Subgroups — #27133869

Order 12256.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259457	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₅₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259446	C₂	1	2	yes	maximal, complemented
4	D₃₀₆₄	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₈₃	—	2	1	no	—
7	Dic₇₆₆	C₂²	1	1	yes	characteristic, complemented
8	D₁₅₃₂	—	2	1	no	—
9	D₁₅₃₂	C₂²	1	2	yes	—
10	D₁₅₃₂	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₃₈₃	—	2	1	no	—
12	C₄×D₃₈₃	—	2	1	no	—
13	C₂×C₄×C₃₈₃	C₂²	1	1	yes	characteristic
14	C₈×C₃₈₃	C₂²	1	2	yes	complemented
15	C₂²×C₃₈₃	D₄	1	1	yes	characteristic
16	C₂×D₃₈₃	—	2	1	no	—
17	C₂×D₃₈₃	—	2	2	no	—
18	C₂×D₃₈₃	—	4	2	no	—
19	C₄×C₃₈₃	D₄	1	1	yes	characteristic
20	C₄×C₃₈₃	C₂³	1	1	yes	characteristic
21	Dic₃₈₃	—	2	1	no	—
22	C₂×C₃₈₃	—	2	1	no	—
23	C₂×C₃₈₃	C₂×D₄	1	1	yes	characteristic
24	D₃₈₃	—	4	1	no	—
25	D₃₈₃	—	4	2	no	—
26	C₃₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	383	1	no	maximal, Sylow
28	#41	—	383	1	no	—
29	C₂×D₄	—	383	1	no	—
30	SD₁₆	—	383	2	no	—
31	D₈	—	383	2	no	—
32	M₁₆	D₃₈₃	1	1	yes	characteristic, complemented
33	C₂³	—	766	1	no	—
34	Q₈	—	383	1	no	—
35	D₄	—	383	1	no	—
36	D₄	—	383	2	no	—
37	D₄	—	766	1	no	—
38	D₄	—	766	1	no	—
39	C₂×C₄	—	766	1	no	—
40	C₂×C₄	C₂×D₃₈₃	1	1	yes	characteristic
41	C₈	C₂×D₃₈₃	1	2	yes	complemented
42	C₂²	—	1532	2	no	—
43	C₂²	—	766	1	no	—
44	C₂²	—	766	2	no	—
45	C₂²	D₁₅₃₂	1	1	yes	characteristic
46	C₄	—	766	1	no	—
47	C₄	D₁₅₃₂	1	1	yes	characteristic
48	C₄	C₂²×D₃₈₃	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	1532	2	no	—
51	C₂	—	1532	1	no	—
52	C₂	C₂×D₁₅₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₅₃₂
commutator, cosocle	C₄×C₃₈₃	C₂³
Frattini	C₄	C₂²×D₃₈₃
Fitting	M₁₆×C₃₈₃	C₂
socle	C₂×C₃₈₃	C₂×D₄
hypercenter	M₁₆	D₃₈₃
hypocenter	C₃₈₃	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₃₈₃ ▷ C₂×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #20259457	C₂	semidirect
2. C₂×D₁₅₃₂	C₂	semidirect
3. #20259446	C₂	semidirect
4. D₃₀₆₄	C₂	semidirect
5. M₁₆×C₃₈₃	C₂	semidirect
7. Dic₇₆₆	C₂²	semidirect
10. D₁₅₃₂	C₂²	semidirect
14. C₈×C₃₈₃	C₂²	semidirect
26. C₃₈₃	AGL(1,ℤ/8)	semidirect
32. M₁₆	D₃₈₃	semidirect
41. C₈	C₂×D₃₈₃	semidirect

Maximal subgroups

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

subgroup	index
#20259457	2
C₂×D₁₅₃₂	2
#20259446	2
D₃₀₆₄	2
M₁₆×C₃₈₃	2
AGL(1,ℤ/8)	383