Subgroups — #10683085

Order 768.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
2	#67587	C₂	1	2	yes	maximal
3	#7544	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₁₆	C₄	1	1	yes	characteristic
7	C₃⋊C₆₄	—	2	2	no	—
8	M₁₂₈	—	3	2	no	—
9	C₄×C₃₂	D₃	1	1	yes	characteristic
10	C₂×C₃×C₁₆	C₈	1	1	yes	characteristic
11	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
12	C₃×C₄×C₈	C₈	1	1	yes	characteristic
13	C₃×C₃₂	—	2	1	no	—
14	C₃×C₃₂	D₄	1	1	yes	characteristic
15	C₃×C₃₂	Q₈	1	1	yes	characteristic
16	C₂×C₃₂	Dic₃	1	1	yes	characteristic
17	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
18	C₄×C₁₆	Dic₃	1	1	yes	characteristic
19	C₆₄	—	6	2	no	—
20	C₂×C₃×C₈	C₁₆	1	1	yes	characteristic
21	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
22	C₃×C₄²	C₁₆	1	1	yes	characteristic
23	C₃×C₁₆	—	2	1	no	—
24	C₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
25	C₃×C₁₆	M₁₆	1	1	yes	characteristic
26	C₂×C₁₆	C₃⋊C₈	1	1	yes	characteristic
27	C₂×C₁₆	C₂×Dic₃	1	1	yes	characteristic
28	C₄×C₈	C₃⋊C₈	1	1	yes	characteristic
29	C₃₂	—	2	1	no	—
30	C₃₂	Dic₆	1	1	yes	characteristic
31	C₃₂	D₁₂	1	1	yes	characteristic
32	C₂×C₃×C₄	—	2	1	no	—
33	C₂×C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
34	C₃×C₈	C₃₂	1	2	yes	—
35	C₃×C₈	C₄⋊C₈	1	1	yes	characteristic
36	C₃×C₈	M₃₂	1	1	yes	characteristic
37	C₂×C₈	C₃⋊C₁₆	1	1	yes	characteristic
38	C₂×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
39	C₄²	C₃⋊C₁₆	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	#208	1	1	yes	characteristic
42	C₁₆	C₄⋊Dic₃	1	1	yes	characteristic
43	C₂²×C₃	M₆₄	1	1	yes	characteristic
44	C₃×C₄	—	2	2	no	—
45	C₃×C₄	C₄⋊C₁₆	1	1	yes	characteristic
46	C₃×C₄	C₂×C₃₂	1	1	yes	characteristic
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	C₂×C₃⋊C₁₆	1	1	yes	characteristic
49	C₈	C₃⋊C₃₂	1	2	yes	—
50	C₈	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
51	C₈	#812	1	1	yes	characteristic
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	C₄⋊C₃₂	1	1	yes	characteristic
54	C₂²	#4499	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	C₄⋊(C₃⋊C₁₆)	1	1	yes	characteristic
57	C₄	C₂×C₃⋊C₃₂	1	1	yes	characteristic
58	C₃	#7544	1	1	yes	characteristic, complemented, Sylow
59	C₂	—	2	1	no	—
60	C₂	C₄⋊(C₃⋊C₃₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	D₁₂
commutator	C₃×C₄	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₃
Fitting	C₃×C₄×C₃₂	C₂
socle	C₂×C₃	C₄⋊C₃₂
prime radical	C₂²×C₃	M₆₄
cosocle	C₂×C₃×C₃₂	C₂²
hypercenter	C₄×C₃₂	D₃
hypocenter	C₃	#7544
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
58. C₃	#7544	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₃₂	2
#67587	2
#7544	3