Subgroups — #28357248

Order 13856.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20294097	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₇₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294086	C₂	1	2	yes	maximal, complemented
4	D₃₄₆₄	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₃₃	—	2	1	no	—
7	Dic₈₆₆	C₂²	1	1	yes	characteristic, complemented
8	D₁₇₃₂	—	2	1	no	—
9	D₁₇₃₂	C₂²	1	2	yes	—
10	D₁₇₃₂	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₄₃₃	—	2	1	no	—
12	C₄×D₄₃₃	—	2	1	no	—
13	C₂×C₄×C₄₃₃	C₂²	1	1	yes	characteristic
14	C₈×C₄₃₃	C₂²	1	2	yes	complemented
15	C₂²×C₄₃₃	D₄	1	1	yes	characteristic
16	C₂×D₄₃₃	—	2	1	no	—
17	C₂×D₄₃₃	—	2	2	no	—
18	C₂×D₄₃₃	—	4	2	no	—
19	C₄×C₄₃₃	D₄	1	1	yes	characteristic
20	C₄×C₄₃₃	C₂³	1	1	yes	characteristic
21	Dic₄₃₃	—	2	1	no	—
22	C₂×C₄₃₃	—	2	1	no	—
23	C₂×C₄₃₃	C₂×D₄	1	1	yes	characteristic
24	D₄₃₃	—	4	1	no	—
25	D₄₃₃	—	4	2	no	—
26	C₄₃₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	433	1	no	maximal, Sylow
28	#41	—	433	1	no	—
29	C₂×D₄	—	433	1	no	—
30	SD₁₆	—	433	2	no	—
31	D₈	—	433	2	no	—
32	M₁₆	D₄₃₃	1	1	yes	characteristic, complemented
33	C₂³	—	866	1	no	—
34	Q₈	—	433	1	no	—
35	D₄	—	866	1	no	—
36	D₄	—	866	1	no	—
37	D₄	—	433	1	no	—
38	D₄	—	433	2	no	—
39	C₂×C₄	—	866	1	no	—
40	C₂×C₄	C₂×D₄₃₃	1	1	yes	characteristic
41	C₈	C₂×D₄₃₃	1	2	yes	complemented
42	C₂²	—	866	1	no	—
43	C₂²	—	866	2	no	—
44	C₂²	—	1732	2	no	—
45	C₂²	D₁₇₃₂	1	1	yes	characteristic
46	C₄	—	866	1	no	—
47	C₄	D₁₇₃₂	1	1	yes	characteristic
48	C₄	C₂²×D₄₃₃	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	1732	2	no	—
51	C₂	—	1732	1	no	—
52	C₂	C₂×D₁₇₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₇₃₂
commutator, cosocle	C₄×C₄₃₃	C₂³
Frattini	C₄	C₂²×D₄₃₃
Fitting	M₁₆×C₄₃₃	C₂
socle	C₂×C₄₃₃	C₂×D₄
hypercenter	M₁₆	D₄₃₃
hypocenter	C₄₃₃	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #20294097	C₂	semidirect
2. C₂×D₁₇₃₂	C₂	semidirect
3. #20294086	C₂	semidirect
4. D₃₄₆₄	C₂	semidirect
5. M₁₆×C₄₃₃	C₂	semidirect
7. Dic₈₆₆	C₂²	semidirect
10. D₁₇₃₂	C₂²	semidirect
14. C₈×C₄₃₃	C₂²	semidirect
26. C₄₃₃	AGL(1,ℤ/8)	semidirect
32. M₁₆	D₄₃₃	semidirect
41. C₈	C₂×D₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
#20294097	2
C₂×D₁₇₃₂	2
#20294086	2
D₃₄₆₄	2
M₁₆×C₄₃₃	2
AGL(1,ℤ/8)	433