	characteristic
5	#16839481	C₄	1	1	yes	characteristic, complemented
6	C₆₁×C₆₄	C₄	1	1	yes	characteristic, complemented
7	#14519846	—	4	2	no	—
8	C₁₆×D₆₁	D₄	1	1	yes	characteristic
9	#14519683	Q₈	1	1	yes	characteristic
10	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
11	C₄⋊#6923	—	8	2	no	—
12	C₈×D₆₁	D₈	1	1	yes	characteristic
13	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
14	#11758185	Q₁₆	1	1	yes	characteristic
15	C₂×#6923	—	16	2	no	—
16	C₄×D₆₁	D₁₆	1	1	yes	characteristic
17	C₈×C₆₁	#107	1	1	yes	characteristic
18	#92418	Q₃₂	1	1	yes	characteristic
19	#7545	—	61	1	no	maximal, Sylow
20	C₂×D₆₁	D₃₂	1	1	yes	characteristic
21	#6923	—	32	2	no	—
22	C₄×C₆₁	#366	1	1	yes	characteristic
23	Dic₆₁	Q₆₄	1	1	yes	characteristic
24	C₂×C₆₄	—	61	1	no	—
25	#1423	—	61	2	no	—
26	C₂×C₆₁	#1423	1	1	yes	characteristic
27	D₆₁	D₆₄	1	1	yes	characteristic
28	D₆₁	Q₁₂₈	1	1	yes	characteristic
29	C₂×C₃₂	—	61	1	no	—
30	#366	—	122	2	no	—
31	C₆₄	—	61	1	no	—
32	C₆₄	#6923	1	1	yes	characteristic, complemented
33	C₆₁	#7545	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₁₆	—	61	1	no	—
35	#107	—	244	2	no	—
36	C₃₂	—	61	1	no	—
37	C₃₂	C₂×#6923	1	1	yes	characteristic
38	C₂×C₈	—	61	1	no	—
39	C₄⋊C₄	—	488	2	no	—
40	C₁₆	—	61	1	no	—
41	C₁₆	C₄⋊#6923	1	1	yes	characteristic
42	C₂×C₄	—	61	1	no	—
43	C₂×C₄	—	976	2	no	—
44	C₈	—	61	1	no	—
45	C₈	#14519846	1	1	yes	characteristic
46	C₂²	—	61	1	no	—
47	C₄	—	61	1	no	—
48	C₄	—	1952	2	no	—
49	C₄	#16840688	1	1	yes	characteristic
50	C₂	—	61	1	no	—
51	C₂	—	61	1	no	—
52	C₂	#21460980	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21460980
commutator	C₃₂×C₆₁	C₂×C₄
Frattini	C₃₂	C₂×#6923
Fitting	C₆₁×C₆₄	C₄
socle	C₂×C₆₁	#1423
prime radical	C₂×D₆₁	D₃₂
cosocle	C₃₂×D₆₁	C₂²
hypercenter	C₆₄	#6923
hypocenter	C₆₁	#7545
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
5. #16839481	C₄	semidirect
6. C₆₁×C₆₄	C₄	semidirect
32. C₆₄	#6923	semidirect
33. C₆₁	#7545	semidirect

Maximal subgroups

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

subgroup	index
#21460980	2
C₆₄×D₆₁	2
#7545	61