Subgroups — #44540478

Order 31712.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30629392	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃₉₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#30629381	C₂	1	2	yes	maximal, complemented
4	D₇₉₂₈	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₉₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₉₉₁	—	2	1	no	—
7	Dic₁₉₈₂	C₂²	1	1	yes	characteristic, complemented
8	D₃₉₆₄	—	2	1	no	—
9	D₃₉₆₄	C₂²	1	2	yes	—
10	D₃₉₆₄	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₉₉₁	—	2	1	no	—
12	C₄×D₉₉₁	—	2	1	no	—
13	C₂×C₄×C₉₉₁	C₂²	1	1	yes	characteristic
14	C₈×C₉₉₁	C₂²	1	2	yes	complemented
15	C₂²×C₉₉₁	D₄	1	1	yes	characteristic
16	C₂×D₉₉₁	—	2	1	no	—
17	C₂×D₉₉₁	—	2	2	no	—
18	C₂×D₉₉₁	—	4	2	no	—
19	C₄×C₉₉₁	D₄	1	1	yes	characteristic
20	C₄×C₉₉₁	C₂³	1	1	yes	characteristic
21	Dic₉₉₁	—	2	1	no	—
22	C₂×C₉₉₁	—	2	1	no	—
23	C₂×C₉₉₁	C₂×D₄	1	1	yes	characteristic
24	D₉₉₁	—	4	1	no	—
25	D₉₉₁	—	4	2	no	—
26	C₉₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	991	1	no	maximal, Sylow
28	#41	—	991	1	no	—
29	C₂×D₄	—	991	1	no	—
30	SD₁₆	—	991	2	no	—
31	D₈	—	991	2	no	—
32	M₁₆	D₉₉₁	1	1	yes	characteristic, complemented
33	C₂³	—	1982	1	no	—
34	Q₈	—	991	1	no	—
35	D₄	—	1982	1	no	—
36	D₄	—	1982	1	no	—
37	D₄	—	991	1	no	—
38	D₄	—	991	2	no	—
39	C₂×C₄	—	1982	1	no	—
40	C₂×C₄	C₂×D₉₉₁	1	1	yes	characteristic
41	C₈	C₂×D₉₉₁	1	2	yes	complemented
42	C₂²	—	3964	2	no	—
43	C₂²	—	1982	1	no	—
44	C₂²	—	1982	2	no	—
45	C₂²	D₃₉₆₄	1	1	yes	characteristic
46	C₄	—	1982	1	no	—
47	C₄	D₃₉₆₄	1	1	yes	characteristic
48	C₄	C₂²×D₉₉₁	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	3964	2	no	—
51	C₂	—	3964	1	no	—
52	C₂	C₂×D₃₉₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃₉₆₄
commutator, cosocle	C₄×C₉₉₁	C₂³
Frattini	C₄	C₂²×D₉₉₁
Fitting	M₁₆×C₉₉₁	C₂
socle	C₂×C₉₉₁	C₂×D₄
hypercenter	M₁₆	D₉₉₁
hypocenter	C₉₉₁	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₉₉₁ ▷ C₂×C₉₉₁ ▷ C₉₉₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #30629392	C₂	semidirect
2. C₂×D₃₉₆₄	C₂	semidirect
3. #30629381	C₂	semidirect
4. D₇₉₂₈	C₂	semidirect
5. M₁₆×C₉₉₁	C₂	semidirect
7. Dic₁₉₈₂	C₂²	semidirect
10. D₃₉₆₄	C₂²	semidirect
14. C₈×C₉₉₁	C₂²	semidirect
26. C₉₉₁	AGL(1,ℤ/8)	semidirect
32. M₁₆	D₉₉₁	semidirect
41. C₈	C₂×D₉₉₁	semidirect

Maximal subgroups

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

subgroup	index
#30629392	2
C₂×D₃₉₆₄	2
#30629381	2
D₇₉₂₈	2
M₁₆×C₉₉₁	2
AGL(1,ℤ/8)	991