Subgroups — #10683097

Order 768.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	M₂₅₆	—	3	1	no	maximal, Sylow
5	D₃×C₃₂	C₄	1	1	yes	characteristic
6	C₃×C₆₄	C₂²	1	1	yes	characteristic
7	C₃⋊C₆₄	C₄	1	1	yes	characteristic
8	C₂×C₆₄	—	3	1	no	—
9	C₁₂₈	—	3	1	no	—
10	C₁₂₈	D₃	1	1	yes	characteristic, complemented
11	D₃×C₁₆	C₈	1	1	yes	characteristic
12	C₃×C₃₂	C₂×C₄	1	1	yes	characteristic
13	C₃⋊C₃₂	C₈	1	1	yes	characteristic
14	C₂×C₃₂	—	3	1	no	—
15	C₆₄	—	3	1	no	—
16	C₆₄	C₂×D₃	1	1	yes	characteristic
17	D₃×C₈	C₁₆	1	1	yes	characteristic
18	C₃×C₁₆	C₂×C₈	1	1	yes	characteristic
19	C₃⋊C₁₆	C₁₆	1	1	yes	characteristic
20	C₂×C₁₆	—	3	1	no	—
21	C₃₂	—	3	1	no	—
22	C₃₂	C₄×D₃	1	1	yes	characteristic
23	C₄×D₃	C₃₂	1	1	yes	characteristic
24	C₃×C₈	C₂×C₁₆	1	1	yes	characteristic
25	C₃⋊C₈	C₃₂	1	1	yes	characteristic
26	C₂×C₈	—	3	1	no	—
27	C₁₆	—	3	1	no	—
28	C₁₆	D₃×C₈	1	1	yes	characteristic
29	C₂×D₃	C₆₄	1	1	yes	characteristic
30	C₃×C₄	C₂×C₃₂	1	1	yes	characteristic
31	Dic₃	C₆₄	1	1	yes	characteristic
32	C₂×C₄	—	3	1	no	—
33	C₈	—	3	1	no	—
34	C₈	D₃×C₁₆	1	1	yes	characteristic
35	C₂×C₃	C₂×C₆₄	1	1	yes	characteristic
36	D₃	—	2	1	no	—
37	C₂²	—	3	1	no	—
38	C₄	—	3	1	no	—
39	C₄	D₃×C₃₂	1	1	yes	characteristic
40	C₃	M₂₅₆	1	1	yes	characteristic, complemented, Sylow
41	C₂	—	6	1	no	—
42	C₂	D₃×C₆₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₃
commutator, socle	C₂×C₃	C₂×C₆₄
Fitting	C₃×C₁₂₈	C₂
prime radical	C₂×D₃	C₆₄
cosocle	C₃×C₆₄	C₂²
hypercenter	C₁₂₈	D₃
hypocenter	C₃	M₂₅₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₆₄ ◁ C₁₂₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₁₂₈	C₂	semidirect
3. C₃⋊C₁₂₈	C₂	semidirect
10. C₁₂₈	D₃	semidirect
40. C₃	M₂₅₆	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₆₄	2
C₃×C₁₂₈	2
C₃⋊C₁₂₈	2
M₂₅₆	3