Subgroups — #16861237

Order 4064.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14523248	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14523237	C₂	1	2	yes	maximal, complemented
4	D₁₀₁₆	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₂₇	—	2	1	no	—
7	Dic₂₅₄	C₂²	1	1	yes	characteristic, complemented
8	D₅₀₈	—	2	1	no	—
9	D₅₀₈	C₂²	1	2	yes	—
10	D₅₀₈	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₁₂₇	—	2	1	no	—
12	C₄×D₁₂₇	—	2	1	no	—
13	C₂×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₈×C₁₂₇	C₂²	1	2	yes	complemented
15	C₂²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×D₁₂₇	—	2	1	no	—
17	C₂×D₁₂₇	—	2	2	no	—
18	C₂×D₁₂₇	—	4	2	no	—
19	C₄×C₁₂₇	D₄	1	1	yes	characteristic
20	C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	Dic₁₂₇	—	2	1	no	—
22	C₂×C₁₂₇	—	2	1	no	—
23	C₂×C₁₂₇	C₂×D₄	1	1	yes	characteristic
24	D₁₂₇	—	4	1	no	—
25	D₁₂₇	—	4	2	no	—
26	C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	127	1	no	maximal, Sylow
28	#41	—	127	1	no	—
29	C₂×D₄	—	127	1	no	—
30	SD₁₆	—	127	2	no	—
31	D₈	—	127	2	no	—
32	M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
33	C₂³	—	254	1	no	—
34	Q₈	—	127	1	no	—
35	D₄	—	127	1	no	—
36	D₄	—	127	2	no	—
37	D₄	—	254	1	no	—
38	D₄	—	254	1	no	—
39	C₂×C₄	—	254	1	no	—
40	C₂×C₄	C₂×D₁₂₇	1	1	yes	characteristic
41	C₈	C₂×D₁₂₇	1	2	yes	complemented
42	C₂²	—	508	2	no	—
43	C₂²	—	254	1	no	—
44	C₂²	—	254	2	no	—
45	C₂²	D₅₀₈	1	1	yes	characteristic
46	C₄	—	254	1	no	—
47	C₄	D₅₀₈	1	1	yes	characteristic
48	C₄	C₂²×D₁₂₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	508	2	no	—
51	C₂	—	508	1	no	—
52	C₂	C₂×D₅₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₅₀₈
commutator, cosocle	C₄×C₁₂₇	C₂³
Frattini	C₄	C₂²×D₁₂₇
Fitting	M₁₆×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₄
hypercenter	M₁₆	D₁₂₇
hypocenter	C₁₂₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #14523248	C₂	semidirect
2. C₂×D₅₀₈	C₂	semidirect
3. #14523237	C₂	semidirect
4. D₁₀₁₆	C₂	semidirect
5. M₁₆×C₁₂₇	C₂	semidirect
7. Dic₂₅₄	C₂²	semidirect
10. D₅₀₈	C₂²	semidirect
14. C₈×C₁₂₇	C₂²	semidirect
26. C₁₂₇	AGL(1,ℤ/8)	semidirect
32. M₁₆	D₁₂₇	semidirect
41. C₈	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#14523248	2
C₂×D₅₀₈	2
#14523237	2
D₁₀₁₆	2
M₁₆×C₁₂₇	2
AGL(1,ℤ/8)	127