Subgroups — #16861242

Order 4064.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14523248	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₅₀₈	C₂	1	2	yes	maximal, complemented
4	#14523237	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₂₅₄	—	2	1	no	—
7	Dic₂₅₄	C₂²	1	2	yes	—
8	Dic₂₅₄	C₂²	1	1	yes	characteristic
9	D₅₀₈	C₂²	1	1	yes	characteristic
10	C₂²⋊D₁₂₇	—	2	1	no	—
11	C₂×Dic₁₂₇	—	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	—
15	C₂²×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×D₁₂₇	—	2	1	no	—
17	C₄×C₁₂₇	D₄	1	1	yes	characteristic
18	C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	Dic₁₂₇	—	2	1	no	—
20	Dic₁₂₇	—	2	2	no	—
21	C₂×C₁₂₇	—	2	1	no	—
22	C₂×C₁₂₇	C₂×D₄	1	1	yes	characteristic
23	D₁₂₇	—	4	1	no	—
24	C₁₂₇	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	127	1	no	maximal, Sylow
26	#41	—	127	1	no	—
27	C₂×Q₈	—	127	1	no	—
28	Q₁₆	—	127	2	no	—
29	SD₁₆	—	127	2	no	—
30	M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
31	Q₈	—	127	2	no	—
32	Q₈	—	127	1	no	—
33	Q₈	—	254	1	no	—
34	D₄	—	127	1	no	—
35	D₄	—	254	1	no	—
36	C₂×C₄	—	254	1	no	—
37	C₂×C₄	—	254	1	no	—
38	C₂×C₄	C₂×D₁₂₇	1	1	yes	characteristic
39	C₈	C₂×D₁₂₇	1	2	yes	—
40	C₂²	—	254	1	no	—
41	C₂²	D₅₀₈	1	1	yes	characteristic
42	C₄	—	254	1	no	—
43	C₄	—	254	2	no	—
44	C₄	D₅₀₈	1	1	yes	characteristic
45	C₄	C₂²×D₁₂₇	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	508	1	no	—
48	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
prime radical	#14523248	C₂
hypercenter	M₁₆	D₁₂₇
hypocenter	C₁₂₇	#137
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
2. C₂×Dic₂₅₄	C₂	semidirect
3. Dic₅₀₈	C₂	semidirect
4. #14523237	C₂	semidirect
5. M₁₆×C₁₂₇	C₂	semidirect
24. C₁₂₇	#137	semidirect
30. M₁₆	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×Dic₂₅₄	2
Dic₅₀₈	2
#14523237	2
M₁₆×C₁₂₇	2
#137	127