Subgroups — #15692283

Order 3232.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13108074	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#13108063	C₂	1	2	yes	maximal, complemented
4	D₈₀₈	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₁	—	2	1	no	—
7	C₂×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
8	C₂²⋊D₁₀₁	—	2	1	no	—
9	D₄₀₄	—	2	1	no	—
10	D₄₀₄	C₂²	1	2	yes	—
11	D₄₀₄	C₂²	1	1	yes	characteristic, complemented
12	C₄×D₁₀₁	—	2	1	no	—
13	Dic₂₀₂	C₂²	1	1	yes	characteristic, complemented
14	C₈×C₁₀₁	C₂²	1	2	yes	complemented
15	C₂²×C₁₀₁	D₄	1	1	yes	characteristic
16	C₂×D₁₀₁	—	2	1	no	—
17	C₂×D₁₀₁	—	2	2	no	—
18	C₂×D₁₀₁	—	4	2	no	—
19	C₄×C₁₀₁	D₄	1	1	yes	characteristic
20	C₄×C₁₀₁	C₂³	1	1	yes	characteristic
21	Dic₁₀₁	—	2	1	no	—
22	C₂×C₁₀₁	—	2	1	no	—
23	C₂×C₁₀₁	C₂×D₄	1	1	yes	characteristic
24	D₁₀₁	—	4	1	no	—
25	D₁₀₁	—	4	2	no	—
26	C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	101	1	no	maximal, Sylow
28	#41	—	101	1	no	—
29	C₂×D₄	—	101	1	no	—
30	SD₁₆	—	101	2	no	—
31	D₈	—	101	2	no	—
32	M₁₆	D₁₀₁	1	1	yes	characteristic, complemented
33	C₂³	—	202	1	no	—
34	Q₈	—	101	1	no	—
35	D₄	—	101	1	no	—
36	D₄	—	101	2	no	—
37	D₄	—	202	1	no	—
38	D₄	—	202	1	no	—
39	C₂×C₄	—	202	1	no	—
40	C₂×C₄	C₂×D₁₀₁	1	1	yes	characteristic
41	C₈	C₂×D₁₀₁	1	2	yes	complemented
42	C₂²	—	404	2	no	—
43	C₂²	—	202	1	no	—
44	C₂²	—	202	2	no	—
45	C₂²	D₄₀₄	1	1	yes	characteristic
46	C₄	—	202	1	no	—
47	C₄	D₄₀₄	1	1	yes	characteristic
48	C₄	C₂²×D₁₀₁	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	404	2	no	—
51	C₂	—	404	1	no	—
52	C₂	C₂×D₄₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄₀₄
commutator, cosocle	C₄×C₁₀₁	C₂³
Frattini	C₄	C₂²×D₁₀₁
Fitting	M₁₆×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂×D₄
hypercenter	M₁₆	D₁₀₁
hypocenter	C₁₀₁	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #13108074	C₂	semidirect
2. C₂×D₄₀₄	C₂	semidirect
3. #13108063	C₂	semidirect
4. D₈₀₈	C₂	semidirect
5. M₁₆×C₁₀₁	C₂	semidirect
11. D₄₀₄	C₂²	semidirect
13. Dic₂₀₂	C₂²	semidirect
14. C₈×C₁₀₁	C₂²	semidirect
26. C₁₀₁	AGL(1,ℤ/8)	semidirect
32. M₁₆	D₁₀₁	semidirect
41. C₈	C₂×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#13108074	2
C₂×D₄₀₄	2
#13108063	2
D₈₀₈	2
M₁₆×C₁₀₁	2
AGL(1,ℤ/8)	101